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UBC Theses and Dissertations

Finite deformation analysis using the finite element method Molstad, Terry Kim 1977

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FINITE DEFORMATION ANALYSIS USING THE FINITE ELEMENT METHOD by TERRY KIM MOLSTAD B.E n g ( C i v i l ) , Royal M i l i t a r y College of Canada, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n 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 BRITISH COLUMBIA July, 1977 c ) Terry Kim Molstad, 1977 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . T .K . M o l s t a d Depar tment o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r , Canada V6T 1W5 Augu s t 1977 i i ABSTRACT An analysis of the f i n i t e deformation of an e l a s t i c body using the f i n i t e element method i s invest igated. The governing nonlinear equations of equi l ibr ium are derived through the p r i n c i p l e of v i r t u a l work using a Lagrangian descr ip t ion . A general incremental v i r t u a l work equation i s obtained, and then l inea r i zed to permit the use of d i rec t so lu t ion techniques. A res idual loading term i s defined which represents the nonsatisfact ion of equi l ibr ium of the so lu t ion obtained at the end of an increment using the l inea r incremental v i r t u a l work equation. The res idual loading term i s used to control the divergence of the l inea r i zed incremental so lu t ion from the exact equi l ibr ium so lu t ion , through the se l f -cor rec t ing solut ion technique. The f i n i t e element method i s introduced i n general for three dimensional ana lys is , and i s then specia l ized for two dimensional, plane e l a s t i c i t y ana lys i s . Two eight degree of freedom rectangular f i n i t e elements are developed using a b i l i n e a r assumed displacement f i e l d . The f i r s t element i s numerically integrated using Gaussian quadrature, while the second employs a nonuniform integrat ion scheme i n order to improve th is element's performance. Four f i n i t e deformation problems are analysed using the pro-cedure presented i n th is thes is , and the resul ts are compared with avai lable closed form solu t ions . The problems analysed are those of a uniformly loaded i n f i n i t e plate s t r i p having e i ther simply supported longi tud ina l edges or f ixed long i tud ina l edges, a cant i lever beam under a uniformly d is t r ibu ted load, and l a s t l y a cant i lever beam with a para-b o l i c a l l y d i s t r ibu ted end load. Excel lent agreement was obtained between the f i n i t e element analysis resul ts and the closed form solut ions . i i i TABLE OF CONTENTS Page ABSTRACT ... i i TABLE OF CONTENTS v . i i i L I ST OF TABLES v i L I ST OF FIGURES v i i NOTATION i x ACKNOWLEDGEMENTS . . x CHAPTER 1. INTRODUCTION 1 1.1 Backg round 1 1.2 P u r p o s e and S cope . 5 1.3 L i m i t a t i o n s 6 2. MATHEMATICAL PRELIMINARIES 7 2.1 G e n e r a l 7 2.2 K i n e m a t i c s o f t h e L a g r a n g i a n D e s c r i p t i o n 7 2.3 S t r e s s i n a L a g r a n g i a n D e s c r i p t i o n . . . 12 3. FORMULATION OF THE EQUILIBRIUM EQUATIONS 18 3.1 G e n e r a l 18 3.2 P r i n c i p l e o f V i r t u a l Work . v . . . . . 19 3.3 V i r t u a l Work U s i n g t h e L a g r a n g i a n D e s c r i p t i o n 22 3.4 I n c r e m e n t a l V i r t u a l Work E q u a t i o n 27 3.5 L i n e a r i z e d I n c r e m e n t a l V i r t u a l Work E q u a t i o n and t h e R e s i d u a l L o a d i n g Term. 32 3.6 S u r f a c e T r a c t i o n s and Body F o r c e s . . . . . . . . 39 3.7 Summary 40 i v CHAPTER Page 4 . SOLUTION OF THE EQUILIBRIUM EQUATIONS 42 4.1 Genera] 42 4 .2 I n c r e m e n t a l Method W i t h o u t E q u i l i b r i u m Checks 43 4 . 3 I n c r e m e n t a l Methods W i t h E q u i l i b r i u m Check s 48 4 . 3 . 1 G e n e r a l 48 4 . 3 . 2 I t e r a t i v e Methods 49 4 . 3 . 3 S e l f - c o r r e c t i n g Method 53 4.4 Summary 56 5. CONSTITUTIVE RELATIONSHIPS ... 57 5.1 G e n e r a l 57 5.2 E l a s t i c . C o n s t i t u t i v e Ten so r 58 6. APPLICATION OF THE F IN ITE ELEMENT METHOD 61 6.1 G e n e r a l 61 6.2 The F i n i t e E l ement Method 61 6.3 I n c r e m e n t a l V i r t u a l Work E q u a t i o n s I n c o r p o r a t i n g t h e F i n i t e E l ement Method 66 6.3.1 The Assumed D i s p l a c e m e n t A p p r o a c h 66 6 .3 .2 The I n c r e m e n t a l V i r t u a l Work E q u a t i o n s 67 6 . 3 . 3 R e s i d u a l L o a d i n g Term 79 6.4 Two D i m e n s i o n a l A n a l y s i s 81 6.4.1 G e n e r a l 81 6 .4 .2 P l a n e S t r a i n 82 6 . 4 . 3 P l a n e S t r e s s . : 89 6.5 E i g h t Degree o f Freedom R e c t a n g u l a r F i n i t e E l e m e n t s 91 6 .5 .1 The Assumed D i s p l a c e m e n t F i e l d 91 6 .5 .2 R e c t a n g u l a r F i n i t e E l ement U s i n g G a u s s i a n Q u a d r a t u r e 95 V CHAPTER Page 6 .5 .3 R e c t a n g u l a r F i n i t e E l ement U s i n g N o n u n i f o r m N u m e r i c a l I n t e g r a t i o n 99 6 .5 .4 P e r f o r m a n c e Compa r i s on o f t h e Two R e c t a n g u l a r F i n i t e E l e m e n t s . . . . 100 7. APPLICATIONS TO NONLINEAR PROBLEMS 112 7.1 G e n e r a l 112 7.2 E l a s t i c . I n f i n i t e P l a t e S t r i p 113 7.2.1 G e n e r a l 113 7 .2 .2 I n f i n i t e P l a t e S t r i p : C l o s e d Form S o l u t i o n . 114 7 .2 .3 I n f i n i t e P l a t e S t r i p : F i n i t e E l ement A n a l y s i s 122 7.3 The E l a s t i c a ... . . 128 7.3.1 G e n e r a l 128 7 .3 .2 C a n t i l e v e r W i t h A V e r t i c a l T i p L o a d : C l o s e d Form S o l u t i o n 131 7 . 3 . 3 C a n t i l e v e r W i t h A V e r t i c a l T i p L o a d : F i n i t e E l ement A n a l y s i s 134 7 .3 .4 C a n t i l e v e r W i t h U n i f o r m L o a d i n g : C l o s e d Form S o l u t i o n 144 7 .3 .5 C a n t i l e v e r W i t h U n i f o r m L o a d i n g : F i n i t e E l emen t A n a l y s i s 146 8. CONCLUSIONS 150 BIBLIOGRAPHY . . . . . . . . . . . 151 APPENDIX A: THREE DIMENSIONAL ANALYSIS OPERATOR MATRICES. 154 APPENDIX B: TWO DIMENSIONAL ANALYSIS OPERATOR MATRICES. . . . . . 159 v i L I ST OF TABLES T a b l e ( s ) Page I C a n t i l e v e r F i n i t e E l ement A n a l y s i s 105 I I I n f i n i t e P l a t e S t r i p F i n i t e E l ement R e s u l t s 110 I I I C l o s e d Form S o l u t i o n R e s u l t s f o r Two I n f i n i t e P l a t e S t r i p s 123 IV F i n i t e E l emen t R e s u l t s f o r t h e I n f i n i t e P l a t e S t r i p w i t h S i m p l y S u p p o r t e d L o n g i t u d i n a l Edges 126 V F i n i t e E l ement R e s u l t s f o r t h e I n f i n i t e P l a t e S t r i p w i t h F i x e d L o n g i t u d i n a l E d g e s . . . . 129 V I A n g l e o f R o t a t i o n and D e f l e c t i o n s o f a C a n t i l e v e r w i t h a T i p Load 135 V I I F i n i t e E l emen t R e s u l t s f o r C a n t i l e v e r End R o t a t i o n 138 V I I I F i n i t e E l emen t R e s u l t s f o r C a n t i l e v e r V e r t i c a l End D e f l e c t i o n s 140 IX F i n i t e E l ement R e s u l t s f o r C a n t i l e v e r H o r i z o n t a l End D e f l e c t i o n s 142 X T i p D e f l e c t i o n s o f a U n i f o r m l y Loaded C a n t i l e v e r : F i n i t e E l emen t R e s u l t s . 147 v i i L I ST OF FIGURES F i g u r e ( s ) Page 1 Th ree C o n f i g u r a t i o n s o f a G e n e r a l Body 8 2 F o r c e V e c t o r s A c t i n g On Deformed and Undeformed C o n f i g u r a t i o n s o f an E l ement o f a S o l i d Body 13 3 L a g r a n g e ' s and K i r c h h o f f ' s R u l e s o f C o r r e s p o n d e n c e f o r F o r c e V e c t o r s 15 4 One D i m e n s i o n a l L o a d - D e f l e c t i o n Graph Showing t h e R e s i d u a l L o a d i n g P a r a m e t e r 38 5 D i v e r g e n c e o f t h e I n c r e m e n t a l Method W i t h o u t E q u i l i b r i u m Checks . . 45 6 Newton-Raphson Method . . . . 50 7 M o d i f i e d Newton-Raphson Method 51 8 S e l f - c o r r e c t i n g Method ' • • • 55 9 E i g h t Degree o f Freedom R e c t a n g u l a r F i n i t e E l ement 92 10 C a n t i l e v e r T e s t •••• 103 11 C a n t i l e v e r F i n i t e E l emen t G r i d s . - 104 12 C a n t i l e v e r : F i n i t e E l emen t C o m p a r i s o n . . . . . . 106 13 S i m p l y S u p p o r t e d I n f i n i t e P l a t e S t r i p Under U n i f o r m L o a d i n g , 108 14 F i n i t e E l emen t G r i d s f o r t h e I n f i n i t e P l a t e S t r i p 110 15 I n f i n i t e P l a t e S t r i p F i n i t e E l ement R e s u l t s I l l -16 ( a ) : S i m p l y S u p p o r t e d I n f i n i t e P l a t e S t r i p 115 (b) : F i x e d Edged I n f i n i t e P l a t e S t r i p '. 115 17 ( a ) : I n f i n i t e P l a t e S t r i p W i t h S i m p l y S u p p o r t e d Edges : F i n i t e E l ement G r i d 125 ( b ) : I n f i n i t e P l a t e S t r i p W i t h F i x e d Edge s : F i n i t e E l ement G r i d 125 v i i i F i g u r e ( s ) Page 18 F i n i t e E l ement R e s u l t s f o r t h e I n f i n i t e P l a t e S t r i p ( S i m p l y S u p p o r t e d ) 127 19 F i n i t e E l ement R e s u l t s f o r t h e I n f i n i t e P l a t e S t r i p ( F i x e d Edges ) 130 20 C a n t i l e v e r W i t h V e r t i c a l T i p Load U n d e r g o i n g L a r g e D i s p l a c e m e n t s 132 21 ( a ) : C a n t i l e v e r w i t h a V e r t i c a l T i p Load 137 ( b ) : C a n t i l e v e r w i t h a V e r t i c a l T i p L o a d : F i n i t e E l e m e n t G r i d 137 22 F i n i t e E l e m e n t R e s u l t s f o r C a n t i l e v e r End R o t a t i o n 139 23 F i n i t e E l ement R e s u l t s f o r C a n t i l e v e r V e r t i c a l End D i s p l a c e m e n t s 141 24 F i n i t e E l ement R e s u l t s f o r C a n t i l e v e r H o r i z o n t a l End D i s p l a c e m e n t s 143 25 ( a ) : C a n t i l e v e r w i t h U n i f o r m L o a d i n g 145 ( b ) : C a n t i l e v e r w i t h U n i f o r m L o a d i n g : F i n i t e E l ement G r i d 145 26 T i p D e f l e c t i o n s o f a U n i f o r m l y Loaded C a n t i l e v e r : F i n i t e E l ement R e s u l t s 148 NOTATION The s p e c i f i c u se and mean ing o f a l l s ymbo l s u sed i s g i v e n i n t h e t e x t o f t h i s t h e s i s where t h e y a r e f i r s t i n t r o d u c e d . The summat ion c o n v e n t i o n h o l d s f o r s u b s c r i p t e d v a r i a b l e s w i t h r e p e a t e d l o w e r c a s e i n d i c e s , i t does n o t a p p l y t o r e p e a t e d upper c a s e i n d i c e s o r t o s u p e r s c r i p t s . The r ange o f summat ion i s n o r m a l l y t h r e e , e x c e p t where s p e c i f i c a l l y i n d i c a t e d t o be o t h e r w i s e . ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o h i s a d v i s o r , D r . M.D. O l s o n , f o r h i s a d v i c e and g u i d a n c e i n t h e p r e p a r a t i o n o f t h i s t h e s i s . He a l s o w i s h e s t o t h a n k D r . N.D. Na than f o r h i s a d v i c e , and f o r r e a d i n g t h e c o m p l e t e d t h e s i s . The f i n a n c i a l s u p p o r t f r o m 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 i n t h e f o r m o f a 1967 S c i e n c e S c h o l a r s h i p i s g r a t e f u l l y a c k n o w l e d g e d . The a u t h o r a l s o w i s h e s t o t hank M r s . E l i z a b e t h W i n t e r f o r d f o r h e r e f f o r t s i n t y p i n g t h e m a j o r i t y o f t h i s t h e s i s . F i n a l l y , t he a u t h o r i s d e e p l y g r a t e f u l t o h i s w i f e Sandy, f o r h e r s u p p o r t and encouragement d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . INTRODUCTION 1.1 Backg round The a n a l y s i s o f f i n i t e d e f o r m a t i o n i s becoming i n c r e a s i n g l y more i m p o r t a n t as s t r u c t u r e s a r e b e i n g d e s i g n e d f o r s e v e r e l o a d i n g c o n d i t i o n s , as new more f l e x i b l e and d u c t i l e m a t e r i a l and s t r u c t u r a l e l e m e n t s a r e emp loyed , and as s t r u c t u r e s a r e b e i n g o p t i m i z e d f o r v a r i o u s c o n s i d e r a t i o n s . The p r i m a r y g e n e r a t o r o f i n t e r e s t i n f i n i t e d e f o r m a t i o n t hu s f a r , has been t h e a e r o s p a c e i n d u s t r y where a c o n s i d e r a b l e amount o f r e s e a r c h has been done. The s e v e r e m e c h a n i c a l and t h e r m a l l o a d s e n c o u n t e r e d , and t h e p e n a l t i e s t o be p a i d f o r e x ce s s w e i g h t have l e d t h e a e r o s p a c e i n d u s t r y t o c o n s i d e r l e s s r i g i d s t r u c t u r e s w i t h t he r e q u i r e m e n t o f an a c c u r a t e a s se s sment o f t h e u l t i m a t e l o a d b e h a v i o u r o f t h e s t r u c t u r e . The u l t i m a t e l o a d b e h a v i o u r o f a s t r u c t u r e may be l a r g e l y g o v e r n e d by f i n i t e d e f o r m a t i o n e f f e c t s , w h i c h a r e n o n l i n e a r , and may be s i g n i f i c a n t l y d i f f e r e n t f r o m t h a t p r e d i c t e d by t he u s u a l s m a l l d e f o r m a t i o n a n a l y s i s . I f t he u l t i m a t e l o a d b e h a v i o u r can be a c c u r a t e l y o b t a i n e d t h r o u g h a f i n i t e d e f o r m a t i o n a n a l y s i s , t h e n t h e s t r u c t u r e may be more e f f i c i e n t l y d e s i g n e d and i t s s a f e t y more r e l i a b l y e s t a b l i s h e d . The re a r e no g e n e r a l methods t o s o l v e n o n l i n e a r bounda r y v a l u e p r o b l e m s , and o n l y a few s p e c i a l i z e d and s i m p l e n o n l i n e a r p r o b l e m s can be s o l v e d by e x a c t methods . The m a j o r i t y o f p r a c t i c a l p r ob l ems a r e u n -a p p r o a c h a b l e by any o f t h e s e methods . Thu s , t h e n o n l i n e a r n a t u r e o f t he g o v e r n i n g e q u a t i o n s o f e q u i l i b r i u m f o r f i n i t e d e f o r m a t i o n make i t 2 n e c e s s a r y t o r e s o r t t o t h e v a r i o u s n u m e r i c a l a p p r o x i m a t i o n schemes a v a i l a b l e . The f i n i t e e l emen t method i s one such t e c h n i q u e w h i c h has been e x t e n s i v e l y u s e d i n l i n e a r a n a l y s i s and i s now b e i n g emp loyed i n n o n l i n e a r • a n a l y s i s [ 1 , 2, 3 ] . O n l y t he deve lopment o f t he h i g h - s p e e d , l a r g e c a p a c i t y , d i g i t a l compute r has made t h i s s o l u t i o n t e c h n i q u e p r a c t i c a l : f o r t h e a n a l y s i s o f a c t u a l s t r u c t u r e s . The n o n l i n e a r i t i e s i n f i n i t e d e f o r m a t i o n can be c o n c e i v e d o f as a r i s i n g f r om two s o u r c e s , g e o m e t r i c n o n l i n e a r i t y and m a t e r i a l n o n -l i n e a r i t y . G e o m e t r i c n o n l i n e a r i t y i s a r e s u l t o f l a r g e d i s p l a c e m e n t s t h a t a l t e r t h e d i s t r i b u t i o n o r magn i t ude o f t he l o a d s on t h e s t r u c t u r e , and t h e manner i n w h i c h t he s t r u c t u r e r e spond s t o t he l o a d i n g . M a t e r i a l n o n l i n e a r i t i e s a r i s e f rom n o n l i n e a r 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 o r f r om n o n c o n s e r v a t i v e d e f o r m a t i o n such as e l a s t o - p l a s t i c o r v i s c o e l a s t i c d e f o r m a t i o n . G e o m e t r i c n o n l i n e a r i t i e s were f i r s t i n c o r p o r a t e d i n s t r u c t u r a l a n a l y s i s t h r o u g h t he use o f an i n i t i a l s t r e s s m a t r i x i n an i n c r e m e n t a l a p p r o a c h . T h i s a c c o u n t e d f o r t h e i n i t i a l s t r e s s e s a t t h e b e g i n n i n g o f each i n c r e m e n t and was o r i g i n a l l y d e r i v e d on t h e b a s i s o f p h y s i c a l i n t u i t i o n . The f i r s t use o f t he i n i t i a l s t r e s s m a t r i x i n a l i n e a r i z e d i n c r e m e n t a l a n a l y s i s was r e p o r t e d by T u r n e r , e t a l [4] f o r - s t r i n g e r s and t r i a n g u l a r membrane e l e m e n t s . G a l l a g h e r and P a d l o g [ 5 ] , d e r i v e d t h e g e o m e t r i c i n i t i a l s t r e s s m a t r i x f r om t h e e x p r e s s i o n o f p o t e n t i a l ene r g y f o r beam co lumns . The deve lopment o f t he i n i t i a l s t r e s s m a t r i x was f i n a l l y d e r i v e d i n a more c o n s i s t e n t manner t h r o u g h t he use o f t h e L a g r a n g i a n o r G r e e n ' s s t r a i n t e n s o r by M a r t i n [ 6 ] . Sub sequent a n a l y s i s by M a r c a l [7] d e m o n s t r a t e d t h e i m p o r t a n c e o f a d d i t i o n a l t e r m s , w h i c h a r e r e p r e s e n t e d by an i n i t i a l d i s p l a c e m e n t m a t r i x i n an i n c r e m e n t a l s o l u t i o n . 3 In c o n s i d e r i n g f i n i t e d e f o r m a t i o n i t i s i m p o r t a n t t o u se a c o n s i s t e n t con t i nuum mechan i c s f o r m u l a t i o n f o r t h e e q u a t i o n s o f e q u i l i b -r i u m , and an e f f e c t i v e and a c c u r a t e s o l u t i o n t e c h n i q u e . In t h e deve lopment o f t h e e q u a t i o n s o f e q u i l i b r i u m two b a s i c a p p r o a c h e s e x i s t , t he L a g r a n g i a n and t he E u l e r i a n d e s c r i p t i o n s . I n t h e L a g r a n g i a n d e s c r i p t i o n t h e s t r e s s e s , s t r a i n s and d i s p l a c e m e n t s a r e r e f e r r e d t o t h e i n i t i a l s t a t e , and t h e n o d a l c o o r d i n a t e s , when c o n s i d e r i n g t h e f i n i t e e l ement method , r e m a i n f i x e d t h r o u g h o u t t he a n a l y s i s . The L a g r a n g i a n d e s c r i p t i o n i s a l s o known as t he m a t e r i a l d e s c r i p t i o n s i n c e any p a r t i c l e o f t he body o r s t r u c t u r e has the same c o o r d i n a t e t h r o u g h o u t t h e d e f o r m a t i o n h i s t o r y . T h i s i s c o n t r a s t e d by t h e E u l e r i a n d e s c r i p t i o n where s t r e s s e s , s t r a i n s and d i s p l a c e m e n t s a r e r e f e r r e d t o t h e c u r r e n t s t a t e , and a c o n v e c t i v e c o o r d i n a t e s y s t e m i s u s e d w i t h t h e n o d a l c o o r d i n a t e s b e i n g u p d a t e d a f t e r each i n c r e m e n t . The E u l e r i a n o r s p a t i a l d e s c r i p t i o n does n o t have a c o n s t a n t c o o r d i n a t e f o r any p a r t i c l e o f t h e body o r s t r u c t u r e u n d e r g o i n g d e f o r m a t i o n . The E u l e r i a n app r oach t y p i f i e d t he e a r l i e r f i n i t e d e f o r m a t i o n a n a l y s e s , w h i l e t h e more r e c e n t a n a l y s e s have t e n d e d t o use t h e L a g r a n g i a n d e s c r i p t i o n . The L a g r a n g i a n d e s c r i p t i o n has been u sed by H i b b i t , M a r c a l and R i c e [ 8 ] , and F e l i p p a and S h a r i f i [ 9 ] ; whereas t h e E u l e r i a n d e s c r i p t i o n has been u sed by S h a r i f i and Popov [ 1 0 ] , and Yaghami and Popov [ 1 1 ] . Once the e q u a t i o n s o f e q u i l i b r i u m have been d e r i v e d , t hen an e f f e c t i v e and a p p r o p r i a t e s o l u t i o n t e c h n i q u e must be u s e d , g i v e n s p e c i f i e d l o a d s and s t r u c t u r a l c h a r a c t e r i s t i c s , , t o s o l v e t h e s e e q u a t i o n s i n o r d e r t o s p e c i f y o r p r e d i c t t h e f i n i t e d e f o r m a t i o n b e h a v i o u r o f t he s t r u c t u r e . U s i n g t h e f i n i t e e l emen t method t o a p p r o x i m a t e t h e s t r u c t u r a l r e s p o n s e f i e l d , t h e s o l u t i o n o f t h e e q u a t i o n s o f e q u i l i b r i u m i s 4 a c c o m p l i s h e d by m i n i m i z a t i o n t e c h n i q u e s , s t a t i c p e r t u r b a t i o n methods , i n c r e m e n t a l methods , i t e r a t i v e p r o c e d u r e s , s e l f - c o r r e c t i n g f o r m u l a s o r p r e d i c t o r ^ c o r r e c t o r methods . An a s se s sment and s u r v e y o f t h e s e t e c h n i q u e s as a p p l i e d t o t h e f i n i t e e l ement method , i n s t a t i c n o n l i n e a r a n a l y s i s , has been made by T i l l e r s o n , S t r i c k l i n and H a i s l e r [ 1 2 ] . Most o f t h e f i n i t e d e f o r m a t i o n a n a l y s e s u se an i n c r e m e n t a l a pp r oach where t he n o n l i n e a r e q u a t i o n s o f e q u i l i b r i u m a re l i n e a r i z e d f o r a s m a l l i n c r e m e n t o f d e f o r m a t i o n . The n o n l i n e a r te rms a r e assumed t o be n e g l i g i b l e f o r t h e s m a l l i n c r e m e n t u s e d , and hence can be n e g l e c t e d . The advan tage s o f u s i n g l i n e a r i z e d i n c r e m e n t a l e q u i l i b r i u m e q u a t i o n s a r e , t h a t a d i r e c t f o r w a r d s o l u t i o n i s a v a i l a b l e , t h e e q u a t i o n s a r e r e a d i l y programmed, and when c o n s i d e r i n g m a t e r i a l n o n l i n e a r i t y an i n c r e m e n t 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 i s e a s i l y i n c o r p o r a t e d . S i n c e t h e i n c r e m e n t a l methods seek t o t r a c e t h e d e f o r m a t i o n b e h a v i o u r i n s m a l l s t e p s , t h i s r e p r e s e n t s t he most r a t i o n a l a pp r oach t o p a t h - d e p e n d e n t p r o b l e m s such as p l a s t i c i t y . A s u r v e y o f t h e deve lopment o f i n c r e m e n t a l methods i n c on t i nuum mechan i c s i s g i v e n b y Yaghami [ 1 3 ] . A t t h e p r e s e n t t i m e t h e r e e x i s t s s e v e r a l g e n e r a l p u r p o s e n o n l i n e a r f i n i t e e l emen t programmes, t h a t have been d e v e l o p e d f o r t h e f i n i t e d e f o r m a t i o n p r o b l e m . Among t h e s e a r e NONSAP f rom t h e U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y [14] and DYPLAS f r o m t h e F r a n k l i n I n s t i t u t e R e s e a r c h L a b o r a t o r i e s [15] , b o t h o f w h i c h use an i n c r e m e n t a l p r o c e d u r e i n a L a g r a n g i a n d e s c r i p t i o n . 5 1.2 Pu rpo se and Scope The p u r p o s e o f t h i s t h e s i s i s t o p r e s e n t t h e deve l opment o f t h e g o v e r n i n g e q u i l i b r i u m e q u a t i o n s f o r s t a t i c , f i n i t e d e f o r m a t i o n i n a L a g r a n g i a n d e s c r i p t i o n , and t h e n t o a p p l y t he f i n i t e e l e m e n t method t o s o l v e t h e i n c r e m e n t a l r e l a t i o n s h i p s o b t a i n e d f r o m t h e s e e q u a t i o n s . The L a g r a n g i a n d e s c r i p t i o n w i l l b e c o n t r a s t e d w i t h t h e E u l e r i a n d e s c r i p t i o n t o show t h e advan t a ge s and d i s a d v a n t a g e s o f e a ch a p p r o a c h . The g o v e r n i n g e q u i l i b r i u m e q u a t i o n s i n b o t h t h e E u l e r i a n and t h e L a g r a n g i a n d e s c r i p t i o n s w i l l be d e v e l o p e d t h r o u g h t h e p r i n c i p l e o f v i r t u a l work . They w i l l be v a l i d f o r a r i b t r a r y magn i t ude s o f d e f o r m a t i o n and s t r a i n , and w i l l n o t be r e s t r i c t e d t o c o n s e r v a t i v e d e f o r m a t i o n . The g o v e r n i n g e q u a t i o n s o f e q u i l i b r i u m t h a t a r e d e v e l o p e d a r e n o n l i n e a r , t h e r e f o r e a l i n e a r i z e d i n c r e m e n t v i r t u a l work r e l a t i o n s h i p w i l l be d e r i v e d be tween two a r b i t r a r y c o n f i g u r a t i o n s o r s t a t e s . A r e s i d u a l l o a d i n g e x p r e s s i o n w i l l a l s o be o b t a i n e d w h i c h r e p r e s e n t s t h e e r r o r o r n o n s a t i s f a c t i o n . o f e q u i l i b r i u m a t t h e end o f e a c h l i n e a r i n c r e m e n t . T h i s r e s i d u a l l o a d i n g t e rm w i l l be u s e d t o e v a l u a t e and c o n t r o l t h e e r r o r i n -v o l v e d i n l i n e a r i z i n g t h e i n c r e m e n t a l v i r t u a l work e x p r e s s i o n . The d e f o r m a t i o n o f t h e s t r u c t u r e i s o b t a i n e d by a n a l y z i n g a s u c c e s s i o n o f i n c r e m e n t a l s t e p s u n t i l t h e d e s i r e d magn i t ude o f l o a d i n g o r d i s p l a c e m e n t i s a c h i e v e d as a sum o f a l l i n c r e m e n t s . The f i n i t e e l ement method w i l l be u sed t o s o l v e t h e l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e x p r e s s i o n , and t o e v a l u a t e t h e r e s i d u a l l o a d i n g t e r m . T h i s , a l o n g w i t h a s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e t o c o n t r o l t h e e r r o r a r i s i n g f r om l i n e a r i z i n g t h e i n c r e m e n t a l v i r t u a l work e x p r e s s i o n , w i l l be u sed t o a n a l y z e s e v e r a l f i n i t e d e f o r m a t i o n p r o b l e m s . The r e s u l t s o f t h e a p p l i c a t i o n o f t h i s method o f a n a l y s i s w i l l be compared t o c e r t a i n a v a i l a b l e c l o s e d - f o r m s o l u t i o n s . 6 1.3 L i m i t a t i o n s A l t h o u g h t h e b a s i c g o v e r n i n g v i r t u a l work e x p r e s s i o n s a r e d e r i v e d , and a r e v a l i d f o r c o n s e r v a t i v e and n o n c o n s e r v a t i v e d e f o r m a t i o n , t h e a t t e n t i o n o f t h i s r t h e s i s w i l l be c o n f i n e d t o c o n s e r v a t i v e d e f o r m a t i o n . F u r t he rmo re^ i t w i l l be assumed t h a t t h e r e e x i s t s a l i n e a r c o n s t i t u t i v e r e l a t i o n s h i p be tween K i r c h h o f f s t r e s s and L a g r a n g i a n s t r a i n . The f i n i t e e l emen t a n a l y s i s w i l l be p e r f o r m e d f o r t h e t w o -d i m e n s i o n a l c a se s o f p l a n e s t r a i n and p l a n e s t r e s s , a l t h o u g h t he f u l l t h r e e - d i m e n s i o n a l p r o c e d u r e w i l l be p r e s e n t e d i n g e n e r a l . 7 MATHEMATICAL PRELIMINARIES 2.1 G e n e r a l The a n a l y s i s o f f i n i t e d e f o r m a t i o n r e q u i r e s a c o n s i s t e n t m a t h e m a t i c a l a pp r oach i n o r d e r t o have a v a l i d f o r m u l a t i o n o f t h e p r o b l e m . The E u l e r i a n and L a g r a n g i a n d e s c r i p t i o n s a r e b o t h v a l i d f o r f i n i t e d e f o r m a t i o n , b u t i n t h i s t h e s i s f o r r e a s o n s g i v e n i n t h e f o l l o w i n g c h a p t e r , t h e L a g r a n g i a n d e s c r i p t i o n w i l l be u s e d . I n t h i s c h a p t e r t h e k i n e m a t i c s o f t h e L a g r a n g i a n d e s c r i p t i o n w i l l be d e m o n s t r a t e d and t h e d e f i n i t i o n o f G r e e n ' s s t r a i n t e n s o r w i l l be g i v e n . The d e f i n i t i o n o f two s t r e s s t e n s o r s i n t h e L a g r a n g i a n d e s c r i p t i o n and t h e i r r e l a t i o n s h i p t o the. n a t u r a l p h y s i c a l c oncep t o f s t r e s s w i l l a l s o be g i v e n . 2.2 K i n e m a t i c s o f t h e L a g r a n g i a n D e s c r i p t i o n In the; L a g r a n g i a n d e s c r i p t i o n t h e m o t i o n o f a t h r e e - d i m e n s i o n a l body i n i t s p a t h o f d e f o r m a t i o n i s d e s c r i b e d i n te rms o f a f i x e d r e c t a n g u l a r c o o r d i n a t e s y s t e m . Th ree c o n f i g u r a t i o n s o f a g e n e r a l body a r e shown i n F i g . 1, t h e unde fo rmed o r r e f e r e n c e c o n f i g u r a t i o n ( ° C ) , t h e c u r r e n t de fo rmed c o n f i g u r a t i o n (^C) , and a n e i g h b o u r i n g de fo rmed c o n f i g u r a t i o n ( 2 C ) w i t h r e s p e c t t o t h e c u r r e n t de fo rmed c o n f i g u r a t i o n OC). In t h i s t h e s i s , l e f t s u p e r s c r i p t s i n d i c a t e t h e c o n f i g u r a t i o n o f t he body t o w h i c h t h e q u a n t i t i e s o r e x p r e s s i o n s r e f e r , w h i l e no l e f t F I G . 1 s u p e r s c r i p t i n d i c a t e s i n c r e m e n t a l q u a n t i t i e s between c o n f i g u r a t i o n s 1C and 2 C . S i n c e t h e c o n f i g u r a t i o n s lC and 2 C a r e c o m p l e t e l y g e n e r a l , t h e i n c r e m e n t r e p r e s e n t s t h e t r a n s i t i o n be tween any two c o n f i g u r a t i o n s and i s c a p a b l e o f a s sum ing any magn i t ude o f d e f o r m a t i o n . A m a t e r i a l p o i n t o f t h e body , i n te rms o f t h e f i x e d r e c t a n g u l a r c o o r d i n a t e s y s t e m , i s d e s c r i b e d b y : 1. t h e m a t e r i a l c o o r d i n a t e s a ^ ( i = 1, 2, 3) i n °C. 2. t h e c o o r d i n a t e s a x ^ i n a C , g i v e n by a x . = a. + V ( i = 1, 2, 3) (2 .1 ) l - i i where a u ^ ( i = 1, 2, 3 ) , a r e t h e v e c t o r components o f t h e t o t a l d i s p l a c e m e n t f r o m t h e r e f e r e n c e c o n f i g u r a t i o n °C t o c o n f i g u r a t i o n a C . D e f o r m a t i o n o f t h e body i s r e p r e s e n t e d b y s t r a i n , and i n t h e L a g r a n g i a n d e s c r i p t i o n t h e s t r a i n t e n s o r u s e d i s t h e L a g r a n g i a n , o r G r e e n ' s s t r a i n tensor . , g i v e n f o r t h e m a t e r i a l p o i n t a^ i n c o n f i g u r a t i o n a C b y a -1-£ i j =2 3 x k 3 x k 8a. 3 a . i j (2 .2 ) where 6 . . i s t h e K r o n e c k e r d e l t a . J-J T h i s s t r a i n t e n s o r i s d e v e l o p e d b y c o n s i d e r i n g t h e t r a n s f o r m a t i o n g i v e n by Eq . 2.1 w h i c h maps t he p o s i t i o n o f a m a t e r i a l p o i n t °P g i v e n by i n c o n f i g u r a t i o n °C, t o a p o i n t P i n c o n f i g u r a t i o n 10 Ct Ct Ct C g i v e n by <x^. Bo th and a ^ a r e r e f e r r e d t o t h e same s e t o f r e c -t a n g u l a r C a r t e s i a n c o o r d i n a t e s . Now c o n s i d e r a m a t e r i a l p o i n t °Q i n c o n f i g u r a t i o n °C i n t he n e i g h b o u r h o o d o f °P , and whose c o o r d i n a t e s a r e g i v e n by a^ + da^ . U s i n g t he t r a n s f o r m a t i o n o f Eq . 2 . 1 , t h e n d a x ^ i s g i v e n by 9 a x . d a x . = d a . (2 .3 ) i da. j Then t he c o o r d i n a t e s o f a Q i n c o n f i g u r a t i o n a C w i l l be a x ^ + d a x ^ . T a k i n g t he d i f f e r e n c e o f the s q u a r e o f t he l e n g t h o f l i n e segments a P a Q and P Q t h e n G r e e n ' s s t r a i n t e n s o r i s d e f i n e d such t h a t d a x . d a x . - da . da . = 2 a e . . da . d a . (2 .4 ) i i i i i ] i j w h i c h g i v e s Eq . 2 . 2 . I t s h o u l d be n o t e d t h a t G r e e n ' s s t r a i n t e n s o r v a n i s h e s o n l y i f t h e r e i s no change i n t h e l e n g t h o f t he l i n e segment i n t h e t r a n s f o r m a t i o n be tween c o n f i g u r a t i o n s °C and a C . Thus t he v a n i s h i n g o f G r e e n ' s s t r a i n t e n s o r i n d i c a t e s t he ab sence o f s t r a i n . I t can a l s o be seen t h a t G r e e n ' s s t r a i n t e n s o r i s s y m m e t r i c f r om E q . 2 . 2 , t h a t i s : e i j - e j i C 2- 5> G r e e n ' s s t r a i n t e n s o r g i v e n i n Eq . 2.2 can a l s o be w r i t t e n as a e - I i j " 2 a u . . + a u . . + a u . a u ... i , j j , i m , i m,3 (2 .6 ) 11 where a comma i n d i c a t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o t he m a t e r i a l c o o r d i n a t e s i n c o n f i g u r a t i o n °C, t h a t i s d U . a 1 u. . (2 .7 ) The i n c r e m e n t a l f o rm o f G r e e n ' s s t r a i n t e n s o r be tween c o n f i g u r a t i o n s *C and 2 C i s d e f i n e d a s : 2 1 i J i j (2 .8 ) S u b s t i t u t i n g f o r u s i n g Eq . 2.6 and r e a r r a n g i n g te rms £ i j = 2 u. . + u . . +. u . u . + u . 1 u 3>i m 5J m , i m , U 1 + — 2 u . u j _ m , i m , i j (2 .9 ) where 2 l u. = u. - u. i i i ( 2 .10) Knowing t h e c o n f i g u r a t i o n X C , and hence d i s p l a c e m e n t s x u ^ and t he r e q u i r e d m a t e r i a l d e r i v a t i v e s , t h e e x p r e s s i o n f o r t h e i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r c an be decomposed i n t o a l i n e a r component e ^ j , and a n o n l i n e a r component n^j f o r t h e i n c r e m e n t o f d e f o r m a t i o n . The two components a r e d e f i n e d as 2e. u. . + u . . + •'•u . u . + u . * u 1J L 1 ^ y m, i m,j m , i m,Tj (2 .11) 2n. u . u _ m , i m , i j (2..12) 12 Thus e. . = e. . + . . ( 2 .13) i j 13 i j I t s h o u l d be n o t e d t h a t t h e i n c r e m e n t a l f o r m o f G r e e n ' s s t r a i n t e n s o r i s s t i l l s y m m e t r i c , and t h a t t h e two components and 5 _ a r e a l s o s y m m e t r i c t e n s o r s . The s p e c i a l case o f s m a l l o r i n f i n i t e s i m a l s t r a i n i s f o u n d f r o m Eq . 2 . 6 , by a s s um ing t h a t d i s p l a c e m e n t s and m a t e r i a l d e r i v a t i v e s a r e s m a l l and t h a t ' t h e p r o d u c t " u . ° u . can be n e g l e c t e d as a h i g h e r r m, i m,3 o r d e r t e r m . 2.3 S t r e s s i n a L a g r a n g i a n D e s c r i p t i o n I n t h e c o u r s e o f a n a l y z i n g t h e d e f o r m a t i o n o f a c o n t i n u u m i t i s n e c e s s a r y t o r e l a t e s t r e s s e s t o s t r a i n s . S i n c e i t i s i n t e n d e d t h a t , t h e Green s t r a i n t e n s o r be u t i l i z e d , and i t r e l a t e s s t r a i n s t o t he o r i g i n a l unde fo rmed o r r e f e r e n c e c o n f i g u r a t i o n , i t w o u l d be c o n v e n i e n t t o d e s c r i b e t h e s t r e s s t e n s o r a l s o w i t h r e s p e c t t o t h e unde fo rmed c o n f i g u r a t i o n . Two s t r e s s t e n s o r s t h a t a r e d e f i n e d w i t h r e s p e c t t o t h e o r i g i n a l undefo rmed c o n f i g u r a t i o n °C, a r e t h e Lagrange and K i r c h h o f f s t r e s s t e n s o r s . They a r e d e f i n e d by a r b i t r a r i l y a s s i g n i n g a r u l e o f c o r r e s p o n d e n c e be tween f o r c e v e c t o r s a c t i n g on t h e s u r f a c e o f an e l emen t o f t h e c o n t i n u u m , i n t h e de fo rmed and unde fo rmed c o n f i g u r a t i o n s . F o l l o w i n g t he deve lopment g i v e n by Fung [ 1 6 ] , c o n s i d e r an e l ement o f a de fo rmed s o l i d w h i c h has a f o r c e v e c t o r o f cLT w h i c h a c t s on s u r f a c e PQRS as shown i n F i g . 2. A c o r r e s p o n d i n g v e c t o r a*T9 a c t s on t h e s u r f a c e P 0 QoRoS 0 i n t h e unde fo rmed c o n f i g u r a t i o n . I f s t r e s s v e c t o r s 1 FORCE VECTORS ACTING ON DEFORMED AND UNDEFORMED CONFIGURATIONS OF AN ELEMENT OF A SOLID BODY F I G . 2 M4 a r e t h e n d e f i n e d i n each c o n f i g u r a t i o n as t he l i m i t o f <§T/a*S and < r r 0 /d 0 S as d ; S a n d d°S go t o z e r o , t h e a r e a s o f t h e s u r f a c e s PQRS and P 0 Q 0 R o S 0 r e s p e c t i v e l y , t h e n s t r e s s t e n s o r s may be d e f i n e d i n each c o n f i g u r a t i o n . F i n a l l y a r u l e o f c o r r e s p o n d e n c e be tween cif and 9t 0 i s n e c e s s a r y , and as s t a t e d b e f o r e t h i s i s a r b i t r a r y b u t must be c o n s i s t e n t . T h e r e f o r e , two r u l e s o f c o r r e s p o n d e n c e a r e u s e d , g i v i n g aT and clTo i n t e rms o f t h e i r component s , by d a T 0 . ( L ) = d a T . (2 .14) 1 1 and d a T ( K ) = j ^ i _ V 9 % . J The r u l e o f c o r r e s p o n d e n c e g i v e n by E q . 2.14 i s t h e L a g r a n g i a n r u l e , and E q . 2.15 g i v e s t he K i r c h h o f f r u l e . The K i r c h h o f f r u l e o f c o r r e s p o n d e n c e u se s t h e same t r a n s f o r m a t i o n as t h a t u s ed f o r a l i n e segment i n g o i n g f r om the de fo rmed t o t h e unde fo rmed c o n f i g u r a t i o n s , as can be seen f rom 9a . da . = — — d a x . (2 .16) 1 9 a x . 3 J The c o r r e s p o n d e n c e o f f o r c e v e c t o r s u s i n g t h e L a g r a n g i a n r u l e and t h e K i r c h h o f f r u l e be tween t h e de fo rmed and undefo rmed c o n f i g u r a t i o n s i s shown i n F i g . 3, f o r t h e t w o - d i m e n s i o n a l c a s e . Th ree d i f f e r e n t s t r e s s t e n s o r s a r e now d e f i n a b l e i n t e rms o f t he f o r c e v e c t o r , t h e e l e m e n t a l s u r f a c e a r e a , and t h e u n i t o u t w a r d ct n o r m a l s a s s o c i a t e d w i t h t he de fo rmed s u r f a c e V/, and t h e unde fo rmed s u r f a c e , V,. a » x2 I ^ a al> xl K i r c h h o f f ' s R u l e o f C o r r e s p o n d e n c e LAGRANGE'S AND KIRCHHOFF 'S RULES.OF CORRESPONDENCE FOR FORCE VECTORS F I G . 3 I f t he de fo rmed c o n f i g u r a t i o n i s c o n s i d e r e d f i r s t , t h e a s s o c i a t e d s t r e s s t e n s o r i s t h e E u l e r s t r e s s t e n s o r , a . . , d e f i n e d as c P T . = aa • a v . c P s ( 2 . 1 7 ) S i m i l a r l y i f t h e Lag range r u l e o f c o r r e s p o n d e n c e i s u s e d , t h e n a L a g r a n g i a n s t r e s s t e n s o r T — J i s d e f i n e d by d a T (L) = a T ° v . d ° S = d a T . (2 .18) l j i j i and i f t h e K i r c h h o f f r u l e i s u s e d , t h e n t h e K i r c h h o f f s t r e s s t e n s o r S . . , i s d e f i n e d by d a T (K) = a s _ _ d o s = da { 2 1 9 ) l ] i i a k v J 3 x k The r e l a t i o n s h i p s be tween t h e t h r e e t e n s o r s a r e g i v e n by o 3 a . a T . . = -£ - i - <*a . (2 .20) P 3 x ™ K m o 3a. 3a. a S . . = -B-—± i - a a , ' (2 .21) 3 1 an a a x a a x m k P 3 \ 3 \ 3a . a S . . = — a T . , (2 .22) 3 \ where p i s t he d e n s i t y i n t h e de fo rmed c o n f i g u r a t i o n and °p i s t he d e n s i t y i n t h e unde fo rmed c o n f i g u r a t i o n . ,=•• , - I t s h o u l d c b e n o t e c f . t h a t L . t h e ' E u l e r i a n s t r e s s t e n s o r i s s y m m e t r i c , t h a t i s '• \ 17 a a (2 .23) a . a . j i b y r e q u i r e m e n t s f o r r o t a t i o n a l e q u i l i b r i u m . Then i t can be s een t h a t t he L a g r a n g i a n s t r e s s t e n s o r i s n o t i n g e n e r a l s y m m e t r i c by e x a m i n i n g Eq. 2 . 20 , w h i l e t h e K i r c h h o f f s t r e s s t e n s o r i s s y m m e t r i c as a con sequence o f t h e symmetry o f t he E u l e r i a n s t r e s s t e n s o r and Eq . 2.21 r e l a t i n g t he two t e n s o r s . The symmetry o f t h e K i r c h h o f f s t r e s s t e n s o r i s r e p r e s e n t e d by The p r o p e r t y o f symmetry makes t h e K i r c h h o f f s t r e s s t e n s o r c o n v e n i e n t t o u se when t h e s t r a i n t e n s o r i s s y m m e t r i c as i n G r e e n ' s s t r a i n t e n s o r , e s p e c i a l l y w i t h t he u se o f a s y m m e t r i c c o n s t i t u t i v e t e n s o r . p h y s i c a l c o n c e p t o f s t r e s s , t h e r e f o r e once t h e de fo rmed c o n f i g u r a t i o n has been d e t e r m i n e d by a L a g r a n g i a n a n a l y s i s t h e n t h e E u l e r i a n s t r e s s e s s h o u l d be c a l c u l a t e d t o g i v e a p h y s i c a l r e p r e s e n t a t i o n o f t h e s t a t e o f s t r e s s . The E u l e r i a n s t r e s s e s may be o b t a i n e d f r o m t h e K i r c h h o f f s t r e s s e s by t h e i n v e r s e o f Eq . 2 . 2 1 , g i v e n by a S. . (2 .24) F i n a l l y , t he E u l e r i a n s t r e s s t e n s o r r e p r e s e n t s t h e a c t u a l a a . j i mk (2 .25) 18 FORMULATION OF THE EQUILIBRIUM EQUATIONS 3.1 . G e n e r a l I n t he a n a l y s i s o f f i n i t e d e f o r m a t i o n i t i s n e c e s s a r y t o f o r m u l a t e t h e e q u i l i b r i u m e q u a t i o n s i n a c o n s i s t e n t manner, b a s e d on e i t h e r t he E u l e r i a n o r t h e L a g r a n g i a n d e s c r i p t i o n . The E u l e r i a n d e s c r i p t i o n i s a t t r a c t i v e f r om t h e p o i n t o f v i e w t h a t t h e a s s o c i a t e d s t r e s s t e n s o r , t h e E u l e r s t r e s s t e n s o r , i s b a s e d on t h e de fo rmed c o n f i g u r a t i o n and i s a n a t u r a l p h y s i c a l c o n c e p t . However, i t i s u s u a l l y d e s i r a b l e t o d e f i n e s t r a i n i n te rms o f t h e o r i g i n a l m a t e r i a l p o i n t s i n t h e unde fo rmed c o n f i g u r a t i o n , and t hu s i t becomes d i f f i c u l t t o r e l a t e s t r e s s t o s t r a i n . The E u l e r i a n d e s c r i p t i o n a l s o p r e s e n t s a p r o b l e m i n an i n c r e m e n t a l v a r i a t i o n a l a n a l y s i s , i n t h a t t he de fo rmed vo lume and s u r f a c e a r e n o t known a t t he b e g i n n i n g o f t he i n c r e m e n t . Thu s , t h e r e q u i r e d s u r f a c e and vo lume i n t e g r a l s may be o n l y a p p r o x i m a t e l y e v a l u a t e d . I n t h e L a g r a n g i a n d e s c r i p t i o n , t h e s t r a i n t e n s o r i s a s s o c i a t e d w i t h t he o r i g i n a l m a t e r i a l p o i n t s b u t t h e a s s o c i a t e d s t r e s s t e n s o r s no l o n g e r r e p r e s e n t t h e n a t u r a l p h y s i c a l c o n c e p t o f s t r e s s . I f a p h y s i c a l r e p r e s e n t a t i o n o f t he s t a t e o f s t r e s s i s r e q u i r e d , t h e n t h e E u l e r s t r e s s must be e v a l u a t e d by means o f a t r a n s f o r m a t i o n ; E q . 2.25 g i v e s t he r e l a t i o n s h i p be tween t h e E u l e r and K i r c h h o f f s t r e s s t e n s o r s . The L a g r a n g i a n d e s c r i p t i o n i n an i n c r e m e n t a l v a r i a t i o n a l a n a l y s i s has t he advan tage t h a t t h e vo lume and s u r f a c e , o v e r w h i c h t he n e c e s s a r y i n t e g r a l s 19 a r e p e r f o r m e d , a r e known a p r i o r i . T h e r e f o r e , t h e r e i s no a p p r o x i m a t i o n i n t r o d u c e d h e r e as i n t h e E u l e r i a n d e s c r i p t i o n . The L a g r a n g i a n d e s c r i p t i o n w i t h i t s c o n s t a n t vo lume and s u r f a c e c h a r a c t e r i s t i c , makes i t p o s s i b l e t h a t s e v e r a l o f t he i n t e g r a l s r e q u i r e d f o r an i n c r e m e n t a l v a r i a t i o n a l a n a l y s i s need be e v a l u l a t e d o n l y once as c o n s t a n t s , o r i n te rms o f an i n c r e m e n t p a r a m e t e r , t hu s s a v i n g c o m p u t a t i o n a l e x p e n s e . I n c r e m e n t a l v a r i a t i o n a l e q u a t i o n s o f e q u i l i b r i u m b a s e d on t h e L a g r a n g i a n d e s c r i p t i o n and t h e p r i n c i p l e o f v i r t u a l d i s p l a c e m e n t s a r e d e v e l o p e d i n t h i s c h a p t e r f o r t h e a n a l y s i s o f f i n i t e d e f o r m a t i o n . The e x p r e s s i o n s d e v e l o p e d w i l l be n o n l i n e a r f o r t he i n c r e m e n t , s o i n o r d e r t o f a c i l i t a t e a d i r e c t s o l u t i o n t h e e q u a t i o n s w i l l be l i n e a r i z e d . A r e s i d u a l l o a d i n g t e rm i s d e v e l o p e d t h a t r e p r e s e n t s t h e n o n s a t i s f a c t i o n o f e q u i l i b r i u m a t t he end o f each i n c r e m e n t c au sed by n e g l e c t i n g the n o n l i n e a r i n c r e m e n t a l v a r i a t i o n a l t e r m s . 3.2 P r i n c i p l e o f V i r t u a l Work The p r i n c i p l e o f v i r t u a l work r e p r e s e n t s a n e c e s s a r y c o n d i t i o n f o r e q u i l i b r i u m o f a body s u b j e c t e d t o p r e s c r i b e d s u r f a c e t r a c t i o n s , body f o r c e s and bounda r y c o n d i t i o n s . The re a r e two app roache s t o t h e p r i n c i p l e o f v i r t u a l work , t h r o u g h t he u se o f v i r t u a l d i s p l a c e m e n t s o r t he use o f v i r t u a l f o r c e s . The v i r t u a l d i s p l a c e m e n t a p p r o a c h i s u s e d t h r o u g h o u t t h i s c h a p t e r . C o n s i d e r a body i n s t a t i c e q u i l i b r i u m when s u b j e c t e d t o s p e c i f i e d s u r f a c e t r a c t i o n s , body f o r c e s and bounda r y c o n d i t i o n s . The s u r f a c e o f t he body i s composed o f two t y p e s o f r e g i o n s ; t h e s u r f a c e o v e r w h i c h s u r f a c e t r a c t i o n s a r e s p e c i f i e d and t h e s u r f a c e S i o v e r w h i c h 20 d i s p l a c e m e n t s a r e s p e c i f i e d . Assume a s e t o f a r b i t r a r y d i s p l a c e m e n t s t h a t v a n i s h on S u , a r e c o m p a t i b l e , and w h i c h do n o t change t h e magn i t ude o r d i r e c t i o n o f t he s u r f a c e t r a c t i o n s o r body f o r c e s . These a r b i t r a r y d i s p l a c e m e n t s a r e c a l l e d k i n e m a t i c a l l y a d m i s s i b l e v i r t u a l d i s p l a c e m e n t s , and t h e i r magn i t ude i s a r b i t r a r y . Then t h e work done by t he s u r f a c e t r a c t i o n s and body f o r c e s as t h e y go t h r o u g h t he v i r t u a l d i s p l a c e m e n t s i s c a l l e d t h e v i r t u a l work and i s g i v e n i n t h e E u l e r i a n d e s c r i p t i o n b y : w <E> v , T. 6u. dS + 'S 1 1 -V pF. Su. dV (3 .1) i i whe re : T^ a r e t h e components o f t he s u r f a c e t r a c t i o n p e r u n i t a r e a o f t he de f o rmed c o n f i g u r a t i o n o f t h e body . ' ; /. " ' ' . F^ a r e t h e components o f t he ?body f o r c e p e r u n i t d e n s i t y . p i s t h e d e n s i t y i n t h e de fo rmed c o n f i g u r a t i o n V i s t h e de fo rmed vo lume o f t h e body S i s t h e de fo rmed s u r f a c e o f t he body . S i n c e t h e v i r t u a l d i s p l a c e m e n t s 6 u^ , t o be k i n e m a t i c a l l y a d m i s s i b l e , v a n i s h on S u t h e s u r f a c e i n t e g r a l i n Eq . 3 .1 need o n l y be e v a l u a t e d on S a . S u b s t i t u t i n g f o r Tj_' i n Eq . 3 . 1 , T... = a . . v: (3 .2 ) i i i j 21 where v>. i s t he o u t w a r d n o r m a l t o t he de fo rmed s u r f a c e and o. . i s t h e E u l e r s t r e s s t e n s o r , and c o n s i d e r i n g o n l y t he s u r f a c e i n t e g r a l f i r s t , t r a n s f o r m by G a u s s ' s t heo rem: TI. 6u. dS i 1 a. . v . Su. dS I J J l ( a . . S u . ) , . dS i j xJ 'j (3 .3 ) Here t h e n o t a t i o n , . = -z— i s u s ed where x . a r e t h e c o o r d i n a t e s o f t h e de fo rmed body . ( T h i s w i l l o n l y be t r u e f o r t h i s s e c t i o n . ) T h e r e f o r e u s i n g t he v i r t u a l work d e f i n i t i o n i n Eq . 3 . 1 , and Eq . 3 . 3 : T - Su. dS + 1 l pF. Su. dV l l (a±. Su^j,. d.v + p F . S u . dV i i a. . , . Su. dV + a . . Su. , . dV + i ] J i j - i j i J V V pF. Su. l l f a . . , . + pF.~| Su. dV + a . . Su. , . dV i j i 1 (3 .4 ) But t h e e q u a t i o n o f e q u i l i b r i u m i n t h e E u l e r i a n d e s c r i p t i o n i s a . . , . + pF. = 0 (3 .5 ) i l J i U s i n g t h e e q u a t i o n o f e q u i l i b r i u m g i v e n by Eq . 3 . 5 , Eq . 3.4 becomes: 22 f f f - T- Su. dS + pF. Su. dV = I a.. 6u.,. dV (3.6) J 1 i j l l j i j i ' j S V V T h i s e x p r e s s i o n , Eq . 3.6, i s t h e v i r t u a l d i s p l a c e m e n t e x p r e s s i o n f o r e q u i l i b r i u m o f a body unde r p r e s c r i b e d s u r f a c e t r a c t i o n s and body f o r c e s i n E u l e r i a n d e s c r i p t i o n . I f t h i s e q u a t i o n i s s a t i s f i e d f o r a l l k i n e m a t i c a l l y a d m i s s i b l e v i r t u a l d i s p l a c e m e n t s t h e n t h e body i s i n s t a t i c e q u i l i b r i u m . The E u l e r i a n d e s c r i p t i o n i s somewhat d i f f i c u l t t o u se i n f i n i t e d e f o r m a t i o n , s i n c e i t i s u s u a l l y d e s i r e d t o f i n d t h e e q u i l i b r i u m c o n f i g u r a t i o n o f a body u n d e r p r e s c r i b e d f o r c e s and t h e r e f o r e t h e de fo rmed s u r f a c e and vo lume w i l l n o t be known a p r i o r i . Then t h e v i r t u a l d i s p l a c e m e n t e x p r e s s i o n can be c a l c u l a t e d o n l y a p p r o x i m a t e l y by a s suming a t r i a l c o n f i g u r a t i o n i n o r d e r t o c a l c u l a t e t h e n e c e s s a r y s u r f a c e and vo lume i n t e g r a l s . The re a r i s e s t h e r e f o r e , t h e n e c e s s i t y o f i t e r a t i n g t o f i n d t he c o r r e c t s o l u t i o n c o n f i g u r a t i o n t o e v a l u a t e t h e v i r t u a l work e x p r e s s i o n p r o p e r l y . I f i n f i n i t e s s i m a l d e f o r m a t i o n i s assumed t h e n t h e d i f f e r e n c e be tween t h e unde fo rmed and de fo rmed c o n f i g u r a t i o n i s s u f f i c i e n t l y s m a l l t h a t t h e i n t e g r a l s may be a d e q u a t e l y e v a l u a t e d u s i n g t he i n i t i a l c o n f i g u r a t i o n . 3.3 V i r t u a l Work U s i n g t h e L a g r a n g i a n D e s c r i p t i o n The v i r t u a l work e x p r e s s i o n i n t h e L a g r a n g i a n d e s c r i p t i o n i s d e v e l o p e d i n t h e same manner as i n t h e E u l e r i a n d e s c r i p t i o n . The v i r t u a l work done by t he s u r f a c e t r a c t i o n s and body f o r c e s g o i n g t h r o u g h an a d m i s s i b l e s e t o f a r t i b r a r y v i r t u a l d i s p l a c e m e n t s must be b a l a n c e d by t he i n t e r n a l v i r t u a l work . 23 In t h e E u l e r i a n d e s c r i p t i o n t h e v i r t u a l work p e r f o r m e d by t h e s u r f a c e t r a c t i o n s and body f o r c e s , was d e f i n e d a s : v a T . 6u. d a S + 1 1 a„ a a F. p 5u. d V l l (3 .7 ) where t h e l e f t s u p e r s c r i p t i n d i c a t e s t h e a p p r o p r i a t e c o n f i g u r a t i o n o f the body . To e x p r e s s t h e v i r t u a l work i n t h e L a g r a n g i a n d e s c r i p t i o n , i t i s n e c e s s a r y t o t r a n s f o r m t h e i n t e g r a l s i n Eq . 3 . 7 , i n t o i n t e g r a l s w i t h r e s p e c t t o t h e unde fo rmed c o n f i g u r a t i o n °C. By u s i n g Eq . 2.18 and 2.19 i t can be seen t h a t a T : d a S = V . ° v . d°S (3 .8 ) and aT i d a s 9 x . * S - V - 5 - ^ °v - d°S Jk 8 ^ j (3 .9) T h e r e f o r e , t h e s u r f a c e t r a c t i o n v i r t u a l work i n t e g r a l c an be e v a l u a t e d w i t h r e s p e c t t o t h e unde fo rmed c o n f i g u r a t i o n by v i r t u e o f t h e f o l l o w i n g e q u i v a l e n c e s : s a T . 6u. d a S = l l 'T.-. °v. Su. d°S (3.10) 24 o r U T . Su. d a S = i 1 1 3 x . V , 6u. d°S j k 3a k j I (3.11) N e x t c o n s i d e r i n g t he body f o r c e v i r t u a l work and r e q u i r i n g t h e c o n s e r v a t i o n o f mass and t h a t t h e body f o r c e p e r u n i t mass r ema in s c o n s t a n t , t h e n F. p 6u. d V = l l a F . °p 6u. d°V l l (3.12) °v v In d e r i v i n g t h e e q u i v a l e n c e o f t h e s e i n t e g r a l s i n t h e two d i f f e r e n t c o n f i g u r a t i o n s , t h e a r b i t r a r y n a t u r e o f t he v i r t u a l d i s p l a c e m e n t s has been u s e d , t h e r e f o r e t h e y do n o t need t o be t r a n s f o r m e d i n any manner. Now r e q u i r i n g t h a t t h e v i r t u a l work p e r f o r m e d by a body be a c o n s t a n t f o r a s p e c i f i e d s e t o f s u r f a c e t r a c t i o n s , body f o r c e s arid bounda r y c o n d i t i o n s r e g a r d l e s s o f t h e d e s c r i p t i o n u s e d t o e v a l u a t e i t , t h a t i s W ™ - w W v v (3.13) t h e n , t h e v i r t u a l work e x p r e s s i o n i n t h e L a g r a n g i a n d e s c r i p t i o n i s g i v e n by e i t h e r o f two e q u i v a l e n t f o r m s : CL) v T . . °v . 6u. d°S + F. °p Su. d°V (3.14) °V and 25 (L) . a 3 x . °S jk 3a - ° v . d°S + F. °p Su. d°V l l 5 V (3 .15) Which o f t he two forms t o be u sed t o e v a l u a t e t h e v i r t u a l work o f t h e s u r f a c e t r a c t i o n s and body f o r c e s , depends on t h e n a t u r e o f t he s u r f a c e t r a c t i o n s p r e s c r i b e d . Bo th forms a r e e q u i v a l e n t b u t one may be more c o n v e n i e n t t o u s e . F o r t h e d e r i v a t i o n o f t he i n t e r n a l v i r t u a l work however , i t i s more c o n v e n i e n t t o u se Eq . 3.15 s i n c e t h e K i r c h h o f f s t r e s s t e n s o r S^ _. i s s y m m e t r i c whereas t he L a g r a n g i a n s t r e s s t e n s o r T\_. i s n o t i n g e n e r a l . F i r s t , a p p l y i n g G a u s s ' s t heo rem t o t h e s u r f a c e i n t e g r a l i n 3 Eq. 3 .15 , and now u s i n g t he n o t a t i o n ,j = - r — : O ci . 3 aS., ax.„ °v. Su. d°S jk i l c j i °S a„ a . . S x . Su. j k l k l d°V a x ; jk x iTc Su. d°V + I a S . , a x . Su. d°V (3 .16) j k l k I j 1 V T h e r e f o r e , s u b s t i t u t i n g f o r t he s u r f a c e i n t e g r a l f r om Eq . 3 .16 , back i n t o Eq . 3 .15 , (L) S jk V k + F. p l Su. d°V + l a s j k a V k S - j d ° v ( 3 - 1 7 ) °v \ 26 But the equation of equi l ibr ium i n terms of the Kirchhoff stress tensor i s given by Fung [17] as a„ a , jk X i ' k + U F . °P = 0 l (3.18) therefore Eq. 3.17 becomes CIO 3V a a . S., X . , v u . , . d°V jk i ' k I ' J (3.19) The v i r t u a l work expression i n the Lagrangian reference frame can thus be expressed by e i ther of the fol lowing two forms, found by combining Eq. 3.19 with Eq. 3.14 and 3.15 respect ive ly : r • a, °S and T. . °v . 6u. d°S + 3 i F. °p 6u. d°V i i a S . , °x-. 6u. , - d°V (3.20) j k i k I j v ^ °V °V a S M a x °v. 6u. d°S + J k i ' k J 1 X F. °p 6u. d°V = l l a a , r <>. O . . X . , . O U - , • j O r . jk i k l j d V •°S 3V 3V (3.21) Both of these v i r t u a l work expressions are exact and v a l i d for a r b i t r a r i l y large deformations and s t r a ins . Sa t i s fac t ion of these expressions i s equivalent to. f inding an equi l ibr ium configuration of the body under the prescribed surface t ract ions and body forces. They are not r e s t r i c t ed to any p a r t i c u l a r cons t i tu t ive re la t ionship and are v a l i d for both conservative and nonconservative deformation. 27 3.4 I n c r e m e n t a l V i r t u a l Work E q u a t i o n Knowing t h e c o n f i g u r a t i o n o f a body unde r any s p e c i f i e d s e t o f s u r f a c e t r a c t i o n s i t i s d e s i r e d t o o b t a i n an i n c r e m e n t a l v i r t u a l work e q u a t i o n t h a t can be u s e d t o f i n d a new n e i g h b o u r i n g c o n f i g u r a t i o n o f the body g i v e n i n c r e m e n t a l s u r f a c e t r a c t i o n s and body f o r c e s . F o r c o n v e n i e n c e t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n w i l l b e w r i t t e n f o r t h e d e f o r m a t i o n o f a body be tween c o n f i g u r a t i o n *C and 2 C (See F i g . 1), b u t i s c o m p l e t e l y g e n e r a l s i n c e t h e two c o n f i g u r a t i o n s u sed a r e a r b i t r a r y . The r e s u l t i n g e x p r e s s i o n can t h e r e f o r e be u s e d t o f i n d t h e d e f o r m a t i o n between any g i v e n c o n f i g u r a t i o n and a new c o n f i g u r a t i o n r e s u l t i n g f r om i n c r e m e n t a l l o a d s . body undergoes a change i n c o n f i g u r a t i o n , due t o an i n c r e m e n t a l change i n s u r f a c e t r a c t i o n s and body f o r c e s , w h i c h r e s u l t s i n t h e body b e i n g i n f o r c e s need n o t be m o n o t o n i c w i t h r e s p e c t t o p a s t i n c r e m e n t a l v a l u e s . To d e r i v e t h e i n c r e m e n t a l e q u a t i o n , w r i t e t h e v i r t u a l work e x p r e s s i o n s o f e q u i l i b r i u m i n each o f t h e two c o n f i g u r a t i o n s 2 C and 1 C , and t h e n fo rm t h e d i f f e r e n c e be tween t h e two. Thus f o r c o n f i g u r a t i o n 2 C , u s i n g Eq . 3 . 20 : I t i s assumed t h a t t h e c o n f i g u r a t i o n C i s known and t h a t t h e c o n f i g u r a t i o n 2 C . The i n c r e m e n t a l v a l u e s o f s u r f a c e t r a c t i o n and body f 2 T . . v . 6u. d°S + 3 i i ' k d°V (3 .22) °V and s i m i l a r l y f o r X C 28 l r T . . ° v . S j . d°S + ^ F . 0 p 6u. d°V 1 1 i s . . 1 x ' 6u. , . d ° V ( 3 . 2 3 ) j k l k I j 3 V Now t a k i n g t h e d i f f e r e n c e o f t h e two e q u a t i o n s above , 2 T . . - lT.. °v. 6u. d°S + L . J 1 . J i J : J 1 -F. - l ¥ . l l 5p Su. d°V I °S °V [ b j k i ' k 1S;ik S ' k j 5 u i ' J d ° V •. (3.24) T h i s e q u a t i o n (Eq. 3 . 2 4 ) , i s made p o s s i b l e by t h e a r b i t r a r y n a t u r e o f t h e a d m i s s i b l e v i r t u a l d i s p l a c e m e n t s , and by t h e f a c t t h a t t h e i n t e g r a l s a r e p e r f o r m e d o v e r t h e same unde fo rmed r e f e r e n c e c o n f i g u r a t i o n s u r f a c e and v o l u m e . These t o g e t h e r a l l o w t h e c o r r e s p o n d i n g i n t e g r a l s f o r each c o n f i g u r a t i o n t o be comb ined . T h i s w o u l d n o t be a v a l i d p r o c e d u r e f o r t h e E u l e r i a n d e s c r i p t i o n s i n c e t h e i n t e g r a l s a r e n o t e v a l u a t e d f o r t h e same domain i n t h e d i f f e r e n t c o n f i g u r a t i o n s . ; Ct Now r e a r r a n g i n g Eq . 3.24 u s i n g t h e d e f i n i t i o n o f g i v e n i n Eq . 2 . 1 , and t he c o n v e n t i o n t h a t no l e f t s u p e r s c r i p t i n d i c a t e s an i n c r e m e n t a l v a l u e be tween c o n f i g u r a t i o n s lC and 2 C : T . . °v . Su. d°S + F. °P 6u. d°V ) J i J i J i i S °V o [ 2 sjk c S ' k + Vk> - l s j k S - k l ' V j d ° v °v S.. 1 x . „ 6 u . , . d°V + 2 S u. Su.,. d°V (3 .25) jk I k I j J jk l k l j ' V °V 29 Examine now, each of the two integrals on the right hand side of Eq. 3.25 in'turn. F i r s t °V S.. *x. 6u. , . d°V jk 1 k I j °V S.. (6.. + lu. „ )6u. , . d°v jk v l k i kJ I j v (3.26) Here S^ i s the Kroenecker delta which has the following properties: 5. = 1 i = k lk 6. j. = 0 i / k lk (3.27) The Kroenecker delta should not be confused with the symbol preceding a v i r t u a l value. Continuing: S., 1x.. Su.,. d°V jk i'k I ' J °V S.. (6.. -+:- 1 u : ) Su. , . _ j k v i k l kJ l j + S, . (6. . + 1u. , .)5u. kj ^  i j 1 2 i k d°V (3.28) by virtue of the interchangeability of dummy indices, that i s , the equation below i s an i d e n t i t y . S.. (6., + *u. )5u. , . = S t. (5. . + xu. , - )5u. j k v ik I kJ i j k j ^ i j i'jJ i'k (3.29) 30 Now e m p l o y i n g t he s y m m e t r i c p r o p e r t y o f t h e K i r c h h o f f s t r e s s t e n s o r and t h e p r o p e r t i e s o f t he K r o e n e c k e r d e l t a : S j k l * . , k 6 u . , . d°V - { \ S . k ( 6 V . + l u . , k fiu.,. + « u . , k °V - °v + 1 u i , j 6 u . , k ) d°V (3 .30) T a k i n g t h e v a r i a t i o n o f Eq . 2 . 1 1 , w h i c h r e p r e s e n t s t h e l i n e a r p o r t i o n o f t h e i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r s : Se. . = -~ 13 2 <5u. i J + Su. J i m I fiu , . m 3 + 6 m i m ]_[ (3 .31) S u b s t i t u t i n g f rom Eq . 3.31 i n t o Eq . 3 . 30 , g i v e s f i n a l l y °V S., 5u . ,'. d°V j k I k 1 3 °V S.. fie.,' d°V Jk j k (3 .32) The s e cond i n t e g r a l on t h e r i g h t hand s i d e o f Eq . 3.25 can be m o d i f i e d , f o l l o w i n g t h e same arguments as u s e d f o r t he f i r s t i n t e g r a l c o n s i d e r e d , t o g i v e : .1 2 S . . u. „ 6u. , . d°V •j j k 1 k 1 3 °V -S u. 6u. , . j k 1 k 1 3 -S,. . u. , . 6u. k j 1 J 1 k d°V 2 S . , [ ^ ( u . , , 5 u . , . + u . , . 6u. d°V j k | _ 2 A 1 k 1 3 1 3 1 k-_| °V (3 .33) 31 Now t a k i n g t h e v a r i a t i o n o f t h e n o n l i n e a r component o f t h e i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r g i v e n i n E q . 2.12 6n. • = -7T (u ,. fiu , . + fiu u , . ) i j 2 v m 1 m j m 1 m y (3 .34) S u b s t i t u t i n g t h i s i n t o E q . 3.33 t h e r e f o r e , 2 S . . u. fiu. , • d°V = j k l k l j 2 S . , fin., d°V j k j k 5 V 5 V (3.35) Now t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n , Eq . 3 .25 , can be w r i t t e n a s : °S T . . ° v . fiu. d°S + J i J 1 J °V F. °p fiu. d°V 1 1 S. r fie.. d°V + Jk j k 2 S . , finM d°V Jk j k 3 V (3 .36) A s i m i l a r e x p r e s s i o n c a n be o b t a i n e d f o r u s i n g a s u r f a c e t r a c t i o n s p e c i f i e d such t h a t t he K i r c h h o f f s t r e s s t e n s o r i s more c o n v e n i e n t t o u s e , by f o l l o w i n g t he same p r o c e d u r e as above b e g i n n i n g w i t h Eq . 3 . 2 1 . T h i s e q u a t i o n w i l l be g i v e n b y : C2Sjk Vk " 1Sjk Vk^j 6 u i d ° S + F. °p fiu. d°V 1 1 °V °V S., fie., d°V + j k j k 2 S . , fin-, d°V j k j k (3 .37) 32 The s u r f a c e t r a c t i o n i n t e g r a l s can be decomposed and r e a r r a n g e d t o g i v e ( ( 2 s j k 2 V k - lsjk Vk^j 6 u i d ° s = S.. 1 x . ° v . 6u. d°S j k 1 k j 1 'S 2 °S S.. u. °v . fiu. d°S (3.38) j k I k j I ^ 1 I t s h o u l d be n o t e d t h a t t h e l a s t i n t e g r a l on t h e r i g h t hand s i d e o f Eq. 3.38 i s dependent on t h e d e f o r m a t i o n e x p e r i e n c e d by t h e s u r f a c e o f t h e body d u r i n g t h e i n c r e m e n t , and t h u s c anno t be c o m p l e t e l y e v a l u a t e d a t t h e b e g i n n i n g o f t h e i n c r e m e n t . The two i n c r e m e n t a l v i r t u a l work e q u a t i o n s g i v e n above , E q . 3.36 and 3.37, a r e e x a c t and v a l i d f o r any magn i t ude o f d e f o r m a t i o n be tween c o n f i g u r a t i o n s lC and 2 C . I f t h e y a r e s a t i s f i e d f o r a r b i t r a r y .o. and a d m i s s i b l e v i r t u a l d i s p l a c e m e n t s , t h e y g i v e t h e i n c r e m e n t a l d e f o r m a t i o n o f t h e body f o r p r e s c r i b e d i n c r e m e n t s o f s u r f a c e t r a c t i o n s and body f o r c e s . Bo th e q u a t i o n s a r e s t i l l n o n l i n e a r and hence t h e y c anno t be s o l v e d d i r e c t l y t o y i e l d t h e i n c r e m e n t a l d e f o r m a t i o n . 3.5 L i n e a r i z e d I n c r e m e n t a l V i r t u a l Work E q u a t i o n and t h e R e s i d u a l L o a d i n g Term  The i n c r e m e n t a l v i r t u a l work e q u a t i o n g i v e n by e i t h e r Eq . 3.36 o r Eq . 3.37 i s n o n l i n e a r and as s uch canno t be s o l v e d d i r e c t l y . I n t h i s s e c t i o n t h e s e two e q u a t i o n s w i l l b e s e p a r a t e d i n t o l i n e a r and n o n l i n e a r t e r m s , and t h e n l i n e a r i z e d by d i s c a r d i n g n o n l i n e a r t e r m s . The l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n s d e r i v e d a r e t h e n n o t e x a c t 33 f o r t h e i n c r e m e n t o f d e f o r m a t i o n b u t i t w i l l be shown t h a t t h e n o n l i n e a r terms d i s c a r d e d a r e o f a h i g h e r o r d e r t h a n t he l i n e a r te rms r e t a i n e d . To examine t h e r e s u l t i n g d e g r e e o f n o n s a t i s f a c t i o n o f e q u i l i b r i u m f o r t h e c o n f i g u r a t i o n o b t a i n e d a t t h e end o f t h e i n c r e m e n t , a r e s i d u a l l o a d i n g t e rm i s d e r i v e d . T h i s r e s i d u a l l o a d i n g t e rm w i l l t h e n be u s e d e i t h e r t o c o r r e c t t he s o l u t i o n , o r u s e d s i m p l y as an i n d i c a t i o n o f t h e e r r o r r e s u l t i n g f r om t h e l i n e a r i z a t i o n . S t a r t i n g w i t h Eq . 3 . 36 , s e p a r a t e t h e l i n e a r and n o n l i n e a r i n t e g r a l t e r m s , and p l a c e t h e n o n l i n e a r te rms i n b r a c k e t s . T . . °v . Su. d°S + JI j 1 F. °p Su. d°V I I °V °v S.. Se.. d°V + Jk j k °V i S . . 6n. , d°V j k j k + t s.. S n d ° v U Jk j k (3 .39) In t h i s c a se t h e r e i s o n l y one n o n l i n e a r i n t e g r a l , and i t s i n t e g r a n d can b e seen t o b e t h e p r o d u c t o f i n c r e m e n t a l K i r c h h o f f s t r e s s e s and i n c r e m e n t s o f t h e n o n l i n e a r p o r t i o n o f G r e e n ' s s t r a i n t e n s o r . U s i n g Eq. 3 .34 , t h i s i s shown t o be S., Sri M = S ~ - ( u , . Su + Su , . u „ ) JK j k jk|_2 m j m k m j m'kJ (3 .40) W r i t i n g t h e i n t e g r a n d s o f t h e o t h e r two i n t e g r a l s on t h e r i g h t hand s i d e o f Eq . 3 .39 , 34 >., <Sn., j k j k _ l S. jk m'k + fiu m j m k;_ (3 .41) and u s i n g Eq . 3.31 S., fie., = S ., J k j k j k 16u, , • + 1 u fiu , . + 6 u . , , + A u , . <5u [ k j m k m j j k m j m k j _ (3 .42) Compa r i n g t h e i n t e g r a n d s i t i s e v i d e n t t h a t s i n c e t h e n o n -l i n e a r t e r m i s a p r o d u c t o f t h e i n c r e m e n t a l K i r c h h o f f s t r e s s and t h e n o n l i n e a r i n c r e m e n t a l p o r t i o n o f G r e e n ' s s t r a i n t e n s o r , i t i s i n g e n e r a l l e s s t h a n t h e i n t e g r a n d r e p r e s e n t e d i n Eq . 3 . 41 . C o n s i d e r i n g t h e i n t e g r a n d i n Eq . 3 .42 , i t i s l a r g e r t h a n t h e n o n l i n e a r i n t e g r a n d s i n c e i t c o n t a i n s te rms t h a t a r e n o t p r o d u c t s o f two m a t e r i a l d e r i v a t i v e s , and a l s o b e c a u s e i t has d e r i v a t i v e s o f t he t o t a l d i s p l a c e m e n t s i n c l u d e d . T h e r e f o r e t h e n o n l i n e a r i n t e g r a l may be>.neg lected w i t h r e s p e c t t o t he o t h e r two l i n e a r i n t e g r a l s . I t s h o u l d be n o t e d t h a t t he c o m p a r i s o n o f t he i n t e g r a n d s , i s made p o s s i b l e by t he f a c t t h a t t he i n t e g r a t i o n s a r e p e r f o r m e d o v e r t h e same domain °V. N e g l e c t i n g t he n o n l i n e a r i n t e g r a l , t h e l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n i s o b t a i n e d a s : T.. ° v . 6u. d°S + F. °p fiu. d°V l l S., fie., d°V + Jk j k °.s °v °v °v ' s , fin., d°v 3k 3k (3 .43) 35 S i m i l a r l y , Eq . 3 . 3 7 can be s e p a r a t e d i n t o l i n e a r and n o n l i n e a r i n t e g r a l te rms a s , S., 1 x . „ V . fiu. d°S + j k i ' k j 1 °S 2S.., u. ,, °v . fiu. d°S + j k i ' k j l F. °p fiu. d°V l l °V S., fie.. d°V + j k j k lS., fin., d°V + j k j k S., fin-, d°V j k i k ( 3 . 4 4 ) where t h e b r a c k e t e d i n t e g r a l i s t h e n o n l i n e a r t e r m , and i t i s t h e same t e r m as i n t h e p r e v i o u s f o r m , Eq . 3 . 3 9 . By t he same argument as advanced a b o v e , t h i s n o n l i n e a r i n t e g r a l may be n e g l e c t e d i n o r d e r t o l i n e a r i z e t h e e q u a t i o n . T h e r e f o r e : °S S., 1x. °v . fiu. d°S + j k I k j I 2 S M u. „ °v . fiu. d°S + j k i ' k j I F. °p fiu. d°V I I °V °v S.. fie., d°V + Jk j k °V lS., fin., d°V Jk j k ( 3 . 4 5 ) In t h e e q u a t i o n above , i t s h o u l d b e n o t e d t h a t t h e s econd s u r f a c e t r a c t i o n i n t e g r a l i s l i n e a r even though i t i s dependent on t h e d e f o r m a t i o n . T h i s i s t r u e b e c a u s e w i l l be p r e s c r i b e d on t h e s u r f a c e a t t h e b e g i n n i n g o f t h e i n c r e m e n t , however t h e i n t e g r a l v a l u e i t s e l f i s n o t known a t ,the b e g i n n i n g o f t h e i n c r e m e n t , and i s dependent on t h e i n c r e m e n t a l d e f o r m a t i o n s o l u t i o n o b t a i n e d . N By l i n e a r i z i n g t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n s a d i r e c t s o l u t i o n may be o b t a i n e d o f t h e f i e l d o r s e t o f i n c r e m e n t a l d i s p l a c e m e n t s 36 unde r p r e s c r i b e d i n c r e m e n t a l s u r f a c e t r a c t i o n s and body f o r c e s . By e x a m i n i n g t h e l i n e a r i z e d e q u a t i o n s i t i s e v i d e n t t h a t t h e c o n f i g u r a t i o n ^ IC must be known s i n c e t h e s o , l u t i o n i s dependent on v a l u e s o f s t r e s se . s and d i s p l a c e m e n t s i n t h a t c o n f i g u r a t i o n . The s e t o f i n c r e m e n t a l d i s p l a c e m e n t s and s t r e s s e s d e r i v e d t h r o u g h t h e use o f a d i r e c t s o l u t i o n t e c h n i q u e w i l l , when added t o t h e c o r r e s « p o n d i n g v a l u e s o f c o n f i g u r a t i o n I C , g i v e t h e a p p r o x i m a t i o n t o c o n f i g u r a t i o n 2 C . T h i s a p p r o x i m a t e s o l u t i o n w i l l t e n d t o d i v e r g e f r om the e x a c t s o l u t i o n depend i n g on t h e n o n l i n e a r i t y o f t h e t r u e p r o b l e m , and t h e s i z e o f i n c r e m e n t i n t h e s u r f a c e t r a c t i o n s and body f o r c e s c h o s e n . I f e r r o r s due t o t h e d i r e c t s o l u t i o n t e c h n i q u e a r e i g n o r e d f o r t h e p r e s e n t , t h e d i v e r g e n c e i n t h e s o l u t i o n w i l l a r i s e f rom two s o u r c e s . The f i r s t s o u r c e o f d i v e r g e n c e i s c au sed b y c o n f i g u r a t i o n lC b e i n g known o n l y a p p r o x i m a t e l y , be cau se i t ha s been d e r i v e d i t s e l f f r om a p r e v i o u s c o n f i g u r a t i o n b y a l i n e a r i z e d i n c r e m e n t a l e q u a t i o n . The s e c o n d s o u r c e o f d i v e r g e n c e o c c u r s w i t h i n t h e i n c r e m e n t f rom *C t o 2 C , and i s a. r e s u l t o f d i s c a r d i n g t h e n o n l i n e a r t e r m i n t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n . As t h i s t e r m becomes s i g n i f i c a n t w i t h r e s p e c t t o t h e l i n e a r te rms r e t a i n e d , t h e s o l u t i o n o b t a i n e d w i l l d i v e r g e f r om t h e t r u e s o l u t i o n . I n g e n e r a l t h e n o n l i n e a r t e r m w i l l become i m p o r t a n t f o r i n c r e m e n t s o f l a r g e d e f o r m a t i o n and f o r h i g h l y n o n l i n e a r p r o b l e m s . I t i s d e s i r a b l e t h a t t h e r e be some i n d i c a t i o n o f t h e a c c u r a c y o f t h e a p p r o x i m a t e s o l u t i o n o b t a i n e d a t t h e end o f an.;: i n c r e m e n t . The deg ree o f n o n s a t l s f a c t i o n o f . t h e e x a c t . v i r t u a l work e q u a t i o n by t h e a p p r o x i m a t e c o n f i g u r a t i o n o b t a i n e d i s c h o s e n , and i s c a l l e d t h e r e s i d u a l l o a d i n g ct t e r m . The r e s i d u a l l o a d i n g t e rm f o r c o n f i g u r a t i o n C i s t h e r e f o r e 37 d e f i n e d as °S T . . °v . 6u. d°S +• F. °p 6u. d°V 1 l 5 V Ci Ct S., x . „ 6u. ,.. d°V j k I k l j 3 V (3 .46) I t can be seen f rom Eq . 3 .20 , t h a t i f upon s u b s t i t u t i n g t he a c o n f i g u r a t i o n o b t a i n e d a t t h e end o f an i n c r e m e n t i n t o E q . 3 .46 , R £ i s z e r o , t h e n t he c o n f i g u r a t i o n r e p r e s e n t s an e q u i l i b r i u m c o n f i g u r a t i o n , ct and hence t h e e x a c t s o l u t i o n . I f R c 1 S n o t e q u a l t o z e r o t h e n e q u i l i b r i u m i s n o t e x a c t l y s a t i s f i e d . Ct S i m i l a r l y , can be d e f i n e d f o r t h e c a s e where a K i r c h h o f f s t r e s s s u r f a c e t r a c t i o n i s u s e d , by °S V. a x . V . , <5u. d°S + j k I k j I a F . °p <5u. d°V 1 I I 3 V °V a S . . a x . S u . , . d°V jk I k I j (3 .47) T h i s p a r a m a t e r a R c > t h e r e s i d u a l l o a d i n g t e r m , somet imes c a l l e d t h e l o a d c o r r e c t i o n t e r m , r e p r e s e n t s i n a s en se t h e u n b a l a n c e d l o a d i n g o f t h e body . By u n b a l a n c e d l o a d i n g , i t i s meant t h e v i r t u a l work done b y t h e s u r f a c e t r a c t i o n s and body f o r c e s on t h e body , t h a t i s n o t b a l a n c e d b y t h e v i r t u a l work p e r f o r m e d by t h e s t r e s s e s and s t r a i n s w i t h i n t h e body i n t h e a p p r o x i m a t e c o n f i g u r a t i o n d e r i v e d . The r e s i d u a l l o a d i n g f o r a s i n g l e deg ree o f f r eedom p r o b l e m has a c l e a r p h y s i c a l c o n c e p t as shown i n F i g . 4. The r e s i d u a l l o a d i n g ^ t e r m h e r e i s t h e d i f f e r e n c e be tween t h e l o a d a p p l i e d and t h e a c t u a l l o a d p 3 o i-4 DEFLECTION ONE-DIMENSIONAL LOAD-DEFLECTION GRAPH SHOWING THE RESIDUAL LOADING PARAMETER F I G . 4 39; r e q u i r e d f o r t h e d e f o r m a t i o n c a l c u l a t e d by t h e l i n e a r i n c r e m e n t (A , ) -I t t h e r e f o r e r e p r e s e n t s t he amount o f l o a d t h a t t h e s t r u c t u r e ha s n o t de fo rmed t o a c c e p t . When t h e p r o b l e m i s m u l t i d i m e n s i o n a l , t h e p h y s i c a l c o n c e p t becomes o b s c u r e b u t t h e m a t h e m a t i c a l c o n c e p t i s s t i l l v a l i d . The r e s i d u a l l o a d i n g t e r m w o u l d b e u s e f u l .even i f i t was o n l y . an a b s t r a c t i n d i c a t i o n o f t h e amount o f e r r o r i n v o l v e d i n t h e a p p r o x i m a t e s o l u t i o n a t t h e end o f an i n c r e m e n t . However , t h i s t e rm can be employed t o c o n t r o l o r r e d u c e t h e e r r o r i n f i n d i n g e q u i l i b r i u m c o n f i g u r a t i o n s . T h i s w i l l be examined i n t h e f o l l o w i n g c h a p t e r . 3.6 S u r f a c e T r a c t i o n s and Body F o r c e s The f o rm o f t h e l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n t h a t i s chosen f o r any p a r t i c u l a r p r o b l e m i s p r i m a r i l y dependent upon t h e ea se w i t h w h i c h t h e s u r f a c e t r a c t i o n s can be r e p r e s e n t e d . I t i s p o s s i b l e t h r o u g h t h e u se o f t h e r e l a t i o n s h i p s between t h e t h r e e d i f f e r e n t t y p e s o f s t r e s s t e n s o r " g i v e n i n Eq . 2 . 20 , 2.21 and 2.22 t o u s e e i t h e r o f t h e two forms d e r i v e d and g i v e n b y E q . 3.43 and 3 . 45 . The t r a n s f o r m a t i o n s be tween t h e s t r e s s t e n s o r s a r e n o t e a s i l y made, and a r e dependent on t h e c o n f i g u r a t i o n o f t h e s u r f a c e a t t h e end o f t h e i n c r e m e n t , t h e r e f o r e t h e y canno t be s p e c i f i e d a t t h e b e g i n n i n g o f t h e i n c r e m e n t . The a l t e r n a t i v e i s t o u s e s u r f a c e t r a c t i o n s t h a t behave d u r i n g d e f o r m a t i o n i n s u ch a manner t h a t t h e y f o l l o w t h e Lag range o r K i r c h h o f f r u l e o f c o r r e s p o n d e n c e as shown i n S e c t i o n 2 .3 . A f o r c e v e c t o r t h a t a c t u a l l y behaves a c c o r d i n g t o t h e Lag range r u l e o f c o r r e s p o n d e n c e has t h e p r o p e r t y t h a t i t ha s a p a r a l l e l l i n e o f a c t i o n w i t h r e s p e c t t o i t s l i n e o f a c t i o n on t h e s u r f a c e ° S , and has t h e 40 same m a g n i t u d e . I t i s t h e r e f o r e ea s y t o r e p r e s e n t any s u r f a c e t r a c t i o n t h a t keeps a p a r a l l e l l i n e o f a c t i o n and t h e same magn i tude u n d e r d e f o r m a t i o n , by a L a g r a n g i a n s t r e s s t e n s o r . T h i s t y p e o f s u r f a c e t r a c t i o n i s t h e most commonly u s e d , i m p l i c i t l y o r e x p l i c i t l y , i n a v a i l a b l e c l o s e d - f o r m s o l u t i o n s f o r l a r g e d e f o r m a t i o n s . A f o r c e v e c t o r t h a t changes a c c o r d i n g t o t h e K i r c h h o f f r u l e o f c o r r e s p o n d e n c e w o u l d be most e a s i l y r e p r e s e n t e d by a K i r c h h o f f s t r e s s t ype o f s u r f a c e t r a c t i o n . ;• x F i n a l l y a s u r f a c e t r a c t i o n t h a t i s n o r m a l t o t h e de fo rmed s u r f a c e and p r o p o r t i o n a l t o i t i s r e p r e s e n t e d by an E u l e r i a n s t r e s s t e n s o r i n t h e de fo rmed c o n f i g u r a t i o n . To be u s e d by e i t h e r o f t he two l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n s t h i s s t r e s s t e n s o r must be t r a n s f o r m e d i n t o e i t h e r a K i r c h h o f f o r Lag range t y p e o f s u r f a c e t r a c t i o n . T h i s t y p e o f l o a d i n g i s t he c o r r e c t r e p r e s e n t a t i o n f o r a p r e s s u r e l o a d i n g o f a s u r f a c e . The use o f t he body f o r c e te rm i n t h e e q u a t i o n s d e s e r v e s some d i s c u s s i o n . I t w o u l d be p o s s i b l e by a p p l y i n g D ' A l e m b e r t ' s p r i n c i p l e t o i n c l u d e i n e r t i a f o r c e s as p a r t o f t h e - b o d y f o r c e s , b u t t h i s l i e s o u t s i d e o f t h e s cope o f t h i s t h e s i s . The body f o r c e s a r e u s u a l l y n e g l e c t e d i n t h e c l o s e d - f o r m s o l u t i o n s a v a i l a b l e , and so w i l l n o t be u sed i n t h e n u m e r i c a l a n a l y s e s p e r f o r m e d f o r c o m p a r i s o n p u r p o s e s where t h i s i s t r u e o f t h e c l o s e d - f o r m s o l u t i o n . 3 . 7 Summary The v i r t u a l work e x p r e s s i o n s ; d e v e l o p e d i n t h i s c h a p t e r u s e t h e L a g r a n g i a n d e s c r i p t i o n . The e q u a t i o n s f o r any p a r t i c u l a r c o n f i g u r a t i o n 41 and t h e n o n l i n e a r i n c r e m e n t a l v i r t u a l work e q u a t i o n a r e e x a c t , v a l i d f o r any magn i t ude o f d e f o r m a t i o n , and a r e i n d e p e n d e n t o f t h e p a r t i c u l a r 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 c h o s e n . They a r e m e r e l y e x p r e s s i o n s o f t h e e q u i l i b r i u m o f a body s u b j e c t e d t o s u r f a c e t r a c t i o n s and body f o r c e s . The n o n l i n e a r i n c r e m e n t a l v i r t u a l work e q u a t i o n was l i n e a r i z e d by n e g l e c t i n g t he n o n l i n e a r i n t e g r a l i n t h e e q u a t i o n . T h i s l i n e a r i z e d i n c r e m e n t a l v i r t u a l work can be u s e d t o p r o v i d e a d i r e c t s o l u t i o n f o r i n c r e m e n t a l d e f o r m a t i o n g i v e n t h e c o n f i g u r a t i o n a t t h e b e g i n n i n g o f t h e i n c r e m e n t and t h e i n c r e m e n t a l v a l u e s o f t he s u r f a c e t r a c t i o n and body f o r c e . T h i s d i r e c t s o l u t i o n thus g i v e s an a p p r o x i m a t i o n t o t h e e x a c t e q u i l i b r i u m c o n f i g u r a t i o n . The a p p r o x i m a t e s o l u t i o n w i l l t e n d t o d i v e r g e f r om t h e e x a c t s o l u t i o n d e p e n d i n g on i n c r e m e n t s i z e and t h e deg ree o f n o n l i n e a r i t y o f t h e p r o b l e m . U s i n g t h e a p p r o x i m a t e s o l u t i o n o b t a i n e d f o r t h e c o n f i g u r a t i o n a t t h e end o f t he i n c r e m e n t , e q u i l i b r i u m can be c h e c k e d by s u b s t i t u t i n g i n t o t h e e x a c t v i r t u a l work e x p r e s s i o n f o r t h e c o n f i g u r a t i o n . The n o n s a t i s f a c t i o n o f t he v i r t u a l work e q u a t i o n , w h i c h i s e q u i v a l e n t t o n o n -s a t i s f a c t i o n o f e q u i l i b r i u m , i s r e p r e s e n t e d by t h e r e s i d u a l l o a d i n g t e r m . The n u m e r i c a l t e c h n i q u e s t o s o l v e t h e e q u a t i o n s d e r i v e d f o r f i n i t e d e f o r m a t i o n a r e g i v e n i n t h e f o l l o w i n g c h a p t e r s . 42 SOLUTION OF THE EQUILIBRIUM EQUATIONS 4.1 G e n e r a l The re a r e v a r i o u s s o l u t i o n s t r a t e g i e s a v a i l a b l e t o s o l v e t h e e q u i l i b r i u m e q u a t i o n s , d e r i v e d i n t h e p r e v i o u s c h a p t e r i n t h e f o rm o f v i r t u a l work e q u a t i o n s . These s t r a t e g i e s w i l l be d i f f e r e n t i a t e d by t he manner i n w h i c h d i v e r g e n c e f r o m the e x a c t e q u i l i b r i u m c o n f i g u r a t i o n s , c au sed by u s i n g a l i n e a r i n c r e m e n t a l v i r t u a l work e q u a t i o n , i s c o n t r o l l e d o r r e d u c e d . By l i n e a r i z i n g t he i n c r e m e n t a l v i r t u a l work e q u a t i o n o n l y a l i n e a r p r e d i c t i o n o r a p p r o x i m a t i o n o f t h e t r u e c o n f i g u r a t i o n a t t h e end o f an i n c r e m e n t i s o b t a i n e d . As was shown i n t h e p r e v i o u s c h a p t e r , t h e deg ree o f n o n s a t i s f a c t i o n o f e q u i l i b r i u m i s e x p r e s s e d by t he r e s i d u a l l o a d i n g t e r m . I f no e f f o r t i s made t o c o n t r o l t he d i v e r g e n c e o f t he s o l u t i o n t h r o u g h t he use o f t h e r e s i d u a l l o a d i n g t e r m , t h e n t h e s o l u t i o n s t r a t e g y i s known as an i n c r e m e n t a l method w i t h o u t e q u i l i b r i u m c h e c k s . C o n v e r s e l y , i f t h e r e s i d u a l l o a d i n g te rm i s e v a l u a t e d and u sed t o m o d i f y t h e s o l u t i o n o b t a i n e d i n o r d e r t o r e d u c e o r c o n t r o l t he d i v e r g e n c e f r om t h e t r u e e q u i l i b r i u m c o n f i g u r a t i o n , t h e n t h e s o l u t i o n s t r a t e g y i s known as an i n c r e m e n t a l method w i t h e q u i l i b r i u m c h e c k s . These may be o f an i t e r a - -r i v e n a t u r e w i t h a t o l e r a n c e l e v e l o f d i v e r g e n c e on some p a r a m e t e r o f t h e s y s t e m , o r t h e y may be what w i l l be c a l l e d s e l f - c o r r e c t i n g " p r o c e d u r e s . No a t t e m p t i s made i n t h i s c h a p t e r t o c o n s i d e r t h e v a r i o u s d i r e c t methods t h a t can be u s e d t o s o l v e t he g o v e r n i n g n o n l i n e a r e q u a t i o n s , 43: o r t he p r e d i c t o r - c o r r e c t o r s t r a t e g i e s s uch as t h e R u n g e - K u t t a and E u l e r p r o c e d u r e s . T h i s c h a p t e r w i l l be r e s t r i c t e d t o t he c o n s i d e r a t i o n o f i n c r e m e n t a l methods w i t h o r w i t h o u t e q u i l i b r i u m c h e c k s . I n t h i s c h a p t e r t h e r e c u r r e n c e r e l a t i o n s f o r t h e v a r i o u s s o l u t i o n s t r a t e g i e s w i l l be w r i t t e n i n m a t r i x f o r m , b u t w i l l be i l l u s t r a t e d g r a p h i c a l l y by r e f e r e n c e t o a s i n g l e deg ree o f f r eedom s y s t e m . The s t i f f n e s s m a t r i x i s u s e d i n a g e n e r a l manner t o r e p r e s e n t t h e r e l a t i o n s h i p be tween an i n p u t and a r e s p o n s e . 4 .2 I n c r e m e n t a l method w i t h o u t E q u i l i b r i u m Checks In t h i s method, t h e s u r f a c e t r a c t i o n s and body f o r c e s a r e a p p l i e d i n a sequence o f i n c r e m e n t s t h a t a r e assumed t o be s u f f i c i e n t l y s m a l l , s uch t h a t t h e body may be assumed t o r e s p o n d l i n e a r l y d u r i n g each i n c r e m e n t . The r e s p o n s e c h a r a c t e r i s t i c s o f t h e body a r e d e t e r m i n e d by t h e c o n f i g u r a t i o n and m a t e r i a l p r o p e r t i e s o f t he body a t t h e b e g i n n i n g o f each i n c r e m e n t . F o r each i n c r e m e n t i n t h e l o a d s on t h e b o d y , i n c r e m e n t a l d i s p l a c e m e n t s , s t r e s s e s and s t r a i n s a r e e v a l u a t e d and u sed t o g e t h e r w i t h t he c o n f i g u r a t i o n o f t h e body a t t h e b e g i n n i n g o f t h e i n c r e m e n t t o d e f i n e t h e c o n f i g u r a t i o n a t t h e end o f t h e l o a d i n c r e m e n t . New r e s p o n s e c h a r a c t e r i s t i c s o f t h e body a r e t h e n c a l c u l a t e d b a s e d on t h e c o n f i g u r a t i o n j u s t o b t a i n e d , and t h e n e x t i n c r e m e n t o f l o a d i s a p p l i e d . T h i s who l e p r o c e s s i s r e p e a t e d u n t i l t h e sum o f t h e l o a d s a p p l i e d i n a l l t h e i n c r e m e n t s e q u a l s t he d e s i r e d t o t a l l o a d f o r w h i c h the s o l u t i o n i s r e q u i r e d . T h i s t e c h n i q u e has t h e d i s a d v a n t a g e t h a t t h e r e i s no r e a l e s t i m a t e o f i t s a c c u r a c y , s i n c e t h e l i n e a r i z e d i n c r e m e n t a l e q u a t i o n s do 44 n o t i n g e n e r a l g i v e a c o n f i g u r a t i o n a t t h e end o f t he l o a d i n c r e m e n t t h a t s a t i s f i e s e q u i l i b r i u m . S i n c e each i n c r e m e n t i s dependent on t he a c c u r a c y o f t he i n i t i a l c o n f i g u r a t i o n and r e p r e s e n t s o n l y a l i n e a r a p p r o x i m a t i o n t o t h e n o n l i n e a r r e s p o n s e i n t h e i n c r e m e n t , t h e s o l u t i o n t e n d s t o " d r i f t " o r d i v e r g e f r om the e x a c t s o l u t i o n . The d i v e r g e n c e o f t he i n c r e m e n t a l method w i t h o u t e q u i l i b r i u m check s i s shown i n F i g . 5, f o r a s i m p l e s i n g l e deg ree o f f r eedom s y s t e m . To e n s u r e t h a t a s o l u t i o n o b t a i n e d i n t h i s manner i s c l o s e t o t h e e x a c t s o l u t i o n , r e c o u r s e must be made t o s o l v i n g t h e same p r o b l e m r e p e a t e d l y w i t h s u c c e s s i v e l y s m a l l e r i n c r e m e n t s u n t i l two s u c c e s s i v e s o l u t i o n s c onve r ge w i t h i n some t o l e r a n c e . By r e d u c i n g t h e i n c r e m e n t s i z e , and hence u s i n g more i n c r e m e n t s t o f o l l o w a n o n l i n e a r r e s p o n s e , i t i s e x p e c t e d t h a t t h e c u m u l a t i v e e r r o r i n t h e s o l u t i o n w i l l d e c r e a s e . I n t h e l i m i t as t he i n c r e m e n t s i z e s h r i n k s t o z e r o and t h e number o f i n c r e m e n t s app roache s i n f i n i t y , t h e s o l u t i o n s h o u l d become e x a c t . S i n c e new r e s p o n s e c h a r a c t e r i s t i c s o f t he body must be c a l c u l a t e d f o r t h e b e g i n n i n g o f each i n c r e m e n t , t h i s p r o c e d u r e can become q u i t e t i m e consuming and c o s t l y f o r r e a s o n a b l y s i z e d p r o b l e m s . The advan tage s o f t h i s method a r e m o s t l y t o be f ound i n t h e a n a l y s i s o f b o d i e s h a v i n g 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 . T h i s i s t r u e p r i m a r i l y o f e l a s t o - p l a s t i c m a t e r i a l s , s i n c e p l a s t i c i t y laws a r e g e n e r a l l y w r i t t e n i n i n c r e m e n t a l f o r m and t hu s a r e e a s i l y i n c o r p o r a t e d i n t o an i n c r e m e n t a l method. S i n c e p l a s t i c d e f o r m a t i o n i s a p a t h dependent phenomenon, i t i s an a t t r a c t i v e f e a t u r e o f t h i s method t h a t by c h o o s i n g i n c r e m e n t s o f l o a d s u f f i c i e n t l y s m a l l , and g i v e n an i adequa te p l a s t i c i t y d e s c r i p t i o n , t h e d e f o r m a t i o n h i s t o r y o f t h e body s h o u l d b e t r a c e a b l e . I n t h i s c a s e , t h i s s o l u t i o n method i s s u p e r i o r t o 45 A i A 2 A 3 DEFLECTION (RESPONSE) DIVERGENCE OF THE INCREMENTAL METHOD WITHOUT EQUILIBRIUM CHECKS F I G . 5 an i t e r a t i v e method w h i c h may e i t h e r o s c i l l a t e a r o u n d t h e c o r r e c t s o l u t i o n o r cau se a f a l s e d e f o r m a t i o n p a t h t o be o b t a i n e d . The two b a s i c s u b d i v i s i o n s o f app r oach w i t h i n t h e i n c r e m e n t a l methods w i t h o u t e q u i l i b r i u m check s a r e t h e t a n g e n t modulus method and t h e i n i t i a l s t r a i n method . A n o t h e r app roach w h i c h i s i n e s s e n c e t h e same as t h e i n i t i a l s t r a i n method i s t h e i n i t i a l s t r e s s method . The t a n g e n t modulus and i n i t i a l s t r a i n methods a r e d i f f e r e n t i a t e d by t he manner i n w h i c h t h e n o n l i n e a r a s p e c t s o f t h e p r o b l e m a r e d e a l t w i t h . F o r t h e t a n g e n t modulus method, c o n s i d e r i n g a g e n e r a l n o n l i n e a r p r o b l e m , t h e r e c u r r e n c e r e l a t i o n s may be w r i t t e n f o r t h e i ^ i n c r e m e n t as { A U } ^ - { A P } i (4 .1 ) NLNT l - l " ^ W u i - i 3 w i t h { u } , = Z { A u } . ( 4 .2 ) j = l 3 I n E q . 4 . 1 , [K^] i s t h e l i n e a r s t i f f n e s s m a t r i x , and [ ^ y ^ ( u ^ _ ^ ) ] and ^ N L G ^ u i - l ^ a r e t ' i e m a t e r i a l a n c * g e o m e t r i c n o n l i n e a r i t y s t i f f n e s s m a t r i c e s r e s p e c t i v e l y . The t e r m i n d i c a t e s t h a t t h e two n o n -l i n e a r m a t r i c e s a r e f u n c t i o n s o f t h e c o n f i g u r a t i o n a t t h e b e g i n n i n g o f t h e i n c r e m e n t . The v e c t o r s {Au;-}.^-and {AP}- a r e t h e i n c r e m e n t a l g e n e r a l i z e d d i s p l a c e m e n t s and f o r c e s r e s p e c t i v e l y , and {uK r e p r e s e n t s the t o t a l g e n e r a l i z e d d i s p l a c e m e n t s a t t h e end o f t h e i ^ i n c r e m e n t . 47 In t h e t a n g e n t modulus method t h e e v a l u a t i o n o f t h e comb ined s t i f f n e s s m a t r i x i n E q . 4.1 f o r each i n c r e m e n t and i t s s ub sequen t i n v e r s i o n , t o s o l v e f o r t h e i n c r e m e n t a l d i s p l a c e m e n t s , cau se s t h i s method t o become v e r y c o s t l y as t h e number o f i t e r a t i o n s r e q u i r e d o r t h e s i z e o f t h e p r o b l e m i n c r e a s e s . I t i s t o r e d u c e t h e c o s t a s s o c i a t e d w i t h i n v e r t i n g a s t i f f n e s s m a t r i x f o r each i n c r e m e n t t h a t t h e i n i t i a l s t r a i n method was d e v e l o p e d . C o n s i d e r i n g a g a i n a g e n e r a l n o n l i n e a r p r o b l e m , t h e r e c u r r e n c e r e l a t i o n s f o r t h e i n i t i a l s t r a i n method a r e w r i t t e n as [ K J {Au}. = {AP}. + ( Q N L G } i _ 1 + { Q ^ } . ^ (4 .3 ) a g a i n u s i n g Eq . 4.2 t o sum t h e i n c r e m e n t a l g e n e r a l i z e d d i s p l a c e m e n t s . The v e c t o r s { Q ^ L G ^ I A N T L ^ N L M ^ i - 1 a r e t ^ i e P s e u d o - l o a d s f o r g e o m e t r i c and m a t e r i a l n o n l i n e a r i t i e s r e s p e c t i v e l y , and a r e b a s e d upon t h e c o n f i g u r a t i o n a t t h e b e g i n n i n g o f t h e i n c r e m e n t . They a r e d e r i v e d by c o n s i d e r i n g t h e i r r e s p e c t i v e s o u r c e o f n o n l i n e a r d e f o r m a t i o n as an i n i t i a l s t r a i n f o r t h e n e x t i n c r e m e n t . The advan tage t o be d e r i v e d f r o m t h e a p p l i c a t i o n o f t h i s method i s t h a t o n l y t he m a t r i x [K^] need be i n v e r t e d , and t h i s i s done o n l y o n c e . The e f f o r t i n e v a l u a t i n g t h e p s e u d o - l o a d v e c t o r s o f E q . 4 .3 i s o f the same o r d e r as t h e e f f o r t r e q u i r e d t o e v a l u a t e t h e n o n l i n e a r s t i f f n e s s m a t r i c e s i n Eq . 4 . 1 , so t h e ma in advan tage i s t o be f ound i n t h e r e q u i r e m e n t t o i n v e r t a s t i f f n e s s m a t r i x o n l y o n c e . The d i s a d v a n t a g e s o f t h i s method as compared t o t he t a n g e n t modulus method , a r e t h a t w i t h a r e d u c t i o n i n l o a d i n c r e m e n t s i z e i t c o n v e r g e s s l o w e r and a l s o t h i s method may have n u m e r i c a l i n s t a b i l i t i e s , e s p e c i a l l y when modera te g e o m e t r i c n o n l i n e a r i t i e s a r e e n c o u n t e r e d [ 18 ] . 48 The t a n g e n t modulus method appea r s t o be more advan tageous t h a n the i n i t i a l s t r a i n method as c o n c e r n s c onve r gence and n u m e r i c a l s t a b i l i t y , b u t t h e c o s t i s h i g h e r f o r a g i v e n number o f i n c r e m e n t s u s ed t o a p p l y t h e f u l l l o a d . 4 .3 I n c r e m e n t a l methods w i t h E q u i l i b r i u m Checks 4 .3 .1 G e n e r a l The i n c r e m e n t a l methods w i t h o u t e q u i l i b r i u m checks do n o t i n g e n e r a l g i v e s o l u t i o n s t h a t s a t i s f y e q u i l i b r i u m . C u m u l a t i v e e r r o r s may become s i g n i f i c a n t a s a r e s u l t , and i t t h e n becomes n e c e s s a r y t o s o l v e t h e p r o b l e m w i t h s m a l l e r l o a d s t e p s , and hence more i n c r e m e n t s , u n t i l c o n v e r g e n c e o f two s u c c e s s i v e s o l u t i o n s p r o v i d e s a deg ree o f c o n f i d e n c e i n t h e r e s u l t s . The i n c r e m e n t a l methods w i t h e q u i l i b r i u m check s were d e v i s e d i n o r d e r t h a t t h e d i v e r g e n c e o f t h e a p p r o x i m a t e s o l u t i o n c o u l d be e v a l u a t e d and t h e n r e d u c e d o r c o n t r o l l e d i n some manner. A t t he same t i m e t h e d e s i r a b l e a s p e c t s o f an i n c r e m e n t a l f o r m u l a t i o n a r e m a i n t a i n e d . Once t h e d i v e r g e n c e o f t he app r o x ima te , s o l u t i o n f rom t h e e x a c t s o l u t i o n i s known i n te rms o f t h e r e s i d u a l l o a d i n g t e r m , t h e n t h e a p p r o x i m a t e s o l u t i o n i s e i t h e r m o d i f i e d by an i t e r a t i v e p r o c e s s , o r e l s e t he r e s i d u a l l o a d i n g t e r m i s u s e d t o m o d i f y t h e n e x t i n c r e m e n t a l s t e p . The l a t t e r a p p r o a c h i s c a l l e d a s e l f - c o r r e c t i n g method . B o t h app roache s w i l l be d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s , and t h e a p p r o p r i a t e r e c u r r e n c e r e l a t i o n s g i v e n . 49 4 . 3 . 2 I t e r a t i v e Methods The i t e r a t i v e methods a r e u s e d t o m o d i f y t h e s o l u t i o n r e p e a t e d l y w i t h i n a p a r t i c u l a r i n c r e m e n t , u n t i l some c o n v e r g e n c e c r i t e r i a i s s a t i s f i e d . The i d e a l a p p r o a c h w o u l d be t o i t e r a t e w i t h i n t h e i n c r e m e n t u n t i l t h e c o n f i g u r a t i o n a c q u i r e d a t t h e end o f t he i n c r e m e n t e x a c t l y s a t i s f i e s ! e q u i l i b r i u m . T h i s wou l d be o b s e r v e d by t h e v a n i s h i n g o f t he r e s i d u a l l o a d i n g t e r m . Such an a p p r o a c h i s r a r e l y p r a c t i c a l howeve r , and t hu s t h e u s u a l t e c h n i q u e * i s t o i t e r a t e u n t i l some c o n v e r g e n c e t o l e r a n c e on t h e magn i t ude o f t he r e s i d u a l l o a d i n g te rm i s s a t i s f i e d . T h i s t o l e r a n c e i s cho sen s u f f i c i e n t l y s m a l l so t h a t t h e a p p r o x i m a t e s o l u t i o n w i l l be c l o s e enough t o t h e e x a c t s o l u t i o n f o r t h e p u r p o s e r e q u i r e d . The c o n v e r g e n c e t o l e r a n c e c o u l d be s p e c i f i e d f o r any p a r a m e t e r o f t he c o n f i g u r a t i o n such as a s t r e s s o r a d i s p l a c e m e n t , b u t : i t i s u s u a l t o u se t h e r e s i d u a l l o a d i n g t e r m s i n c e an e q u i l i b r i u m c o n f i g u r a t i o n i s s o u g h t . Two d i f f e r e n t i t e r a t i v e methods t h a t may be a d o p t e d a r e t he Newton-Raphson method shown i n F i g . 6, and t he M o d i f i e d Newton-Raphson method shown i n F i g . 7. The d i f f e r e n c e be tween t h e s e two methods i s i n t h e manner i n w h i c h t h e s t i f f n e s s m a t r i x i s h a n d l e d d u r i n g t h e i t e r a t i o n s t o f i n d t h e c o n f i g u r a t i o n a t t h e end o f an i n c r e m e n t . I n t h e Newton-Raphson method , f o r each i t e r a t i o n t h e r e s i d u a l l o a d i n g t e rm i s e v a l u a t e d and t h e s t i f f n e s s m a t r i x u p d a t e d . Then t he r e s i d u a l l o a d i n g t e rm i s a p p l i e d t o f i n d an i n c r e a s e i n t h e i n c r e m e n t a l d e f o r m a t i o n s t o b r i n g t he a p p r o x i m a t e c o n f i g u r a t i o n c l o s e r t o t h e e x a c t s o l u t i o n . T h i s i s r e p e a t e d u n t i l t h e r e s i d u a l l o a d i n g t e r m i s s u f f i c i e n t l y s m a l l t o s a t i s f y t he t o l e r a n c e c r i t e r i a . The p r o c e d u r e can be e x p r e s s e d f o r t h e f i r s t s t e p i n t h e i t e r a t i o n by 5 0 DEFLECTION NEWTON-RAPHSON METHOD F I G . 6 51 F I G . 7 52 + _L KNLK-I) {Au}:1 = {AP}. 1 1 j = 1 (4 .4 ) and f o r each i t e r a t i o n w i t h i n t h e i ^ i n c r e m e n t t h e r e a f t e r by NL u. , + Au. l - l l j - 1 {Au}? =' {R } j " 1 j = 2, 3. l c J (4 .5 ) where j i s t h e number o f t h e i t e r a t i o n w i t h i n t h e i n c r e m e n t and •n J k Au? = E Au. * k= l 1 (4 .6 ) {R } j = {P}, -c 1 T h i s p r o c e d u r e c ea se s when t h e r e s i d u a l l o a d i n g t e rm s a t i s f i e s t h e t o l e r a n c e r e q u i r e d . Then t h e n e x t i n c r e m e n t o f l o a d i n g i s a p p l i e d . The m a t r i x [ K ^ ] i n c l u d e s b o t h g e o m e t r i c and m a t e r i a l n o n l i n e a r i t y . The M o d i f i e d Newton-Raphson method has t h e same f o r m a t as above e x c e p t t h a t Eq. 4 .5 i s r e p l a c e d by L . u . NL l - l v+'-.Au? l {u. • + Au3. } l - l l (4 .7) NL i-1 {Au}? = { R c } j - l j = 2, 3, (4.8) As can be seen, this method does not use an updated stiffness matrix after each i terat ion, but rather uses the stiffness matrix derived at the beginning of the increment for a l l i terations. This saves calculation effort since the same stiffness matrix in Eq. 4.4 and 4.8 need only be inverted once for a l l iterations within the increment. In an extension to this method, the stiffness matrix for one increment is used for several successive increments unt i l convergence deteriorates, then a new stiffness matrix is evaluated and inverted. The Newton-Raphson method generally should converge in fewer iterations than the Modified Newton-Raphson method, but the latter may be more computationally eff icient by not requiring the stiffness matrix to be evaluated and inverted after each i terat ion. Both methods would appear to give more informative results than an incremental method without equilibrium checks, simply because of the control on the degree of nonsatisfaction of equilibrium permitted. This is true at least for conservative deformation; in nonconservative deformation however, the i terative nature of the methods may adversely influence the solution by causing an osc i l la t ion of the iterations about the true solution. 4.3.3 Self-correcting Method The self-correcting method is derived in the attempt to obtain the advantages of having an equilibrium check that is used to control the divergence of the solution, without the disadvantages of i terat ing. The particular procedure used in this thesis can be expressed by the 54 r e c u r r e n c e r e l a t i o n as {Au}. = {AP}. + {R }. . (4 .9 ) 1 1 c l - l v J where ( R c ' i - l • ( P J i - l -and where t he t o t a l d i s p l a c e m e n t s a r e summed u s i n g Eq . 4 .2 as b e f o r e , and t h e t o t a l l o a d s a r e summed s i m i l a r l y . The p r o c e d u r e i s shown g r a p h i c a l l y i n F i g . 8, f o r a s i n g l e deg ree o f f reedom s y s t e m . The s t i f f n e s s m a t r i x i n Eq . 4 .9 i s d e r i v e d on t h e b a s i s o f t he c o n f i g u r a t i o n a t t h e b e g i n n i n g o f t he i n c r e m e n t , t h e c o n t r i b u t i o n [ K N ^ ( u ^ _ j ) ] b e i n g a f u n c t i o n o f t h e d e f o r m a t i o n a t t h e s t a r t o f t h e i n c r e m e n t . The d i s t i n g u i s h i n g f e a t u r e o f t h i s method i s t h a t t h e r e s i d u a l l o a d i n g te rm f rom t h e p r e v i o u s i n c r e m e n t , w h i c h i s e v a l u a t e d u s i n g Eq . 4 . 1 0 , i s added t o t he l o a d s p e c i f i e d f o r t he p r e s e n t i n c r e m e n t . T h i s p r o c e d u r e may be t h o u g h t o f as a o n e - s t e p i t e r a t i o n o f t he Newton-Raphson method f o r t h e p a s t i n c r e m e n t , added i n t o t he p r e s e n t i n c r e m e n t o f d e f o r m a t i o n . S t r i c k l i n , e t a l [19] m u l t i p l y t h e r e s i d u a l l o a d i n g t e r m i n Eq . 4.9 by a s c a l a r w i t h a r ange o f 1.0 t o 1.2, i n an a t t e m p t t o imp rove t he conve r gence o f t h i s method . The r a t i o n a l e b e h i n d t h i s w o u l d appea r t o be t h a t t h e t r e n d i n n o n l i n e a r i t y o v e r t h e p r e s e n t i n c r e m e n t o f d e f o r m a t i o n w i l l be s i m i l a r i n f o rm t o t h e p a s t i n c r e m e n t , so t h e f a c t o r a p p l i e s an i n c r e a s e d r e s i d u a l l o a d i n g t e rm t o a n t i c i p a t e t h i s . T h i s t e c h n i q u e o f e m p l o y i n g a s c a l a r a m p l i f i e r w i t h t h e r e s i d u a l l o a d i n g te rm w i l l n o t be u sed o r e v a l u a t e d i n t h i s t h e s i s . i - l | K N t U i - 1 {u} i - l ( 4 .10) 55 F I G . 8 56 4.4 Summary Solution strategies for solving the linearized incremental virtual work equations have been presented. These are c lass i f ied by whether or not there is a procedure to control or reduce the non- .. satisfaction of equilibrium caused by l inearizing the v ir tua l work equations. Incremental methods with equilibrium checks include i terative and self-correcting procedures. The self-correcting procedure presents a combination of equilibrium control that is desirable without the cost and disadvantages of i terat ing. The self-correcting procedure w i l l be used in this thesis in the numerical solutions presented in Chapter 7. 57 CONSTITUTIVE RELATIONSHIPS 5.1 G e n e r a l I n o r d e r t o a n a l y z e t h e d e f o r m a t i o n o f any body u n d e r p r e s c r i b e d s u r f a c e t r a c t i o n s , body f o r c e s and bounda r y c o n d i t i o n s u s i n g t he i n c r e m e n t a l v i r t u a l work e q u a t i o n s d e v e l o p e d i n C h a p t e r 3, t h e K i r c h h o f f s t r e s s t e n s o r w i t h i n t h e body must be r e l a t e d t o t h e G r e e n ' s s t r a i n t e n s o r . W i t h o u t s uch a c o n s t i t u t i v e r e l a t i o n s h i p , an a n a l y s i s c anno t p r o c e e d , s i n c e t h e r e w o u l d be t h e n an i n f i n i t e number o f p o s s i b l e s o l u t i o n s f o r t h e d e f o r m a t i o n o f t he body . The c h o i c e o f a 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 h o u l d be such t h a t t h e model:, w h i c h i s d e f i n e d by t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n s , s h o u l d r e p r e s e n t a c c u r a t e l y t h e c h a r a c t e r i s t i c s o f t h e r e a l body . To t h i s e n d , many 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 have been p r o p o s e d . These i n c l u d e e l a s t o - p l a s t i c r e l a t i o n s h i p s o f v a r i o u s k i n d s , v i s c o e l a s t i c i t y , h y p e r e 1 a s t i c i t y , h y p o e l a s t i c i t y , and e l a s t i c 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 . I n t h i s t h e s i s , i t i s i n t e n d e d t h a t a l i n e a r e l a s t i c c o n s t i t u t i v e r e l a t i o n s h i p be tween K i r c h h o f f s t r e s s and G r e e n ' s s t r a i n t e n s o r s be a d o p t e d . F u r t h e r m o r e t h e r e l a t i o n s h i p w i l l be assumed t o be t h a t o f H o o k e ' s law f o r an i s o t r o p i c m a t e r i a l . The deve lopment i n t h e f o l l o w i n g s e c t i o n w i l l be f o r a g e n e r a l e l a s t i c i t y c a s e , i n c l u d i n g t he l i n e a r e l a s t i c , i s o t r o p i c r e l a t i o n s h i p as a s p e c i a l c a s e . 58 5.2 E l a s t i c C o n s t i t u t i v e T e n s o r A s suming o n l y e l a s t i c d e f o r m a t i o n f o r t h e p r e s e n t , t h e K i r c h h o f f s t r e s s t e n s o r and G r e e n ' s s t r a i n t e n s o r may be r e l a t e d b y where a C . . . . i s t h e e l a s t i c c o n s t i t u t i v e t e n s o r f o r c o n f i g u r a t i o n a C . F o r t h e i lC, t h e n n c r e m e n t a l K i r c h h o f f s t r e s s , , be tween c o n f i g u r a t i o n s 2 C and 1, i j ijki ki ijki ki 2 C . .. e. „ + ( 2 C . .. . - 1 C . ., 0)1e, „ (5 .2 ) 13 ki ijki ijkiJ ki K J The i n c r e m e n t 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 g i v e n i n E q . 5.2 i s c a p a b l e o f r e p r e s e n t i n g n o n l i n e a r and n o n i s o t h e r m 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 . The use o f t h i s r e l a t i o n s h i p w o u l d however r e q u i r e i t e r a t i o n s i n c e t h e c o n f i g u r a t i o n 2 C i s n o t known a p r i o r i , and hence i t s e l a s t i c c o n s t i t u t i v e t e n s o r ^ C . ., „ i s unknown. A r e a s o n a b l e s i m p l i f i c a t i o n t o overcome t h i s p r o b l e m i s t o assume t h a t t h e change i n t h e c o n s t i t u t i v e t e n s o r f o r t h e i n c r e m e n t i s n e g l i g i b l e , t h e n S. . = 1 C . .. . e. . ( 5 .3 ) i ] xjki ki ^ 1 The r e s i d u a l l o a d i n g t e r m c o u l d t h e n be emp loyed t o l i m i t t h e e r r o r i n t r o d u c e d by e v a l u a t i n g t he s t r e s s e s w i t h i n t h e body a t t h e end o f t he i n c r e m e n t u s i n g Eq . 5 . 1 . 59 A f u r t h e r s i m p l i f i c a t i o n s t i l l needs t o be made, s i n c e e, i s n o n l i n e a r f o r t h e i n c r e m e n t o f d e f o r m a t i o n . By i n t r o d u c i n g E q . 5.3 i n t o t he l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n s t h e y w o u l d a g a i n become n o n l i n e a r . T h e r e f o r e o n l y t h e l i n e a r p o r t i o n o f t h e i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r i s u s e d t o e v a l u a t e t h e i n c r e m e n t a l K i r c h h o f f s t r e s s , t hu s m o d i f y i n g E q . 5.3 and n e g l e c t i n g t he n o n l i n e a r t e rm as b e i n g o f s m a l l e r o r d e r S i j = l G i j k £ e k £ < 5 : 5 ) F o r t h e p r o b l e m s a n a l y z e d i n t h i s t h e s i s , t h e c o n s t i t u t i v e r e l a t i o n s h i p w i l l be t a k e n as b e i n g l i n e a r l y e l a s t i c . T h e r e f o r e t h e d i f f e r e n t i a t i o n be tween c o n s t i t u t i v e t e n s o r s i n d i f f e r e n t c o n f i g u r a t i o n s d i s a p p e a r s . The o n l y a p p r o x i m a t i o n t h a t r ema in s i s t h e n e c e s s a r y employment o f o n l y t h e l i n e a r p o r t i o n o f t he i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r i n Eq . 5 . 5 . T h i s i s r e q u i r e d i n o r d e r t o keep t he i n c r e m e n t a l e q u a t i o n s l i n e a r , so t h a t a d i r e c t s o l u t i o n may be o b t a i n e d . In t h e r e s i d u a l l o a d i n g te rm howeve r , t h i s w i l l be c o r r e c t e d f o r by u s i n g t h e f u l l G r e e n ' s s t r a i n t e n s o r t h e r e . The s t r e s s t e n s o r 1 S ^ . i n t h e s e c o n d i n t e g r a l on t h e r i g h t hand s i d e o f E q . 3.43 w i l l a l s o be e v a l u a t e d u s i n g t h e f u l l G r e e n ' s s t r a i n t e n s o r and t h e c o n s t i t u t i v e r e l a t i o n s h i p g i v e n i n Eq . 5 . 1 . T h i s i s p o s s i b l e b e c a u s e c o n f i g u r a t i o n i s known a t t he b e g i n n i n g o f t he i n c r e m e n t . Now the choice of the elements of the constitutive tensor is s t i l l undetermined. It is intended that they be chosen so as to represent an isotropic material corresponding to Hooke's law. This can be expressed in ind ic ia l notation as C i jk& 6., 6 . . + l k j £ t 5 i £ 6jkJ 6 i k 6 k £ vE |_(l+v)(l-2v) J (5.6) This gives a symmetric constitutive relationship, where G is the shear modulus, E is Young's modulus, and v is Poisson's rat io . 61 APPLICATION OF THE F IN ITE ELEMENT METHOD 6.1 G e n e r a l The f i n i t e e l emen t method i s i n t r o d u c e d i n t h i s c h a p t e r and then a p p l i e d t o the i n c r e m e n t a l v i r t u a l work e q u a t i o n d e r i v e d i n C h a p t e r 3, i n o r d e r t o p r o v i d e a p r o c e d u r e f o r n u m e r i c a l a n a l y s i s . The i n c r e m e n t a l v i r t u a l work e q u a t i o n s w i l l be r e c a s t i n m a t r i x f o rm u s i n g t h e f i n i t e e l ement method f o r t he g e n e r a l t h r e e d i m e n s i o n a l c a s e , t hen s p e c i a l i z e d f o r two d i m e n s i o n a l a n a l y s i s . 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 w i l l be d e v e l o p e d f o r p l a n e s t r a i n and p l a n e s t r e s s a n a l y s i s , by a s suming a l i n e a r e l a s t i c r e l a t i o n s h i p between t h e K i r c h h o f f s t r e s s t e n s o r and G r e e n ' s s t r a i n t e n s o r . The r e l a t i o n s h i p u s e d w i l l be H o o k e ' s law f o r an i s o t r o p i c m a t e r i a l . Two e i g h t deg ree o f f reedom r e c t a n g u l a r f i n i t e e l e m e n t s w i l l be d e r i v e d u s i n g t he assumed d i s p l a c e m e n t a p p r o a c h . These e l e m e n t s w i l l be fo rmed u s i n g n u m e r i c a l i n t e g r a t i o n , and o n l y d i f f e r f r om each o t h e r i n the i n t e g r a t i o n scheme a d o p t e d . The f i r s t w i l l be i n t e g r a t e d u s i n g G a u s s i a n q u a d r a t u r e and t h i s e l e m e n t i s t h e n j u s t t he Me lo sh r e c t a n g l e [ 20 ] . The s e c o n d f i n i t e e l emen t w i l l u se a n o n u n i f o r m i n t e g r a t i o n scheme i n an a t t e m p t t o imp rove t h e a c c u r a c y o b t a i n e d , f o r g i v e n c o m p u t a t i o n a l e f f o r t , o v e r t h a t o f t h e Me lo sh r e c t a n g l e . 6.2 The F i n i t e E l ement Method The f i n i t e e l ement m e t h o d . i s a t e c h n i q u e whereby a c o n t i n u o u s s o l u t i o n o f a p r o b l e m o v e r a s p e c i f i e d domain may be r e p r e s e n t e d by 62 p i e c e w i s e c o n t i n u o u s a p p r o x i m a t i o n s . I t i s a p u r e l y t o p o l o g i c a l a p p r o x i m a t i o n and i s i n d e p e n d e n t o f t he v a r i a t i o n a l p r i n c i p l e s u s e d , and the t e c h n i q u e s u s e d t o m i n i m i z e t h e e r r o r be tween t h e a p p r o x i m a t i o n and t h e a c t u a l c o n t i n u o u s s o l u t i o n . The t o p o l o g i c a l n a t u r e o f t he f i n i t e e l e m e n t method was p r e s e n t e d by Oden [ 2 1 ] , and t h e b a s i c s t e p s i n v o l v e d i n t h e method were d e v e l o p e d i n t h e same p a p e r . The f i r s t s t e p i n any f i n i t e e l emen t a n a l y s i s i s t o r e p l a c e t h e s p e c i f i e d domain o f the c o n t i n u o u s s o l u t i o n b y a n o t h e r domain w h i c h can be d i v i d e d i n t o a f i n i t e number o f subdomains c a l l e d e l e m e n t s . These e l e m e n t s do n o t o v e r l a p . I t i s d e s i r a b l e t o have t h i s new domain be e q u i v a l e n t t o the domain o f t he c o n t i n u o u s s o l u t i o n , b u t t h i s may n o t i n a l l c a se s be p o s s i b l e . I f t h i s i s t he c a s e t h e n t he d i f f e r e n c e be tween t h e two doma in s , c a l l e d t h e e r r o r doma in , s h o u l d be k e p t as s m a l l as p o s s i b l e . T h i s w i l l be a p r o b l e m e n c o u n t e r e d when t r y i n g t o model complex domain b o u n d a r i e s w i t h e l e m e n t s h a v i n g a s i m p l e r / g e o m e t f i c f o r m . F o r e xamp le , i n r e p l a c i n g t h e domain o f t he c o n t i n u o u s s o l u t i o n where t h i s domain has c u r v e d b o u n d a r i e s by a domain made up o f f i n i t e e l e m e n t s h a v i n g o n l y s t r a i g h t b o u n d a r i e s , the new domain w i l l o n l y a p p r o x i m a t e t h a t o f t he c o n t i n u o u s s o l u t i o n . By u s i n g more e l e m e n t s o f s m a l l e r d i m e n s i o n a l o n g t h e b o u n d a r y , t he e r r o r domain may be made t o be as s m a l l as d e s i r e d . In t he s e cond s t e p , t he i n d i v i d u a l e l emen t s a r e assumed t o be c o n n e c t e d t o a d j a c e n t e l e m e n t s o n l y , and t o t h e s e e l e m e n t s a t o n l y a f i n i t e number o f d i s c r e t e p o i n t s c a l l e d nodes . These nodes a r e g e n e r a l l y l o c a t e d on t h e b o u n d a r i e s o f t he e l ement s i n c e i n t e r i o r nodes may n o t be c o n n e c t e d t o any a d j a c e n t e l ement w i t h o u t c a u s i n g t h e e l e m e n t s t o o v e r l a p i n t h e i r domains . I n t e r i o r nodes i f t h e y o c c u r a r e u s u a l l y s t a t i c a l l y condensed o u t o f the e l e m e n t . Now t h e v a l u e o f t h e s o l u t i o n and t he d e r i v a t i v e s , i f a p p l i c a b l e , a t t he nodes w i l l be t h e 63 unknown p a r a m e t e r s o f t he p r o b l e m . I f t h e a c t u a l c o n t i n u o u s s o l u t i o n on t h e s p e c i f i e d domain i s to, t hen an a p p r o x i m a t e s o l u t i o n u i s d e f i n e d u n i q u e l y i n each o f t h e e l e m e n t s f o r any p o i n t p w i t h i n t he e l emen t by w 6 ( p ) = aJ <|>?(p) j = 1, . . . . N (6 .1 ) where <J>j(p) a r e known c o o r d i n a t e f u n c t i o n s , o r i n t e r p o l a t i o n f u n c t i o n s , d e f i n e d i n t h e domain o f t he e l ement c o n c e r n e d o n l y , a . a r e t h e n o d a l v a l u e s o f t he s o l u t i o n , and N i s t he number o f n o d a l v a l u e s , o r deg ree s o f f r eedom o f the e l e m e n t . The c o o r d i n a t e f u n c t i o n s a r e r e q u i r e d t o s a t i s f y t h e c o n d i t i o n t h a t i f t h e n o d a l c o o r d i n a t e s o f node n a r e g i v e n by x " , t h e n •<be ( x n ) = S: (6 .2 ) By s a t i s f y i n g t h i s c o n d i t i o n , t h e c o o r d i n a t e f u n c t i o n s w i l l be l i n e a r l y i n d e p e n d e n t t h r o u g h o u t t h e e l ement doma in . The t h i r d s t e p i s t o combine a l l t h e e l emen t s t o fo rm t he a p p r o x i m a t e domain by e q u a t i n g t he n o d a l p a r a m e t e r s o f t he c o r r e s p o n d i n g nodes on t h e i n t e r e l e m e n t b o u n d a r i e s o f a d j a c e n t e l e m e n t s . The a p p r o x i m a t i o n u i s now s p e c i f i e d o v e r t h e who le o f t h e a p p r o x i m a t e •> doma in . By c o m b i n i n g t he e l e m e n t s , t h e r e r e s u l t s a s y s t em o f M l i n e a r l y i n d e p e n d e n t c o o r d i n a t e f u n c t i o n s , and M unknown n o d a l p a r a m e t e r s . The v a l u e M i s n u m e r i c a l l y e q u a l t o the number o f d i s t i n c t nodes l e f t i n t h e p r o b l e m a f t e r c o m b i n i n g t h e e l e m e n t s , m u l t i p l i e d by t h e number o f deg ree s o f f reedom p e r node, minus t h e c o n s t r a i n e d deg ree s o f f reedom r e p r e s e n t i n g 64 bounda r y c o n d i t i o n s . : Then t h e e q u a t i o n s f o r d e t e r m i n i n g t he M unknown n o d a l p a r a m e t e r s may be o b t a i n e d by u s i n g w e i g h t e d r e s i d u a l methods s u ch as t h e G a l e r k i n method, l e a s t s q u a r e s method, c o l l e c t i o n method , o r i n t h e c a s e o f a v i r t u a l work f o r m u l a t i o n by t h e M i n d e p e n d e n t e q u a t i o n s t h a t can be g e n e r a t e d . I n u s i n g t he f i n i t e e l emen t method , i t i s h i g h l y d e s i r a b l e t h a t some f o r m o f c onve r gence o f t h e a p p r o x i m a t e s o l u t i o n t o t h e r e a l c o n t i n u o u s s o l u t i o n be known t o e x i s t i f c e r t a i n r e a d i l y i d e n t i f i a b l e c r i t e r i a a r e s a t i s f i e d . The two b a s i c c r i t e r i a upon w h i c h c o n v e r g e n c e p r o o f s have been d e v e l o p e d a r e t h e c o m p l e t e n e s s o f i n t e r p o l a t i o n f u n c t i o n s w i t h i n t h e e l e m e n t s , arid i n t e r - e l e m e n t c o m p a t a b i l i t y . The n e c e s s i t y and/ o r s u f f i c i e n c y o f t h e s e two c r i t e r i a t o e n s u r e c o n v e r g e n c e , has been a t o p i c o f i n t e r e s t and d i s p u t e s i n c e t h e f i r s t i n v e s t i g a t i o n s were made i n t o t he c onve r gence o f t h e f i n i t e e l e m e n t method. The c r i t e r i a o f c o m p l e t e n e s s a r i s e s f r om t h e r e q u i r e m e n t t h a t t he h i g h e s t d e r i v a t i v e o f t h e i n t e r p o l a t i o n f u n c t i o n s i n v o l v e d i n t h e f o r m u l a t i o n o f t h e e n e r g y o r v i r t u a l work i n t e g r a l s must be c o n t i n u o u s , and be a b l e t o t a k e f i n i t e v a l u e s w i t h i n t h e e l e m e n t . T h i s was e n u n c i a t e d by O l i v e i r a [22, 23] i n r e q u i r i n g d i s p l a c e m e n t p o l y n o m i a l i n t e r p o l a t i o n f u n c t i o n s t o be c o m p l e t e t o o r d e r p, where p i s t h e o r d e r o f t h e maximum d e r i v a t i v e o f d i s p l a c e m e n t s e n c o u n t e r e d i n t h e ene r gy i n t e g r a l s . O l i v e i r a [23] i n one o f t he f i r s t c onve r gence p r o o f s f o r t h e f i n i t e e l e m e n t method , c o n c l u d e d t h a t o n l y c o m p l e t e n e s s o f t h e i n t e r p o l a t i o n f u n c t i o n s was n e c e s s a r y t o g u a r a n t e e c onve r gence i n t h e l i m i t as t h e e l e m e n t mesh i s r e f i n e d . The n e c e s s i t y o f c o m p l e t e n e s s f o r c onve r gence gave a t h e o r e t i c a l b a s i s f o r an e a r l i e r i n t u i t i v e r e q u i r e m e n t advanced by M e l o s h [ 2 4 ] , t h a t e l e m e n t s s h o u l d be c a p a b l e o f e x a c t l y r e p r e s e n t i n g r i g i d body modes and c o n s t a n t s t r a i n s t a t e s . The n o n - c o n v e r g e n c e o f some 65 c o m p l e t e , b u t i n c o m p a t i b l e e l e m e n t s , i n d i c a t e s t h a t more i s r e q u i r e d o f an e l e m e n t t o en su r e c o n v e r g e n c e , t h a n j u s t c o m p l e t e n e s s . Thus c o m p l e t e -ness i s a n e c e s s a r y b u t n o t a s u f f i c i e n t c o n d i t i o n f o r c o n v e r g e n c e o f t h e f i n i t e e l e m e n t a p p r o x i m a t i o n t o t h e e x a c t c o n t i n u o u s s o l u t i o n . Some, u se has been made o f i n c o m p l e t e e l e m e n t s , and a l t h o u g h t h e y w i l l n o t u l t i m a t e l y c o n v e r g e t o t h e e x a c t c o n t i n u o u s s o l u t i o n , t h e y may conve r ge t o a s o l u t i o n w h i c h i s o n l y s l i g h t l y i n e r r o r . [ 2 5 ] . The c r i t e r i a o f c o m p a t a b i l i t y may be app roached f r om d i f f e r e n t v i e w p o i n t s , p e r h a p s t h e most p o p u l a r o f w h i c h i s t h r o u g h t h e use o f t h e theo rem o f minimum p o t e n t i a l e ne r g y when u s i n g d i s p l a c e m e n t i n t e r p o l a t i o n f u n c t i o n s . U s i n g t h i s a p p r o a c h a c o m p a t i b l e e l e m e n t i s one w h i c h ha s a s u f f i c i e n t deg ree o f i n t e r - e l e m e n t c o n t i n u i t y , s uch t h a t t h e t o t a l p o t e n t i a l ene r g y o f t h e s y s t em b e i n g a n a l y s e d conve r ge s m o n o t o n i c a l l y t o a minimum as t h e s u b d i v i s i o n o f t h e domain o r mesh i s r e f i n e d . The deg ree o f c o m p a t a b i l i t y s u f f i c i e n t t o e n s u r e - t h i s convergence," \ " "g iven comp le tene s s , h a s been o b t a i n e d f r om conve r gence p r o o f s . F o r c o m p a t a b i l i t y , i f a dependent v a r i a b l e e n t e r s t h e ene r gy e x p r e s s i o n s w i t h t h e h i g h e s t d e r i v a t i v e o f o r d e r q (q > 0 ) , t h e n t h e q - 1 d e r i v a t i v e o f t h a t v a r i a b l e must be c o n t i n u o u s a c r o s s i n t e r - e l e m e n t b o u n d a r i e s [ 2 6 ] . F o r an e l ement t h a t s a t i s f i e s b o t h c o m p a t a b i l i t y and c o m p l e t e n e s s r e q u i r e m e n t s as o u t l i n e d h e r e , m o n o t o n i c c o n v e r g e n c e t o t h e c o r r e c t minimum p o t e n t i a l ene rgy i s a s s u r e d as the f i n i t e e l e m e n t mesh i s r e f i n e d . The re i s t h e r e f o r e a g r e a t advan tage i n u s i n g an e l ement o f t h i s t y p e , and an o r d e r o f a c c u r a c y a n a l y s i s i s a v a i l a b l e t o g i v e an e x p e c t e d conve r gence r a t e , a t l e a s t w i t h r e s p e c t t o p o t e n t i a l e n e r g y . McLay p r e s e n t s t he b a s i s o f s uch an a n a l y s i s b e g i n n i n g w i t h T a y l o r ' s t heo rem [27,]. 66 The use o f c omp le te b u t i n c o m p a t i b l e e l e m e n t s has been shown t o have many a d v a n t a g e s , s i n c e many o f t h e s e e l e m e n t s appea r t o p e r f o r m b e t t e r t h a n c o m p a t i b l e e l e m e n t s . U n t i l r e c e n t l y howeve r , t h e i r c onve r gence c o u l d n o t be e x p e c t e d , s i n c e no conve r gence p r o o f e x i s t e d . Now i t w o u l d appea r t h a t i f t h e e l emen t s a t i s f i e s t h e p a t c h t e s t , t h e n c onve r gence can be e x p e c t e d a l t h o u g h no o r d e r o f a c c u r a c y a n a l y s e s y e t e x i s t . S t r a n g and F i x s t a t e t h a t s u c c e s s f u l p e r f o r m a n c e o f an i n c o m p a t i b l e e l emen t i n t h e p a t c h t e s t i s b o t h n e c e s s a r y and s u f f i c i e n t f o r c o n v e r g e n c e [28]. The a p p l i c a t i o n o f t he f i n i t e e l emen t method has t h e r e f o r e , the p r i m a r y advantage o f r e d u c i n g t he p r o b l e m o f o b t a i n i n g a c o n t i n u o u s s o l u t i o n h a v i n g an i n f i n i t e number o f deg ree s o f f r eedom, t o t h a t o f an a p p r o x i m a t e s o l u t i o n e x p r e s s e d i n te rms o f a f i n i t e number o f deg ree s o f f r eedom. T h i s method p e r m i t s t h e a c q u i s i t i o n o f s o l u t i o n s , a l t h o u g h o n l y a p p r o x i m a t e , t o p r ob l ems t h a t a r e o t h e r w i s e i n t r a c t a b l e . 6.3 I n c r e m e n t a l V i r t u a l Work E q u a t i o n s I n c o r p o r a t i n g t he F i n i t e E l ement Method  6.3.1 The Assumed D i s p l a c e m e n t App roach The a p p l i c a t i o n o f t he f i n i t e e l emen t method t o t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n s d e v e l o p e d i n C h a p t e r 3, w i l l be a c c o m p l i s h e d by a d o p t i n g t he assumed d i s p l a c e m e n t a p p r o a c h . In t h i s app roach t he c o n t i n u u m i s f i r s t d i s c r e t i z e d i n t o a number o f f i n i t e e l e m e n t s . W i t h i n each e l emen t i n t e r p o l a t i o n f u n c t i o n s a r e p r e s c r i b e d w h i c h u n i q u e l y d e f i n e t h e g e n e r a l i z e d d i s p l a c e m e n t s w i t h i n t h e e l e m e n t i n t e rms o f t he g e n e r a l i z e d n o d a l d i s p l a c e m e n t s . T h i s i s e x p r e s s e d by 67 "iCp) = N . j (p) u? (6 .3 ) where u i ( p ) = g e n e r a l i z e d d i s p l a c e m e n t a t a p o i n t p i n t h e domain o f t h e f i n i t e e l emen t = n o d a l g e n e r a l i z e d d i s p l a c e m e n t s o f t he f i n i t e e l e m e n t . I\U (p) = i n t e r p o l a t i o n o r shape f u n c t i o n s w h i c h a r e f u n c t i o n s o f t he p o i n t p, f o r t h e e l e m e n t . A d o p t i n g m a t r i x n o t a t i o n , t h i s may be a l t e r n a t i v e l y w r i t t e n as {u} = [N(p) ] { u } 6 (6 .4 ) where ( u ) = v e c t o r o f g e n e r a l i z e d d i s p l a c e m e n t s a t p o i n t p. [N(p)] = m a t r i x o f i n t e r p o l a t i o n f u n c t i o n s f o r p o i n t p. {u} = v e c t o r o f n o d a l g e n e r a l i z e d d i s p l a c e m e n t s f o r t h e e l e m e n t 6 .3 .2 The I n c r e m e n t a l V i r t u a l Work E q u a t i o n s The i n c r e m e n t a l v i r t u a l work e q u a t i o n s d e v e l o p e d i n C h a p t e r 3 and g i v e n by e i t h e r E q . 3.36 o r 3 .38 , a r e e x a c t e x p r e s s i o n s f o r t h e i n c r e m e n t a l d e f o r m a t i o n o f a body . They a r e however n o n l i n e a r , and t h e r e f o r e t h e i n c r e m e n t a l e x p r e s s i o n s were l i n e a r i z e d i n o r d e r t o be a b l e t o employ a d i r e c t s o l u t i o n method . The r e s u l t i n g l i n e a r i z e d 68 e q u a t i o n s a r e g i v e n by Eq . 3.43 and 3.45, and i t i s t h e s e e q u a t i o n s t h a t w i l l be u s e d a l o n g w i t h t h e f i n i t e e l emen t p rocedure- . F i r s t , Eq . 3.43 w i l l be r e w r i t t e n i n m a t r i x f o rm and t h e n t h e f i n i t e e l ement a p p r o x i m a t i o n w i l l be i n t r o d u c e d i n g e n e r a l f o r t h e t h r e e -d i m e n s i o n a l p r o b l e m . Th roughou t t h i s c h a p t e r , i t w i l l be assumed t h a t t h e d e f o r m a t i o n i s f o r t h e i n c r e m e n t be tween two n e i g h b o u r i n g a r b i t r a r y c o n f i g u r a t i o n s , f r om c o n f i g u r a t i o n ^ t o c o n f i g u r a t i o n 2 C . T h i s i s c o n s i s t e n t w i t h t h e i n c r e m e n t o f d e f o r m a t i o n cho sen i n C h a p t e r 3. D e f i n e t h e i n c r e m e n t a l K i r c h h o f f s t r e s s v e c t o r a s { S } T = < S H S 1 2 S ] 3 S n S 2 2 S 2 3 S 3 1 S 3 2 S 3 3 > (6.5) and t h e i n c r e m e n t a l l i n e a r component o f G r e e n ' s s t r a i n v e c t o r as { 6 } T = * e l l 612 613 e21 e22 e23 e31 e32 e33 > ( 6 - 6 ) i S i m i l a r l y t h e i n c r e m e n t a l n o n l i n e a r component o f G r e e n ' s s t r a i n v e c t o r i s d e f i n e d as {n>T = < n n n 1 2 n1 3 n2 1 n2 2-n2 3 n3 1 n 3 2 n 3 3 > (6.7) Now t h e l i n e a r component o f G r e e n ' s s t r a i n v e c t o r f o r t h e i n c r e m e n t , {e} , can be s e p a r a t e d i n t o two v e c t o r s , one i n c l u d i n g t h e s t r i c t l y l i n e a r te rms ( e^ } , and t h e o t h e r r e p r e s e n t i n g t h e n o n l i n e a r e f f e c t s o f t h e i n i t i a l d i s p l a c e m e n t s f o r t h e i n c r e m e n t { e ^ } - B o t h {e^} and {e^j^} a r e l i n e a r f o r t h e i n c r e m e n t , b u t t h e f o r m e r i s i n d e p e n d e n t o f p r e v i o u s d e f o r m a t i o n , w h i l e t h e l a t t e r depends on t h e de fo rmed 69 c o n f i g u r a t i o n a t t he b e g i n n i n g o f t h e i n c r e m e n t . Thus {e} = { e L } + { e N L } (6.8) where b o t h {e^} and { e j ^ } a r o d e f i n e d i n an a h a l a g o u s manner t o {e} i n Eq . 6 .6 . The e l e m e n t s o f t he v e c t o r s {e T } and {e X T T } a r e g i v e n by L NL & 1 (e. . ) . = - k u . . + u . .) (6.9) 13 7 L 2*- 1,3 3 , 3 / y } and C e i j k = i ( l u k , i \,j + u k , i l u k j ) ( 6 - 1 0 ^ T h i s can be seen t o f o l l o w f rom t h e use o f Eq . 2.11 and Eq . 6 .8 . Now r e l a t i n g t h e s t r a i n v e c t o r s d e f i n e d above, t o t h e i n c r e m e n t a l t h r e e - d i m e n s i o n a l d i s p l a c e m e n t v e c t o r d e f i n e d as T {u} = < u i U 2 U 3 > (6.11) i t i s n e c e s s a r y t o d e f i n e two m a t r i x o p e r a t o r s [B^] and [ B ^ ( l u ) ] b y t h e r e l a t i o n s h i p s , {e L > = [ B L ] {u} (6.12) { e N L } = [ B N L ( 1 u ) ] / { u } ^ [ 6 A 5 ) The n o t a t i o n [ B ^ (M] s i g n i f i e s t h a t t h i s o p e r a t o r m a t r i x i s a f u n c t i o n o f t h e i n i t i a l d i s p l a c e m e n t s f o r t h e i n c r e m e n t . Bo th t h e s e o p e r a t o r s a r e g i v e n i n f u l l i n A p p e n d i x A , and a r e d e r i v e d f r o m Eq . 6.9 and 6.12 f o r [ B L ] and Eq . 6.10 and 6.13 f o r [ B ^ L ( x u ) ] . A r e l a t i o n s h i p be tween t h e s e two o p e r a t o r s i s a l s o exam ined , t o r e d u c e c o m p u t a t i o n a l e f f o r t , i n A p p e n d i x A. 70 F i n a l l y , by virtue of the decision to use a linear symmetric constitutive relationship as expressed i n Eq. 5.5, the constitutive equation in, matrix form i s {S} = [C] {e} (6.14) where [C] i s the constitutive matrix. Considering each in t e g r a l of Eq. 3.43 i n turn, rewrite the equation i n matrix form beginning with / S . v 6e., d°V ,= / {S}T {6e} d°V = / ( e } T [C-]T {Se} d°V / {<Se}T [C] {e} d°V (6.15) °V Using Eq. 6.14 and 6.8, and the relationships given i n Eq. 6.12 and 6.13 this can be written as { y S j k 6 6 j k d ° V " o y (6u} T [B L ] T [C] [BL] {u} d°V + Q/ {6u} T [ B ^ u ) ] 1 [C] [ B J {u} d°V + o y { 6 U } T [ B L ] T [ C ] 1 \ L ( 1 U ~ ) ] { U > D ° V + o v {6u} T [B N L ( l u ) ] T [C] [B N L ( 1 U ) ] {u} d°V (6.16) 71 Considering for the moment only one f i n i t e element dbmain\ then by using the expression i n Eq. 6.4, and substituting this into Eq. 6.16, / S &e d°V={6u} e T / [N] T [B ] T [C] [B.] [N] d°V {u} 6 O y J i>- J K O y ^ + {6u} G 1 f [N] {B^CM] [C] [BL] [N] d°V {u} 6 + {6u} e T on.NlT [ B J 1 [C|. lB^. 1 ;( 3rOI '[XI d°V {uf + {6u} e T / [N] T [ B ^ C 1 ^ ] 1 [C] [B^C 1 ^] [N] d ° V { u } e (6.17) where the righ t hand superscript e i s used to indicate elemental nodal displacements as introduced i n Eq. 6.4. The above equation can also be expressed i n the form o v ; S . k 6e. k d°V= {6u} e T [K L] {u} 6 + {6u} e T [ K ^ ] {u} e + {6 u ) e T [ K N L 2 ] {-u-}6 + {6u} e T [ K N L 3 ] {u} 6 (6.18) where [K L] = / [N] T [ B J 1 [C] [B L] [N] d°V (6.19) [ KNL1 ] = o ; v [ N ] T [ B N L ( l u ) ] T [ C ] [ B L ] [ N ] d ° V C 6 - 2 0 ) [ K N L 2 ] = / [N] T [ B T ] T [C] D W V ) ] [N] d°V (6.21) °V 72 = 0 { f M T ^ N L ^ ^ M &NL^ tN] d°V (6 .22) The symbo l = i s u s ed h e r e t o i n d i c a t e t h e a p p r o x i m a t i o n o f t he e x a c t v a r i a t i o n a l t e r m , o n l y i n t h e s en se t h a t t h e f i n i t e e l emen t method i n t r o d u c e s an a p p r o x i m a t i o n . I t s h o u l d be n o t e d t h a t [K^] g i v e n by Eq . 6.19 i s t h e u s u a l s m a l l s t r a i n , s m a l l d i s p l a c e m e n t s t i f f n e s s m a t r i x . A l s o i t i s w o r t h n o t i n g t h a t s i n c e [C] i s s y m m e t r i c . T h i s means t h a t o n l y one o f t h e s e two s t i f f n e s s m a t r i c e s need be e v a l u a t e d , and t h e n j u s t t r a n s p o s e d t o g i v e t h e o t h e r . S i n c e [K^] and [ K ^ ^ l a r e b o t h s y m m e t r i c t h e a d d i t i o n Of t h e f o u r m a t r i c e s w i l l g i v e a s y m m e t r i c m a t r i x . C o n s i d e r n e x t t he s e cond i n t e g r a l on t h e r i g h t hand s i d e o f Eq . 3 . 43 , The v e c t o r o f K i r c h h o f f s t r e s s e s f o r c o n f i g u r a t i o n 1 C , { 1 S } , c an be e v a l u a t e d u s i n g t he t o t a l G r e e n ' s s t r a i n v e c t o r f o r t he same c o n f i g u r a t i o n , { 1 e } . F r o m E q . 5 . 1 , a s sum ing t h e c o n s t i t u t i v e r e l a t i o n s h i p t o be c o n s t a n t t h r o u g h o u t d e f o r m a t i o n , i n m a t r i x f o rm [ K N L 1 ] = [ K N L 2 ] T ( 6 .23 ) fin., d°v = fOs}1 {5n> d°v (6 .24) {lS} = [C] {.M (6 .25) 73 where { l £ } T = < l £ l l l £ 1 2 l £ 1 3 S i S 2 S 3 S i S 2 S 3 > ( 6 - 2 6 ) and f r o m Eq . 2.6 1, e. . = -k1!!. , . + lu.,. + V ,. V , . ) ( 6 .27) i i 2 I j J i , k i k y v S e p a r a t i n g { x e } , i t can be s een t h a t Oe} = ( 1 e L } + Or)} (6 .28) where f 1 ^ } and { 1 n} a r e d e f i n e d a n a l o g o u s l y t o t h e i r i n c r e m e n t a l c o u n t e r p a r t s , o n l y i n s t e a d o f i n c r e m e n t a l d i s p l a c e m e n t s u^ , t h e t o t a l d i s p l a c e m e n t s f o r c o n f i g u r a t i o n lC, 1 u ^ , a r e u s e d . The e l e m e n t s o f { 1 T } } a r e g i v e n b y By c o m p a r i n g { J n } t o {e^.j } i t can be seen t h a t { l n > = \ [ B N L ( l u ) ] { l u } ( 6 - 3 0 ) and ^ e ^ } i s g i v e n by { 1 e L > = [ B L ] Ou] (6 .31) 74 T h e r e f o r e , Eq . (6.25) becomes { IS} = [C] [ B L ] { l u } + i [ C ] [ B N L ( l u ) ] ' { l u } (6 .32) and a p p l y i n g t h e f i n i t e e l e m e n t method { IS} = [C] [Bj ] [N] { H i } 6 + i [ C j [ B N L ( l u ) ] [N] { l u } 6 ( 6 .33) Thus know ing t he n o d a l d i s p l a c e m e n t s f o r a c o n f i g u r a t i o n "C o f t h e f i n i t e e l e m e n t m o d e l , t h e f u l l G r e e n ' s s t r a i n and c o r r e c t K i r c h h o f f s t r e s s v e c t o r s may be o b t a i n e d . T R e t u r n i n g t o Eq . 6 .24 , { ! S } i s now a v a i l a b l e t h r o u g h t h e u se o f Eq. 6 .33 b u t t h e v e c t o r {6n} 1 S n o t r e a d i l y c a l c u l a b l e . The p r o d u c t o f t he two v e c t o r s has t o be r e c o m b i n e d i n a more r e a d i l y u s e a b l e f o r m . To do t h i s , d e f i n e t h e v e c t o r o f m a t e r i a l d e r i v a t i v e s o f t h e i n c r e m e n t a l d i s p l a c e m e n t s as T {u. ,.} = < u. u , 0 u, _ u 0 , u 0 0 u„ _ u_ , u„ 0 u_ , > (6 .34) L l j J 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 and d e f i n e an o p e r a t o r m a t r i x [L] s uch t h a t •{u. .} = [L] {u} (6.35) 1} J The o p e r a t o r m a t r i x [L] i s g i v e n f u l l y i n A p p e n d i x A. Now a l s o d e f i n e a new m a t r i x composed o f t he e l e m e n t s o f { 1S} and c a l l e d [ 1 S T ] , where t h i s m a t r i x i s a r r a n g e d such t h a t 75 { i S } 1 {6n) = {Su. . } T [ i ST ] {u. .} i , J i , J (6 .36) The m a t r i x [1ST] i s a l s o g i v e n f u l l y i n A p p e n d i x A. Then i n s e r t i n g t h e r e s u l t o f Eq . 6.36 i n t o Eq . 6 .24 , and a l s o u s i n g Eq . 6.35 /S., 6 n - t d°V = / { 6 u } T [ L ] T pST] [L] {u} d°V O y J K J K ° y (6 .37) N e x t c o n s i d e r i n g j u s t one f i n i t e e l ement domain / S j k 6 n j k d°V = { S u } 6 o { [ N ] T [ L ] T [1ST] [L] [N] d°V {u} ( { S u } 6 [ K I S ] { u } 6 (6 .38) where [ K j g ] i s c a l l e d t h e i n i t i a l s t r e s s m a t r i x , and i s g i v e n by [ K I S ] = / [ N ] T [ L ] T [1ST] [L] [N] d°V °V (6 .39) T h i s m a t r i x i s c a l l e d t h e i n i t i a l s t r e s s m a t r i x , s i n c e i t r e p r e s e n t s t h e e f f e c t o f t h e s t r e s s e s p r e s e n t a t t h e b e g i n n i n g o f t h e i n c r e m e n t on t h e i n c r e m e n t a l d e f o r m a t i o n . No te a l s o t h a t [K^g] i s s y m m e t r i c . Now t h e r i g h t hand s i d e o f Eq . 3 . 43 , u s i n g t h e f i n i t e e l e m e n t method may b e g i v e n , f o r a s i n g l e e l e m e n t , as fS., Se . . d°V + / l o . ,o, r ^ r r i e > v Jk j k °v s j k 6 n j k d V = { 6 u } &i + [ W + [ K N L 2 ] + [ K N L 3 ] + ^ (6 .40) 76 S i n c e [K L1> [ K N L 3 - ' a n c l [Kis^ a r e a 1 1 s y m m e t r i c , and a d d i n g [ K N L 1 ] t o i t s t r a n s p o s e [ K N L 2 ] g i v e s a symmetr ic ; m a t r i x , t h e s t i f f n e s s m a t r i x r e s u l t a n t i s s y m m e t r i c . T h i s i s e x p e c t e d , s i n c e t he s t i f f n e s s m a t r i x f o r any l i n e a r i n c r e m e n t s h o u l d be s y m m e t r i c by B e t t i ' s t heo rem. C o n t i n u i n g w i t h t he l o a d i n g i n t e g r a l s g i v e n on t h e l e f t hand s i d e o f Eq. 3 . 43 , by d e f i n i n g an i n c r e m e n t a l body f o r c e v e c t o r {F} as { F } T = < F i F 2 . F 3 > (6 .41) t h e n .,/. °p F. fiu. d°V = / ° p ( F } T { f i u } d°V (6 .42) V °V s i m i l a r l y , c o n s i d e r i n g t he s u r f a c e t r a c t i o n i n t e g r a l , d e f i n e f i r s t an i n c r e m e n t a l s u r f a c e t r a c t i o n v e c t o r { T ^ } . where { T ( L ) } T = < T i ( L ) T 2 ( L ) T 3 ( L ) > ( 6 < 4 3 ) and t h e e l e m e n t s o f t h i s v e c t o r a r e g i v e n by T . ^ = T . . °v . (6 .44) 1 J i : Then t h e s u r f a c e t r a c t i o n i n t e g r a l becomes TT. . °v- fiu. d°S = / { T ( L ) } T {fiu} d°S (6 .45) 3 g J i 3 1 o s 77 I n t r o d u c i n g t h e f i n i t e e l emen t method o f a p p r o x i m a t i o n , t h e n t h e l e f t hand s i d e o f Eq . 3.43 i s r e p r e s e n t e d as •T ST.. °v . Su. d°S + f°p F. 6u. d°V = { 6 u } 6 / [ N ] T ' { T C L ) } d°S O g J l J 1 O y 1 1 O g T + { 6 u } 6 /°p [ N ] T {F} d°V (6 .46) °V by u s i n g Eq. 6.42 and 6 .45 , f o r a s i n g l e e l e m e n t . F i n a l l y t h e l i n e a r i z e d i n c r e m e n t a l v i r t u a l work e q u a t i o n i s w r i t t e n i n m a t r i x f o r m , u t i l i z i n g t h e f i n i t e e l ement method , by c o m b i n i n g the r e s u l t s o f Eq . 6.40 and 6.46 t o g i v e , f o r a s i n g l e e l e m e n t , T { 6 u } e T f [ N ] T { T ( L ) } d°S + { 6 u } 6 / ° p [ N ] T v { F } d°V °S °V T Uuf" |~ [KT] + [ K N L ] J { u } 6 (6 .47) where tV = [ K N L J + [ K N L 2 I + [ K NL3^ + r v f 6 - 4 8 ^ A s i m i l a r a p p r o a c h c o u l d have been a d o p t e d s t a r t i n g w i t h Eq . 3.45 w h i c h i s e s s e n t i a l l y t h e same e q u a t i o n e x c e p t f o r a s u r f a c e t r a c t i o n d e f i n e d i n d i f f e r e n t t e r m s . D e f i n i n g a d i f f e r e n t i n c r e m e n t a l s u r f a c e t r a c t i o n v e c t o r , t h a n t he one i n Eq. 6.43, as { T ( K ) } T = < ^ ( K ) T 2 ( K ) T 3 ( K ) ; > ( 6 _ 4 9 ) 78 where t h e e l e m e n t s o f t he v e c t o r a r e g i v e n by T. ( K ) = S.. °v. 1 J i J (6 .50) and a l s o d e f i n i n g S , l S , 2 \,3 ^u 2,1 2,2 2,3 ^u 3,1 3,2 u 3 , 3 t hen Eq . 3.45 can be w r i t t e n f o r a s i n g l e e l ement as (6.51) { 6 u ' } e T / [ N ] T { T ( K ) } d°S + { 6 u } e / [ N ] T [ i u . .] { T ( K ) } d ° S 1,3" + { 6 U } 6 I °p [ N ] T {F} d°V = {SuV (6 .52) The e q u a t i o n s g i v e n above f o r t h e i n c r e m e n t a l d e f o r m a t i o n be tween c o n f i g u r a t i o n s *C and 2 C , e m p l o y i n g t h e f i n i t e e l ement method , a r e w r i t t e n f o r a s i n g l e e l e m e n t . The e q u a t i o n s r e p r e s e n t i n g t he f i n i t e e l ement model o f t h e who l e body a r e d e r i v e d by c o m b i n i n g t h e e l e m e n t a l e q u a t i o n s . These e q u a t i o n s a r e comb ined by e q u a t i n g t h e g e n e r a l i z e d d i s p l a c e m e n t s and summing t h e g e n e r a l i z e d f o r c e s a t c o r r e s p o n d i n g node s . The r e s u l t i n g g l o b a l s y s t e m o f e q u a t i o n s i s t h e n s o l v e d f o r t h e i n c r e m e n t a l d e f o r m a t i o n o f t h e body . 79 6 . 3 . 3 R e s i d u a l L o a d i n g Term The r e s i d u a l l o a d i n g t e r m was d e v e l o p e d i n S e c t i o n 3.5 as a r e p r e s e n t a t i o n o f t h e n o n s a t i s f a c t i o n o f e q u i l i b r i u m . I n g e n e r a l t h e c o n f i g u r a t i o n o f a ' b o d y a t t h e end o f a l o a d i n c r e m e n t as f ound by u s i n g a l i n e a r i n c r e m e n t a l v i r t u a l work e q u a t i o n w i l l n o t e x a c t l y s a t i s f y e q u i l i b r i u m . The r e s i d u a l l o a d i n g te rm when known can t h e n b e employed w i t h any o f t h e s o l u t i o n p r o c e d u r e s shown i n C h a p t e r 4 t h a t come unde r t h e h e a d i n g o f i n c r e m e n t a l methods w i t h e q u i l i b r i u m c h e c k s . The r e s i d u a l •N l o a d i n g te rm i s g i v e n b y e i t h e r Eq . 3.46 o r Eq . 3 .47 , t h e c h o i c e o f w h i c h fo rm t o u s e b e i n g d e c i d e d b y t h e p a r t i c u l a r i n c r e m e n t a l v i r t u a l work e q u a t i o n e m p l o y e d . I f t h e p r o b l e m i s s uch t h a t a L a g r a n g i a n s u r f a c e t r a c t i o n v e c t o r i s f o u n d t o be e a s i e r t o u s e , t h e n t h e l i n e a r i n c r e m e n t a l v i r t u a l work e q u a t i o n w i l l be t h a t g i v e n i n Eq . 3 .43 . The accompany ing r e s i d u a l l o a d i n g t e r m i s t h e n g i v e n b y Eq . 3 .46. R e w r i t i n g t h i s e q u a t i o n f o r t h e r e s i d u a l l o a d i n g te rm i n m a t r i x f o r m , f o r c o n f i g u r a t i o n lC, and i n t r o d u c i n g t h e f i n i t e e l ement method f o r a s i n g l e e l e m e n t , T X X R e = { 6 u } 6 / [ N ] T { 1 T ( L ) } d°S +' { 6 u } e •"/ °p [ N ] T i1?} d°V c- o g o y - ( 6 U } C / [ N ] T [ L ] T C'S) d°V °V I T - { 6 u ) e / [ N ] T [ L ] T [ X ST] [L] [N] d°V {luf ^ ( 6 .53) °V In t h i s e q u a t i o n , ^ S } i s e v a l u a t e d t h r o u g h t h e u se o f E q . 6 . 3 3 . I t s h o u l d be n o t e d t h a t t h e l a s t i n t e g r a l t e rm on t h e r i g h t hand s i d e o f 80 Eq . 6.53 i s t h e i n i t i a l s t r e s s m a t r i x [ K j g ] f o r t h e n e x t i n c r e m e n t o f l o a d i n g o i n g f r o m c o n f i g u r a t i o n lC t o 2 C . Thus t h e e v a l u a t i o n o f t h e r e s i d u a l l o a d i n g te rm c o n t a i n s one t e r m t h a t w o u l d have t o be e v a l u a t e d f o r t h e n e x t l o a d i n c r e m e n t i n any c a s e . S i m i l a r l y , i f t he p r o b l e m i s such t h a t t h e s u r f a c e t r a c t i o n may be more e a s i l y e x p r e s s e d i n te rms o f a K i r c h h o f f s t r e s s v e c t o r , t h e n t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n i s g i v e n b y Eq . 3 . 45 . The c o r r e s p o n d i n g r e s i d u a l l o a d i n g t e r m i s t h e n g i v e n by Eq . 3 . 47 . P r o c e e d i n g as above T T !R 6 = { S u } 6 / [N] T " ' { l T ( K ) } d°S + {6uf ' / [ N ] T [1U. .] { i T ^ } d°S C o g O g 1 , J + i&uf / °p [ N ] T OF] d°V - {<5u}e $ [ N ] T [•E] T ' { 1 S} d°V °V °V T - { 6 u } 6 / [ N ] T [ L ] T [ i ST ] [L] [N] d°V { x u } e ( 6 .54) °V In Eq . 6.53 and 6 . 54 , t h e v a l u e s o f t h e t o t a l s t r e s s e s , d i s p l a c e m e n t s , s u r f a c e t r a c t i o n s ,and body f o r c e s d e r i v e d b y t h e i n c r e m e n t a l a n a l y s i s a r e u s e d . However , t h e v i r t u a l d i s p l a c e m e n t s a r e c o m p l e t e l y a r b i t r a r y , and t h e y need n o t be a s s o c i a t e d w i t h t h e c o n f i g u r a t i o n f o r w h i c h t he r e s i d u a l l o a d i n g te rm i s b e i n g e v a l u a t e d . I t i s n e c e s s a r y though t h a t t h e s p e c i f i e d d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s a r e n o t c h a n g i n g as t h e body deforms f o r t h i s t o be t r u e . 81 6.4 Two D i m e n s i o n a l A n a l y s i s 6.4.1 G e n e r a l , The a n a l y s i s o f t he g e n e r a l d e f o r m a t i o n o f a t h r e e d i m e n s i o n a l body i s a c o m p u t a t i o n a l l y e x p e n s i v e a p p r o a c h t o any g i v e n p r o b l e m . T h i s app roach may n o t be j u s t i f i a b l y n e c e s s a r y f o r a r a t h e r l a r g e c l a s s o f p r o b l e m s w h e r e , u s i n g c e r t a i n s i m p l i f y i n g a s s u m p t i o n s , t h e o r i g i n a l t h r e e d i m e n s i o n a l p r o b l e m may be r e d u c e d t o a two d i m e n s i o n a l one . The two b a s i c a s s u m p t i o n s t h a t a r e made a r e t h o s e o f a p l a n e s t r a i n c o n d i t i o n o r o f a p l a n e s t r e s s c o n d i t i o n . F o r p l a n e s t r a i n a n a l y s i s i t i s assumed t h a t t h e s t r a i n i n t h e o u t - o f - p l a n e d i m e n s i o n i s r e s t r a i n e d , and t h a t t h e d e f o r m a t i o n o f the body i s a f u n c t i o n o f p l a n a r c o o r d i n a t e s o n l y . On t h e o t h e r hand , f o r p l a n e s t r e s s a n a l y s i s i t i s t h e o u t - o f - p l a n e n o r m a l s t r e s s t h a t i s assumed t o be z e r o , o r a t l e a s t n e g l i g i b l e , and t h a t o n l y t he p l a n a r s h e a r and n o r m a l s t r e s s e s s p e c i f y the s t a t e o f s t r e s s . Bo th o f t h e s e s i m p l i f y i n g a s s umpt i on s l e a d t o t h e a n a l y s i s o f a s i g n i f i c a n t l y r e d u c e d p r o b l e m f rom t h a t o f a g e n e r a l t h r e e d i m e n s i o n a l p r o b l e m . Care must be t a k e n t o choose t h e p r o p e r a s s u m p t i o n c o r r e s p o n d i n g t o t he p r o b l e m b e i n g a n a l y s e d . A c t u a l p h y s i c a l p r o b l e m s w i l l i n g e n e r a l f a l l be tween t h e two ex t remes r e p r e s e n t e d by t h e s e a s s u m p t i o n s . In C h a p t e r 7 t he p l a n e s t r a i n a s s u m p t i o n w i l l be u s ed f o r a p l a t e h a v i n g an i n f i n i t e l e n g t h , and t h e p l a n e s t r e s s a s s u m p t i o n w i l l be u s e d f o r a c a n t i l e v e r beam. 82 These s i m p l i f y i n g a s s umpt i on s w i l l be shown more c l e a r l y i n t he f o l l o w i n g s e c t i o n s , and t h e f i n i t e e l emen t i n c r e m e n t a l v i r t u a l work e q u a t i o n s and r e s i d u a l l o a d i n g t e rm w i l l be s u i t a b l y m o d i f i e d f o r two d i m e n s i o n s . 6 .4 .2 P l a n e S t r a i n I n a p l a n e s t r a i n a n a l y s i s t h e a s s u m p t i o n made i s t h a t t h e o u t -o f - p l a n e s t r a i n i s r e s t r a i n e d o r p r e v e n t e d and t h a t t h e d e f o r m a t i o n o f t he body i s a f u n c t i o n o f p l a n a r c o o r d i n a t e s o n l y . T h e r e f o r e t a k i n g t h e o u t - o f - p l a n e d i m e n s i o n as b e i n g a s s o c i a t e d w i t h the s u b s c r i p t 3, t h e n °Vi = " u . 3 = 0 (6 .55) R e f e r r i n g t o Eq . 2.6 w h i c h d e f i n e s G r e e n ' s s t r a i n t e n s o r , t h i s g i v e s % , i = % , 3 = 0 C 6 - 5 6 ) and t h e n u s i n g Eq . 2 . 8 , w h i c h d e f i n e s the i n c r e m e n t a l G r e e n ' s s t r a i n , £ 3 , i = £ i , 3 = ° C 6 - 5 7 ) As a consequence o f t he p l a n e s t r a i n a s s u m p t i o n g i v e n i n Eq . 6 . 55 , and t h e d e f i n i t i o n o f an i n c r e m e n t a l q u a n t i t y t h e n (6 .58) 83 Now u s i n g Eq . 6.58 and t h e d e f i n i t i o n s o f e „ and n_„ , g i v e n by Eq . 2.11 and Eq . 2.12 r e s p e c t i v e l y , t h e n 6 i 3 = 6 3 i = * i 3 = ^ 3 i = ° ( 6 - 5 9 > S i n c e b o t h e . . and S . . a r e s y m m e t r i c t e n s o r s , i t becomes e c o n o m i c a l t o c a l c u l a t e o n l y one o f e 1 2 and e 2 1 , and a l s o o n l y one o f S 1 2 and S 2 1 . T h e r e f o r e , d e f i n e t h e v e c t o r {e"}, where t h i s v e c t o r i n p l a i n s t r a i n a n a l y s i s c o n t a i n s a l l t h e n o n - z e r o e ^ t e r m s , as T {e} = < e n e 2 2 2 e 1 2 > (6 .60) and t h e n d e f i n e t he v e c t o r {S} as ( S } T = < S n S 2 2 S12 > (6 .61) The v e c t o r ( S i c o n t a i n s a l l t h e s t r e s s components t h a t a r e m u l t i p l i e d by n o n - z e r o s t r a i n s i n t h e v i r t u a l work e x p r e s s i o n s . By d e f i n i n g { S i and (e"} as above , t h e v e c t o r p r o d u c t i s c o r r e c t f o r t h e i n t e g r a l t e rm i n t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n shown be l ow / o v ( S } T {Se} d°V = / o v { S } T {6e} d°V (6 .62) The s c a l a r f a c t o r o f two p r e c e e d i n g t h e t e r m e 1 2 i n Eq . 6.60 i s r e q u i r e d i n o r d e r t o c o r r e c t l y a c c o u n t f o r t h e t e r m e 2 1 , w h i c h i s n o t e v a l u a t e d t o s ave c o m p u t a t i o n a l e f f o r t . T h i s can be shown, by c o n s i d e r i n g a l l t he n o n - z e r o terms i n t h e v e c t o r p r o d u c t s o f t h e 84 i n t e g r a n d s o f Eq . 6.62 as f o l l o w s , { S } 1 {Se} = S 11 6 e l l + S 12 5 e 1 2 + S 21 5 e 2 1 + S 22 5 e 22 = S 1 : L Se-Q + S 2 2 S e 2 2 + S 1 2 ( 2 6 e i 2 ) *{S}T {<Se} (6 .63) On l y t h e s y m m e t r i c p r o p e r t i e s o f SV. and have been u s e d t o d e m o n s t r a t e t h i s r e s u l t . R e l a t i n g t he i n c r e m e n t a l s t r e s s v e c t o r {S} t o t he i n c r e m e n t a l s t r a i n v e c t o r {e"} i n a s i m i l a r manner t o Eq . 6 .14 , {S} = [ C ] {e} (6 .64) whe re , as b e f o r e , t he i n c r e m e n t a l s t r e s s e s a r e r e l a t e d t o o n l y t h e l i n e a r p o r t i o n o f t h e i n c r e m e n t a l G r e e n ' s s t r a i n t e n s o r . In t he r e s i d u a l l o a d i n g te rm howeve r , t h e s t r e s s e s w i l l be e v a l u a t e d u s i n g t h e c o m p l e t e G r e e n ' s s t r a i n t e n s o r . E m p l o y i n g H o o k e ' s law f o r an i s o t r o p i c m a t e r i a l , t h e n t h e c o n s t i t u t i v e m a t r i x [ C ] f o r a p l a n e s t r e s s a n a l y s i s i s g i v e n by t h e s y m m e t r i c m a t r i x , r 1 TTTV) 0 it} = E ( 1 - v ) — -L U J (1 + v ) (1 - 2v) (1 - v ) (6 .65) 1 - 2v 2 (1 - v) 85 As b e f o r e i n t h e t h r e e d i m e n s i o n a l a n a l y s i s , t h e v e c t o r {e} may be decomposed i n t o t h e sum o f two v e c t o r s {¥} = {eL} +' { i " N L } (6.66) where {e^} i s i n d e p e n d e n t o f p r e v i o u s d e f o r m a t i o n , and { e ^ } i s dependent on t h e c o n f i g u r a t i o n o f t h e body a t t h e b e g i n n i n g o f t h e p a r t i c u l a r i n c r e m e n t b e i n g c o n s i d e r e d . I n an a n a l o g o u s manner t o Eq . 6.12 and , E q . 6 . 1 3 , two m a t r i x o p e r a t o r s a r e d e f i n e d such t h a t {e L > = [ B L ] {u} " (6.67) { e N L } = [ B N L ( l u ) ] ( 6 - 6 8 ) where _ T {u} = < ul u 2 > (6 .69) The two m a t r i x o p e r a t o r s [ B L ] and [ B^ L (lu) ] a r e shown i n f u l l i n A p p e n d i x B. The v e c t o r o f K i r c h h o f f s t r e s s e s f o r c o n f i g u r a t i o n lC, d e n o t e d by OS} where t h i s v e c t o r i s d e f i n e d by { 1S} T = < 1Sn ^ 2 2 ^ 1 2 > (6 .70) 86 i s evaluated u s i n g ; the f u l l Green's s t r a in vector for the same configuration, {le"}, defined as 07}'1 = < 1eu lz22 2 1 e 1 ? > (6.71) They are re la ted by the same const i tu t ive matrix used i n Eq. 6.64. The f u l l Green's s t r a in vector i s evaluated by {1 £} = [ B L ] {lu} + i [ B N L ( xu) ] Ou} (6.72) and hence the Kirchhoff stress vector {^S} i s given by {IS} = [ C ] [ B L ] ' {lu} + j [ C ] [ B N L ( lu ) ] ' {lu} ' (6.73) where {lu} i s defined i n an analogous manner to {u"}. The development of the above two equations, p a r a l l e l s the development i n the three dimensional case, given i n Eq. 6.28 through Eq. 6.32, of the corresponding three dimensional vectors. Next, define i n a s i m i l a r manner to Eq. 6.34, the two dimensional vector of material derivat ives of the incremental displacements as — T {u. .} = < u • u u u > (6.74) 1 1,3 1,1 1,-2 2 ; 1 2,2 K J and then the matrix operator [ L ] , defined s i m i l a r l y to [L] i n Eq. 6.35, i s given such that 87 {u. .} = [ L J {u} (6.75) - 1- > J Now t h e - m a t r i x [^ST] i s a s s emb l ed f rom t h e e l e m e n t s o f { 1 S } i n o r d e r t h a t t he i n t e g r a n d p r o d u c t shown b e l o w , o f t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n , i s c o r r e c t l y e v a l u a t e d . Tha t i s { X S } T {6n} = ( Su . } T [I'ST] {u. .} (6.76) The m a t r i x o p e r a t o r [ L ] and t h e m a t r i x [ X ST ] d e f i n e d above a r e shown f u l l y i n A p p e n d i x B. F i n a l l y , a two d i m e n s i o n a l body f o r c e v e c t o r i s d e f i n e d as { a F } T = < a ¥ 1 a F 2 > v (6.77) and t h e s u r f a c e t r a c t i o n v e c t o r i s g i v e n by e i t h e r p T ( L ) } T = < a T x ( L ) a T 2 ( L ) > (6.78) o r | a T ( K ) T T . = . < a ? i ( K ) a^CK) > ( 6 7 9 ) d e p e n d i n g on t h e p a r t i c u l a r p r o b l e m b e i n g a n a l y z e d . The i n c r e m e n t a l fo rms o f t h e s e v e c t o r s a r e s i m i l a r l y d e f i n e d . The i n c r e m e n t a l v i r t u a l work e q u a t i o n and t h e r e s i d u a l l o a d i n g t e r m f o r two d i m e n s i o n a l p l a n e s t r a i n a n a l y s i s may now be w r i t t e n by s u b s t i t u t i n g t he two d i m e n s i o n a l m a t r i x o p e r a t o r s , m a t r i c e s , and v e c t o r s 88 defined i n this section for t h e i r corresponding three dimensional counterparts i n the equations given i n sections 6.3.2 and 6.3.3. Thus the incremental v i r t u a l work equations for plane s t r a i n analysis are given, for a single element, by adapting Eq. 6.47 as T T {6u} 6 / o s [ N ] T {T ( L ) } d°S + {6u} 6 / o v ° p [ N ] T { F } d°V {&uf | [ K L ] + [ K N L ] (6.80) where [ ] and [ ] are formulated i d e n t i c a l l y to [K ^ j J and [K^], only with the corresponding two dimensional matrix operators, matrices, and vectors substituted for th e i r three dimensional counterparts. By f i r s t defining the matrix [lu^ ^ ] as [lu. .] 1 , 1 1 , 2 l u l u 2 , 1 2 , 2 (6.81) then Eq. 6.52, can be s i m i l a r l y adapted to two dimensional plane s t r a i n as T T {Su} 6 / o „ [ N ] T {T ( K )} d°S'+ {Su} 6 / 0 c ; [ N ] T [lu. .]{T ( K )} d°S +>{6-u-}e / o v [ N ] T {F} d°V uf [ [ K J + [ K N L ] r — T e { U } (6.82) 89 The e x p r e s s i o n f o r t h e r e s i d u a l l o a d i n g te rms a r e s i m i l a r l y a d a p t e d f rom Eq . 6.53 and 6 .54 , and a r e n o t r e p r o d u c e d h e r e . 6 . 4 . 3 P l a n e S t r e s s I n p l a n e s t r e s s a n a l y s i s i t i s assumed t h a t t h e o u t - o f - p l a n e n o r m a l s t r e s s i s z e r o , o r a t l e a s t n e g l i g i b l e , and t h a t t h e s t a t e o f s t r e s s i s c o m p l e t e l y s p e c i f i e d by t h e two p l a n a r no rma l s t r e s s e s and t h e i n - p l a n e s h e a r s t r e s s . The s t a t e o f s t r e s s i s assumed n o t t o be a f u n c t i o n o f t h e o u t - o f - p l a n e d i m e n s i o n . T h e r e f o r e t a k i n g t h e o u t - o f -p l a n e d i m e n s i o n as b e i n g a s s o c i a t e d w i t h t h e s u b s c r i p t 3 , t h e n a S i 3 = % . = 0 (6.83) and as a consequence S i 3 = S 3 . = 0 (6 .84) Now t h e v e c t o r s { S } and {^S} as d e f i n e d i n t h e p r e v i o u s s e c t i o n c o n t a i n a l l t he n o n - z e r o i n c r e m e n t a l K i r c h h o f f s t r e s s e s and t o t a l K i r c h h o f f s t r e s s e s f o r c o n f i g u r a t i o n ^ C , r e s p e c t i v e l y . A l t h o u g h t h e r e a r e o n l y t h r e e e l emen t s i n t h e s e v e c t o r s , and f o u r n o n - z e r o s t r e s s e s , a l l t h e s t r e s s e s a r e r e p r e s e n t e d s i n c e S j 2 and S 2 1 a r e e q u a l by symmetry . I t i s t h e r e f o r e e c o n o m i c a l t o r e t a i n o n l y one o f them and a c c o u n t f o r t h e o t h e r ' s i n f l u e n c e by t he i n c l u s i o n o f an a p p r o p r i a t e s c a l a r f a c t o r . The v e c t o r { e } as d e f i n e d i n t h e p r e v i o u s s e c t i o n c o n t a i n s a l l t he s t r a i n components t h a t a r e m u l t i p l i e d by n o n - z e r o s t r e s s components 90 i n the formation of the v i r t u a l work expression. By re t a i n i n g the fac t o r of two preceeding e ^ , the vector product of the i n t e g r a l term as shown below, i s c o r r e c t l y evaluated. / (S} T {<Se} d°V = f {S}T {6e} d°V (6.85) °V °V By using the same stress and s t r a i n vectors as for plane s t r a i n a nalysis, plane stress analysis i s then i d e n t i c a l with the exception that a d i f f e r e n t c o n s t i t u t i v e matrix i s used. Relating incremental Kirchhoff stresses to the incremental l i n e a r p o r t i o n of Green's s t r a i n analogously to Eq. 6.64 i n the plane s t r a i n a n a l y s i s , then {S} = [ E ] { e } 6.86 where [E ] i s the plane stress c o n s t i t u t i v e matrix. U t i l i z i n g Hooke's law f or an i s o t r o p i c material, the c o n s t i t u t i v e matrix i s given by (6.87) where E i s Young's modulus and v i s Poisson's r a t i o . By using the same d e f i n i t i o n of stress and s t r a i n vectors as used i n plane s t r a i n a n a l y s i s , the plane stress analysis uses the same equations as derived i n the previous section, with the exception that the [ E ] (1 -^2) 1 v 0 v 1 0 0 0 (1 - v) 2. 9 1 c o n s t i t u t i v e m a t r i x [ E ] i s s u b s t i t u t e d f o r [ C ] . Thus a compute r p rog ram can be u s e d e a s i l y f o r b o t h p l a n e s t r e s s and p l a n e s t r a i n a n a l y s i s by s i m p l y i n p u t t i n g o r c h o o s i n g t he a p p r o p r i a t e c o n s t i t u t i v e m a t r i x . 6.5 E i g h t Degree o f Freedom R e c t a n g u l a r F i n i t e E l emen t s 6 .5 .1 The Assumed D i s p l a c e m e n t F i e l d The b a s i c e i g h t deg ree o f f reedom r e c t a n g u l a r f i n i t e e l e m e n t , f rom w h i c h t he two d i f f e r e n t n u m e r i c a l l y i n t e g r a t e d e l e m e n t s w i l l be d e r i v e d , i s shown i n F i g . 9 . The e lement i s d e r i v e d i n te rms o f a l o c a l c o o r d i n a t e s y s t e m ( £ , n ) w i t h t h e o r i g i n l o c a t e d a t t h e c e n t r o i d o f t h e e l e m e n t . d i s p l a c e m e n t s u^ and u 2 a r e g i v e n i n t e r m s o f g e n e r a l i z e d d i s p l a c e m e n t c o e f f i c i e n t s a . , as shown b e l o w , A b i l i n e a r d i s p l a c e m e n t f i e l d i s cho sen where t h e g e n e r a l i z e d I u 2 = a 5 + a 6 c l + a 7 n + ctg^n (6 .88) o r i n m a t r i x f o rm H n 0 0 0 0 {u} = (6 .89) 0 0 0 0 l 5 n 5n where 92 EIGHT DEGREE OF FREEDOM RECTANGULAR FINITE ELEMENT FIG. 9 93 T {cu} = < a 2 a 3 a g a g a ? a 8 > (6 .90) In u t i l i z i n g t h e f i n i t e e l emen t method however , i t i s n e c e s s a r y t o have t h e g e n e r a l i z e d d i s p l a c e m e n t s {u} r e l a t e d t o t h e n o d a l g e n e r a l i z e d — e d i s p l a c e m e n t s {u} , n o t t h e c o e f f i c i e n t s {a^} as i n Eq . 6 . 89 . F i r s t d e f i n e t h e v e c t o r o f e l ement n o d a l d i s p l a c e m e n t s as - e T 1 1 2 2 3 3 i + i + {u} = < Uj u 2 u x u 2 Uj u 2 u : u 2 > (6 .91) where t h e r i g h t hand s u p e r s c r i p t i s now u sed t o i n d i c a t e t h e node number. The e l e m e n t s o f {u} and t h e i r p o s i t i v e d i r e c t i o n a r e as shown i n F i g . 9 . — e Now {u} c an be e x p r e s s e d i n te rms o f {cu } by a t r a n s f o r m a t i o n m a t r i x [T] s uch t h a t { u } 6 = [T] { a i } (6 .92) t h e n o b t a i n i n g t he i n v e r s e o f [ T ] , {a±} = [ T " 1 ] { u } e (6 .93) S u b s t i t u t i n g f o r {cu} i n E q . 6 . 89 , and c a r r y i n g ou t t h e r e q u i r e d m a t r i x m u l t i p l i c a t i o n , t h e r e s u l t i s {u} = [N] { u } e (6 .94) where 94 L N.] N 0 N 0 N 0 N 0. 1 2 3 ^ • 0 Nj 0 N 2 0 N 3 0 N 4 (6.95) and t h e i n t e r p o l a t i o n f u n c t i o n s , N^, a r e g i v e n by (b - g) (c - n) 4b c = (b + g) (c + n) 4b c (b + g) (c - n) 4b c (6 .96) (b - g) (c + n) 4b c T h i s i s t he Me l o sh r e c t a n g l e , and t h e i n t e r p o l a t i o n f u n c t i o n s g i v e n i n Eq . 6.96 r e p r e s e n t t h e a p p l i c a t i o n o f L a g r a n g e ' s i n t e r p o l a t i o n f o r m u l a f o r two d i m e n s i o n s [29].. The e l ement as f o r m u l a t e d s a t i s f i e s b o t h c o m p l e t e n e s s and c o m p a t i b i l i t y r e q u i r e m e n t s f o r a two d i m e n s i o n a l a n a l y s i s o f e i t h e r p l a n e s t r a i n o r p l a n e s t r e s s . I n e v a l u a t i n g t h e e l e m e n t a l s t i f f n e s s m a t r i c e s , l o a d v e c t o r s , and r e s i d u a l l o a d i n g v e c t o r s t he v i r t u a l work i n t e g r a n d s a r e f u n c t i o n s o f t he l o c a l c o o r d i n a t e s g and n. 1 ° e v a l u a t e t h e i n t e g r a l s r e q u i r e d f o r t h e f i n i t e e l emen t a n a l y s i s , two n u m e r i c a l i n t e g r a t i o n schemes a r e p r o p o s e d and e v a l u a t e d i n t h e f o l l o w i n g s e c t i o n s . The f i r s t i n t e g r a t i o n scheme u se s G a u s s i a n q u a d r a t u r e t o e v a l u a t e t h e i n t e g r a l s . The second scheme p r o p o s e d i s a n o n u n i f o r m o r l o w e r o r d e r i n t e g r a t i o n , w h i c h i s u s ed i n an a t t e m p t t o imp rove t h e r e s u l t s o b t a i n e d f o r a g i v e n number o f e l e m e n t s . 95 6 .5 .2 R e c t a n g u l a r F i n i t e E l ement U s i n g G a u s s i a n Q u a d r a t u r e The i n t e g r a t i o n o f t he e x p r e s s i o n s r e q u i r e d i n t h e v i r t u a l work e q u a t i o n s , may be a c c o m p l i s h e d t h r o u g h the use o f v a r i o u s n u m e r i c a l i n t e g r a t i o n schemes. O f t h e s e , G a u s s i a n q u a d r a t u r e r e q u i r e s t h e l e a s t number o f i n t e g r a t i o n p o i n t s t o c o n s t r u c t and e x a c t l y i n t e g r a t e a p o l y n o m i a l o f a g i v e n o r d e r . The e v a l u a t i o n o f t he i n t e g r a n d a t each i n t e g r a t i o n p o i n t , when u s i n g t he f i n i t e e l emen t method , i n v o l v e s a c o n s i d e r a b l e amount o f complex c a l c u l a t i o n s . T h e r e f o r e , t h e f e w e r i n t e g r a t i o n p o i n t s w h i c h a r e r e q u i r e d i n u s i n g G a u s s i a n g u a d r a t u r e , makes t h i s method advan tageou s t o u s e . G a u s s i a n q u a d r a t u r e i s p e r f o r m e d by e v a l u a t i n g t h e i n t e g r a n d a t each i n t e g r a t i o n p o i n t , m u l t i p l y i n g t h i s by a w e i g h t i n g v a l u e and t h e n a d d i n g t he r e s u l t t o t h e r e s u l t s f r om the o t h e r i n t e g r a t i o n p o i n t s . C o n s i d e r i n g f i r s t o n l y one d i m e n s i o n , u s i n g n i n t e g r a t i o n p o i n t s a p o l y n o m i a l o f deg ree 2n - 1 may be c o n s t r u c t e d and e x a c t l y i n t e g r a t e d u s i n g G a u s s i a n q u a d r a t u r e . The a c c u r a c y o f any n u m e r i c a l t e c h n i q u e i s dependent on how w e l l t he s u b s t i t u t e d p o l y n o m i a l f i t s t he r e a l i n t e g r a n d . In t h i s ca se i f t he r e a l i n t e g r a n d i s o f e q u a l o r l o w e r o r d e r t h a n 2n - 1, t h e n i t w i l l be e x a c t l y i n t e g r a t e d , i f n o t t h e n t h e e r r o r i s on t h e o r d e r 2n o f ( A ) where A i s t h e s p a c i n g o f t he i n t e g r a t i o n p o i n t s . [30, 31] The i n t e g r a l s r e q u i r e d f o r t h e v i r t u a l work e q u a t i o n a r e e i t h e r s u r f a c e o r vo lume i n t e g r a l s . F i r s t c o n s i d e r i n g t h e vo lume i n t e g r a l s , w h i c h i n c l u d e t he l i n e a r and n o n l i n e a r s t i f f n e s s m a t r i c e s and t h e body f o r c e i n t e g r a l , t h e y a r e o f t he fo rm I = / f ( £ , n) d°V °V (6 .97) 96 T a k i n g t h e t h i c k n e s s o f each e l ement a s b e i n g c o n s t a n t , s ay t , t h e n c b I = t f f f ( £ , n) d K dn (6 .98) - c -b F o r t h e b i l i n e a r e i g h t deg ree o f f reedom r e c t a n g l e o u t l i n e d i n t h e p r e v i o u s s e c t i o n , t h e i n t e g r a n d w i l l be composed o f te rms whose h i g h e s t o r d e r i s q u a d r a t i c . S i n c e one i n t e g r a t i o n p o i n t w i l l e x a c t l y i n t e g r a t e a l i n e a r i n t e g r a n d , and two i n t e g r a t i o n p o i n t s i n each c o o r d i n a t e d i r e c t i o n w i l l e x a c t l y i n t e g r a t e a c u b i c p o l y n o m i a l i n t e g r a n d ; two i n t e g r a t i o n p o i n t s a r e r e q u i r e d i n each d i r e c t i o n . The i n t e g r a l i n Eq . 6 .98 i s e v a l u a t e d u s i n g G a u s s i a n q u a d r a t u r e by 2 2 I = ( t b c) -l J. H H f ( £ n . ) ( 6 .99) i = l j = l J x J where t h e i n t e g r a t i o n p o i n t s ( g^ , r i j ) and t h e w e i g h t s 1L o r hL , a r e g i v e n by q = ( - 1 ) 1 (b) r h = ( - 1 ) J (c ) Hi := "H, = 1.0 r 1 /3 / J | (6 .100) The s u r f a c e i n t e g r a l s a r e a l s o n u m e r i c a l l y i n t e g r a t e d u s i n g G a u s s i a n q u a d r a t u r e . The i n t e g r a l s o v e r t he s u r f a c e a r e o f t h e f o rm I = / q ( 5 , n ) d°S (6.101) °S 97 A g a i n a s suming c o n s t a n t e l emen t t h i c k n e s s t , and a l s o t h a t s u r f a c e t r a c t i o n s a r e a p p l i e d o n l y t o t he b o u n d a r i e s o f t he e l emen t t h e n t h e i n t e g r a l i n Eq. 6.101 can be w r i t t e n f o r t h e b i l i n e a r r e c t a n g u l a r f i n i t e e l e m e n t as b - c -b I = t / g(g, c) dg + t / g(b, n) d n + t / gCg, - c ) dg -b c b c + t / g ( - b , n) d n (6 .102) - c I f t he i n t e g r a n d g ( ? , n) i s l i n e a r w i t h i n a p a r t i c u l a r i n t e g r a l o f Eq . 6.102 t h e n o n l y one Gauss i n t e g r a t i o n p o i n t i s r e q u i r e d a t m i d - s i d e ; i f t he i n t e g r a n d i s e i t h e r q u a d r a t i c o r c u b i c t h e n two Gauss i n t e g r a t i o n p o i n t s a r e r e q u i r e d on t h e p a r t i c u l a r edge o f the e l e m e n t . C o n s i d e r i n g j u s t one edge o f t h e e l emen t t o have a n o n - z e r o s u r f a c e t r a c t i o n , s a y t h e edge where n = c and -b - £ * b, t h e n E q . 6.102 r e d u c e s t o b t / g(5, c) d? (6 .103) -b I f t he i n t e g r a n d i s l i n e a r t h e n G a u s s i a n q u a d r a t u r e g i v e s I = t ( 2 b ) g ( 0 , c) (6 .104) and i f t h e i n t e g r a n d i s q u a d r a t i c o r l i n e a r t h e n 2 I = t b I H. gU c) (6 .105) i = l 98 where £^ and 1L a r e as g i v e n i n E q . 6 .100 , When e m p l o y i n g t he L a g r a n g i a n s u r f a c e t r a c t i o n i n t e g r a l as i n Eq. 6 . 80 , s i n c e t h e m a t r i x o f i n t e r p o l a t i o n f u n c t i o n s [ N ] v a r i e s l i n e a r l y a l o n g an edge, a one p o i n t G a u s s i a n q u a d r a t u r e r u l e w i l l be r e q u i r e d i f t he s u r f a c e t r a c t i o n { T ^ } i s c o n s t a n t a l o n g t h a t edge, and two i n t e g r a t i o n p o i n t s w i l l be r e q u i r e d i f { T ^ } i s l i n e a r o r q u a d r a t i c a l o n g t h a t edge . The same c r i t e r i a a p p l y t o t he f i r s t s u r f a c e t r a c t i o n i n t e g r a l u s i n g t h e K i r c h h o f f s t r e s s t e n s o r i n Eq . 6 . 82 , as f o r t h e L a g r a n g i a n i n t e g r a l d e s c r i b e d above . The s e cond s u r f a c e t r a c t i o n i n t e g r a l i n t h i s e q u a t i o n howeve r , c o n t a i n s t he p r o d u c t o f [N ] w h i c h i s l i n e a r on an edge , and t h e m a t r i x [ x u . .] w h i c h i s a l s o l i n e a r a l o n g an edge. The i n t e g r a n d i s o b v i o u s l y t h e n q u a d r a t i c when a c o n s t a n t s u r f a c e t r a c t i o n a l o n g t he edge i s s p e c i f i e d , and c u b i c when a l i n e a r l y v a r y i n g s u r f a c e t r a c t i o n i s u s e d . B o t h o f t h e s e ca se s r e q u i r e t h e u se o f a two p o i n t G a u s s i a n q u a d r a t u r e r u l e . F o r a p a r a b o l i c a l l y c h a n g i n g s u r f a c e t r a c t i o n a t h r e e p o i n t G a u s s i a n q u a d r a t u r e must be u s e d . U s i n g two i n t e g r a t i o n p o i n t s i n each c o o r d i n a t e d i r e c t i o n i n a G a u s s i a n q u a d r a t u r e scheme, t h e l i n e a r s t i f f n e s s m a t r i x f o r an e l ement was e v a l u a t e d . T h i s n u m e r i c a l l y o b t a i n e d s t i f f n e s s m a t r i x was compared t o t he s t i f f n e s s m a t r i x f o r t h e same e l ement o b t a i n e d f rom a c l o s e d - f o r m i n t e g r a t i o n o f t he r e q u i r e d i n t e g r a l . The two m a t r i c e s were f o u n d t o be i d e n t i c a l t o a t l e a s t t e n s i g n i f i c a n t f i g u r e s f o r a l l e n t r i e s i n t h e m a t r i c e s . The v a l u e o f t e n s i g n i f i c a n t f i g u r e s was t he a r b i t r a r y l i m i t o f ag reement t h a t was c h e c k e d , and s h o u l d n o t be c o n s t r u e d as t he a c t u a l l i m i t o f agreement o f t he n u m e r i c a l l y i n t e g r a t e d scheme r e s u l t s t o t h e c l o s e d f o rm r e s u l t s . 99 The n u m e r i c a l l y i n t e g r a t e d e l e m e n t a l s t i f f n e s s m a t r i x was c h e c k e d i n an e i g e n v a l u e r o u t i n e where t h r e e z e r o e i g e n v a l u e s were f o u n d , c o r r e s p o n d i n g t o t h r e e z e r o ene r gy d e f o r m a t i o n modes, o r r i g i d body modes as r e q u i r e d . T h i s e l e m e n t a l s o s u c c e s s f u l l y p a s s e d t h e p a t c h t e s t . 6 . 5 . 3 R e c t a n g u l a r F i n i t e E l emen t U s i n g N o n u n i f o r m N u m e r i c a l I n t e g r a t i o n The m o t i v a t i o n t o d e v e l o p an a l t e r n a t e i n t e g r a t i o n t e c h n i q u e t o u t i l i z e i n p l a c e o f G a u s s i a n q u a d r a t u r e , a r i s e s f r om the p o o r p e r f o r m a n c e o f t h e r e c t a n g u l a r e l e m e n t i n p u r e b e n d i n g when t h e s t i f f n e s s m a t r i x i s i n t e g r a t e d u s i n g G a u s s i a n q u a d r a t u r e . Cook [ 3 2 ] , i n d i s c u s s i n g t he p o o r p e r f o r m a n c e o f l i n e a r i s o p a r a m e t r i c e l e m e n t s , o f w h i c h t he r e c t a n g u l a r b i l i n e a r e l ement i s a s p e c i a l c a s e , shows t h a t t h e p r o b l e m a r i s e s f r om what he te rms " p a r a s i t i c " s h e a r a t t he Gauss i n t e g r a t i o n p o i n t s . A p p l y i n g n o d a l d i s p l a c e m e n t s c o r r e s p o n d i n g t o p u r e b e n d i n g , s h e a r s t r a i n s s h o u l d be z e r o t h r o u g h o u t t h e e l e m e n t . F o r t h e r e c t a n g u l a r f i n i t e e l e m e n t o f S e c t i o n 6 . 5 . 1 , s h e a r s t r a i n s w i l l however , e x i s t e ve r ywhe re w i t h i n t h e e l e m e n t e x c e p t a t t h e c e n t r o i d o f t he e l ement ( £ = 0 , n=0). T h i s " p a r a s i t i c " s h e a r e x i s t s t h e n a t t h e Gauss i n t e g r a t i o n p o i n t s o f any q u a d r a t u r e r u l e , o t h e r t h a n a o n e - p o i n t r u l e , and makes t he n u m e r i c a l l y i n t e g r a t e d e l ement t oo s t i f f i n p u r e b e n d i n g b e c a u s e t h e d e f o r m a t i o n p a t t e r n f o r b e n d i n g r e q u i r e s t h e s t o r a g e o f s h e a r s t r a i n ene r gy as w e l l as no rma l s t r a i n e n e r g y . T h e r e f o r e , t o overcome t h i s p r o b l e m o f " p a r a s i t i c " s h e a r a t t h e Gauss i n t e g r a t i o n p o i n t s , a n o n u n i f o r m i n t e g r a t i o n scheme i s p r o p o s e d . In t h i s n o n u n i f o r m i n t e g r a t i o n a l l terms u s e d i n t h e f o r m a t i o n o f t h e s t i f f n e s s m a t r i c e s t h a t a r e a s s o c i a t e d w i t h s h e a r s t r a i n , a r e e v a l u a t e d 100 a t t he c e n t r o i d o f t h e e l e m e n t . A l l o t h e r te rms a r e e v a l u a t e d a t t h e u s u a l Gauss i n t e g r a t i o n p o i n t s . S u r f a c e i n t e g r a l s a r e i n t e g r a t e d as i n t h e p r e v i o u s s e c t i o n . The e l emen t l i n e a r s t i f f n e s s m a t r i x as f o r m u l a t e d u s i n g t h i s n o n u n i f o r m i n t e g r a t i o n scheme, was c hecked i n an e i g e n v a l u e r o u t i n e where t h r e e z e r o e i g e n v a l u e s were f o u n d c o r r e s p o n d i n g t o t he r e q u i r e d r i g i d body modes. T h i s e l emen t a l s o s u c c e s s f u l l y p a s s e d t h e p a t c h t e s t . The r e c t a n g u l a r f i n i t e e l e m e n t , when i n t e g r a t e d u s i n g t h i s n o n u n i f o r m i n t e g r a t i o n may be t h o u g h t o f as an e l ement w h i c h i s c a p a b l e o f o n l y r e p r e s e n t i n g c o n s t a n t s h e a r s t r a i n t h r o u g h o u t t h e e l e m e n t doma in . Bu t i n t h e l i m i t , as t h e mesh i s s u c c e s s i v e l y r e f i n e d , t h e a c t u a l r e q u i r e d s h e a r s t r a i n d i s t r i b u t i o n w i t h i n an e l e m e n t domain w i l l a p p r o a c h a c o n s t a n t v a l u e as i s r e p r e s e n t e d by t h i s e l ement u s i n g n o n u n i f o r m i n t e g r a t i o n . 6.5.4 P e r f o r m a n c e Compa r i s on o f t h e Two R e c t a n g u l a r F i n i t e E l e m e n t s The two n u m e r i c a l i n t e g r a t i o n schemes p r o p o s e d i n t h e p r e c e e d i n g s e c t i o n s g i v e r e c t a n g u l a r f i n i t e e l e m e n t s t h a t have t h e r e q u i r e d r i g i d body modes, and t h a t a l s o s u c c e s s f u l l y p a s s t h e p a t c h t e s t . Conve rgence t o t h e c o r r e c t s o l u t i o n u s i n g e i t h e r o f t h e s e two f i n i t e e l e m e n t s i s t h e r e f o r e a s s u r e d , s i n c e b o t h a l s o s a t i s f y t h e c o m p l e t e n e s s r e q u i r e m e n t s . I n c h o o s i n g between t h e s e two e l e m e n t s t h e r e f o r e , t h e c r i t e r i a s h o u l d be t h e deg ree o f s o l u t i o n a c c u r a c y t h a t i s o b t a i n e d f o r g i v e n c o m p u t a t i o n a l e f f o r t . That i s , t h e more d e s i r a b l e o f t h e two f i n i t e e l e m e n t s wou l d be t h e one t h a t g i v e s t h e c l o s e s t a p p r o x i m a t i o n t o t h e e x a c t s o l u t i o n when e a c h i s u s ed w i t h t h e same deg ree o f mesh r e f i n e m e n t on a g i v e n p r o b l e m . 101 The c o m p u t a t i o n a l e f f o r t r e q u i r e d i n o r d e r t o o b t a i n t h e f i n i t e e l emen t s o l u t i o n t o a g i v e n p r o b l e m may be decomposed i n t o two p a r t s . The f i r s t p a r t i s t h e c o m p u t a t i o n a l expense r e q u i r e d t o f o r m t h e r e q u i r e d e l e m e n t a l s t i f f n e s s m a t r i c e s . The f i n i t e e l e m e n t d e v e l o p e d u s i n g t h e n o n u n i f o r m i n t e g r a t i o n scheme r e q u i r e d s l i g h t l y l e s s c o m p u t a t i o n t h a n t h e G a u s s i a n i n t e g r a t e d e l emen t i n t h i s r e g a r d , s i n c e te rms a s s o c i a t e d w i t h s h e a r s t r a i n s a r e e v a l u a t e d o n l y once a t t h e c e n t r o i d o f t h e e l e m e n t w i t h n o n u n i f o r m i n t e g r a t i o n , and n o t a t e a c h o f t h e f o u r Gauss p o i n t s s e p a r a t e l y . The s a v i n g i n o v e r a l l c o m p u t a t i o n a l expense i s m i n i m a l howeve r , s i n c e t h e second p a r t o f t h e p r o c e s s o f o b t a i n i n g t h e f i n i t e e l emen t s o l u t i o n , t h a t o f a s s e m b l i n g t h e g l o b a l s t i f f n e s s m a t r i x and s o l v i n g t h e g l o b a l s y s t e m o f e q u a t i o n s , r e p r e s e n t s by f a r t h e g r e a t e s t p o r t i o n o f c o m p u t a t i o n r e q u i r e d . T h i s second p a r t o f t h e c o m p u t a t i o n a l expense i s a f u n c t i o n o n l y o f t h e mesh geometry and t h e t o t a l number o f d e g r e e s o f f r e e d o m , s i n c e b o t h o f t h e f i n i t e e l e m e n t s have t h e same e l e m e n t a l geomet ry and d e g r e e s o f f r e e d o m . Thus t h e two e l e m e n t s may be assusmed t o r e q u i r e e s s e n t i a l l y t h e same c o m p u t a t i o n a l e f f o r t f o r a s p e c i f i e d f i n i t e e l ement mesh i n a g i v e n p r o b l e m . Compa r i s on o f t h e p e r f o r m a n c e o f t h e two e l e m e n t s s h o u l d t h e r e f o r e be on a b a s i s o f t h e a c c u r a c y o f t h e s o l u t i o n s o b t a i n e d w i t h each e l emen t u s i n g t h e same mesh a r r a n g e m e n t , when compared t o t h e e x a c t s o l u t i o n f o r t h e p r o b l e m b e i n g c o n s i d e r e d . The p e r f o r m a n c e o f t h e two f i n i t e e l e m e n t s was compared i n t h e l i n e a r a n a l y s i s o f two d i f f e r e n t p r o b l e m s ; a c a n t i l e v e r w h i c h ha s a p a r a b o l i c a l l y v a r y i n g end s h e a r , and an i n f i n i t e p l a t e s t r i p s u b j e c t e d t o a u n i f o r m p r e s s u r e . These two t e s t examples were s e l e c t e d becau se t h e y a r e s i m i l a r t o t h e p r o b l e m s w h c i h w i l l be a n a l y s e d f o r n o n l i n e a r 102 d e f o r m a t i o n . The f i n i t e e l ement d i s p l a y i n g t h e b e s t p e r f o r m a n c e i n t h e s e t e s t s s h o u l d be u t i l i z e d i n t h e n o n l i n e a r a n a l y s e s , s i n c e w i t h o u t a t l e a s t a good r e p r e s e n t a t i o n o f t h e l i n e a r r e s p o n s e t h e n o n l i n e a r r e s p o n s e w i l l n o t be a c c u r a t e l y d e t e r m i n e d u s i n g an i n c r e m e n t a l a n a l y s i s . The d i m e n s i o n s , l o a d i n g , and m a t e r i a l p r o p e r t i e s o f t h e c a n t i l e v e r u sed t o compare t h e p e r f o r m a n c e o f t h e two r e c t a n g u l a r f i n i t e e l e m e n t s , a r e as shown i n F i g . 10. T h i s p a r t i c u l a r c a n t i l e v e r was cho sen becau se s o l u t i o n s t o t h i s p r o b l e m u s i n g o t h e r f i n i t e e l e m e n t s a r e a v a i l a b l e i n t h e l i t e r a t u r e , i n a d d i t i o n t o a c l o s e d f o r m s o l u t i o n f r o m beam t h e o r y t h a t g i v e s an upper bound f o r t h e t i p d e f l e c t i o n o f t h e c a n t i l e v e r . The c a n t i l e v e r shown i n F i g . 10 was a n a l y s e d u s i n g each o f t h e two f i n i t e e l e m e n t s under c o n s i d e r a t i o n , i n t h r e e d i f f e r e n t g r i d s . These t h r e e f i n i t e e l ement g r i d s a r e shown i n F i g . 1 1 , a l o n g w i t h t h e bounda r y c o n d i t i o n s impo sed . A l l nodes a t t h e f i x e d end o f t h e c a n t i l e v e r were f i x e d a g a i n s t e i t h e r v e r t i c a l o r h o r i z o n t a l d i s p l a c e m e n t s , so as t o c o r r e s p o n d w i t h t h e f i n i t e e l ement s o l u t i o n s g i v e n i n t h e l i t e r a t u r e f o r v a r i o u s e l e m e n t s . The e x a c t p l a n e s t r e s s e l a s t i c i t y s o l u t i o n does n o t e x i s t f o r t h i s c a s e howeve r , ' b e c a u s e o f t h e r e s t r a i n t a t t h e bounda ry i n t h e v e r t i c a l d i r e c t i o n . The c l o s e d f o r m s o l u t i o n t o w h i c h t h e r e s u l t s a r e compared i s d e r i v e d f r o m beam t h e o r y , and c an be shown t o p r o v i d e an upper bound t o t h e t i p d i s p l a c e m e n t o f t h e c a n t i l e v e r w i t h t h e bounda ry c o n d i t i o n s a s imposed on t h e f i n i t e e l ement a n a l y s i s . The n u m e r i c a l r e s u l t s u s i n g t h e two r e c t a n g u l a r e l e m e n t s i n e ach o f t h e t h r e e g r i d s o f F i g . 1 1 , a r e t a b u l a t e d i n T a b l e I. The com-p a r i s o n i s b a sed on t h e t i p d e f l e c t i o n , s i n c e i n t h e n o n l i n e a r a n a l y s i s t i p d e f l e c t i o n w i l l be u sed t o a s s e s s t h e a c c u r a c y o f t h e method o f a n a l y s i s p r e s e n t e d i n t h i s t h e s i s . The r e s u l t s a r e a l s o shown i n F i g . 12 , 0 103 y i 12 ' \ — * - r n n 48 E = 30 ,000 k s i v = 0.25 t = 1.0 i n ( t i s t h e t h i c k n e s s o f t h e beam ) P = 40 k i p s CANTILEVER TEST F I G . 10 G R I D # 1 G R I D # 2 l b G R I D # 3 C A N T I L E V E R F I N I T E E L E M E N T G R I D S F I G . 11 105 TABLE I CANTILEVER F IN ITE ELEMENT ANALYSIS E l ement I n t e g r a t i o n G r i d Degrees o f Freedom (Net ) T i p D e f l e c t i o n ••-^ ( i n . ) w % Of Theo r y % E r r o r G a u s s i a n 1 16 0.242424 68.1 31.9 N o n u n i f o r m 1 16 0.328333 92.3 7.7 G a u s s i a n 2 32 0.304762 85.7 14.3 N o n u n i f o r m 2 32 0.332083 93.3 6.7 G a u s s i a n 3 96 0.336642 94.6 5.4 N o n u n i f o r m 3 96 0.351087 98.7 1.3 CST - 48 0.198 55 .6 44 .4 CST - 160 0.30556 85 .9 14.1 LST - 48 0.34872 98 .0 2 .0 LST - 160 0.355066 99.8 0.2 THEORY 0.35583 106 w w (Upper Bound) = 0.35583 • Deg rees o f Freedom CANTILEVER: F IN ITE ELEMENT COMPARISON F I G . 12 107 where results for the same problem using the constant stress triangle and linear stress triangle are included for comparison purposes. The comparison is based on the total number of degrees of freedom used in obtaining the solution, and while this is val id for the two rectangular elements in comparing them to each other, i t should be borne in mind when comparing the other two elements to them that other factors need to be considered to determine the amount of computation expended. As can be seen from Table I and F ig . 12, the rectangular f in i te element using nonuniform integration exhibits s ignif icantly better performance than the Gaussian integrated element, especially when a coarse grid of elements was used. Both rectangular elements perform better than the constant stress tr angle but not as well as the l inear strain triangle, when compared on the basis of total number of degrees of freedom used in obtaining the solution. This comparison does not con-sider the computational expense of forming the stiffness matrices or the solution of the set of resulting equations to be different when using the different types and geometries of elements, which of course i t is in general. The comparison of the two rectangular elements' performances on the basis of the number of degrees of freedom is val id for reasons outlined above. The second problem analysed to compare the performance of the two rectangular elements was an inf in i te plate strip with simply supported longitudinal edges, subjected to a uniform pressure on the upper surface. The plate strip along with the dimensions, material properties and pressure used is shown in F ig . 13. The closed form linear solution to this problem is developed by Timoshehko and Woinowsky-Krieger [33]. 108 A" L = 10.0 in . t = 0.2 in . E v q 10 4 ks i 0.25 5.0 psi w, MAX = 0.457764 Not to scale SIMPLY SUPPORTED INFINITE PLATE STRIP UNDER UNIFORM LOADING FIG. 13 109 By arguements of symmetry, only half the width of the plate needs to be analysed. The two grids used for each of the two rectangular f in i te elements are shown in F ig . 14, with the boundary conditions imposed. The numerical results using the two different rectangular f in i te elements are tabulated in Table II , and are also shown in Fig . 15. As can be seen, the nonuniformly integrated rectangular f in i te element again performs signif icantly better than the element that is integrated using the complete Gaussian quadrature. On the basis of the results of these two test problems for comparing the performance of the two rectangular elements, the nonuniformly integrated f in i te element was chosen to be used in the nonlinear analyses of the following chapter. 1> f c — ——< GRID # 1 1 f < GRID Ik 2 F IN ITE ELEMENT GRIDS FOR THE INF IN ITE PLATE STRIP F I G . 14 TABLE I I INF IN ITE PLATE STRIP F IN ITE ELEMENT RESULTS E l ement I n t e g r a t i o n G r i d Deg rees o f Freedom (Net ) C D e f l e c t i o n t % Of Theory % E r r o r G a u s s i a n 1 61 0.135474 29.6 70.4 N o n u n i f o r m 1 61 0.442357 96.6 3.4 G a u s s i a n 2 121 0.283161 61.9 38.1 N o n u n i f o r m 2 121 0.443819 96.9 3.1 Theo ry 0.457764 C L D e f l e c t i o n E x a c t = 0.457764 0.20 - f -7^ 0 G a u s s i a n i n t e g r a t e d r e c t a n g u l a r e l emen t A, N o n u n i f o r m l y i n t e g r a t e d r e c t a n g u l a r e l emen t —I 1 1 1 1 1 H—l H 10 20 30 40 50 100 Deg rees o f Freedom INF IN ITE PLATE STRIP F IN ITE ELEMENT RESULTS F I G . 15 112 APPLICATIONS TO NONLINEAR PROBLEMS 7.1 G e n e r a l To e v a l u a t e t h e a b i l i t y o f t h e l i n e a r i z e d v i r t u a l wo rk e q u a t i o n s , t h r o u g h t h e u se o f t h e f i n i t e e l ement method and a s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e , i n a c c u r a t e l y r e p r e s e n t i n g n o n l i n e a r d e f o r m a t i o n , s e v e r a l t e s t p r o b l e m s a r e s o l v e d . These t e s t p r o b l e m s were cho sen becau se a c o r r e s p o n d i n g a n a l y t i c a l c l o s e d f o r m s o l u t i o n i s a v a i l a b l e f o r c o m p a r i s o n . The t e s t p r ob l ems u sed i n t h i s c h a p t e r g e n e r a l l y f a l l i n t o one o f two c a t e g o r i e s . F i r s t l y , t h e r e a r e t h o s e p r ob l ems i n w h i c h t h e n o n l i n e a r n a t u r e o f t h e p r o b l e m a r i s e s p r i m a r i l y f r o m t h e b u i l d up o f l a r g e s t r a i n s , and hence c o r r e s p o n d i n g l y h i g h s t r e s s e s , w i t h o u t v e r y l a r g e d i s p l a c e m e n t s o f t h e body . The d e f o r m a t i o n b e -h a v i o u r o f t h i n p l a t e s , where t h e maximum d e f l e c t i o n i s o n l y on t h e o r d e r o f t h e t h i c k n e s s o f t h e p l a t e i s an example o f t h i s t y p e . S e c o n d l y , t h e n o n l i n e a r i t y may a r i s e f r o m l a r g e d i s p l a c e m e n t s , b u t where t h e body s t i l l has o n l y r e l a t i v e l y s m a l l s t r a i n s , a s i n t h e c l a s s i c p r o b l e m o f t h e e l a s t i c a . F o r e a c h t e s t p r o b l e m , t h e c l o s e d f o r m s o l u t i o n a v a i l a b l e w i l l be p r e s e n t e d and d i s c u s s e d on t h e b a s i s o f i t s p a r t i c u l a r p o s t u l a t e s and s i m p l i f i c a t i o n s . Then t h e f i n i t e e l ement s o l u t i o n w i l l be o b t a i n e d and compared t o t h e c o r r e s p o n d i n g c l o s e d f o r m s o l u t i o n . I t i s i m p o r t a n t t o u n d e r s t a n d t h a t i n c o m p a r i n g t h e c l o s e d f o r m and f i n i t e e l emen t s o l u t i o n s , t h e d i s c r e p a n c i e s between t h e two 113 s o l u t i o n s f o r a g i v e n p r o b l e m may a r i s e f r o m s e v e r a l s o u r c e s . F i r s t l y , t h e c l o s e d f o r m s o l u t i o n may have been o b t a i n e d t h r o u g h t h e u se o f s i m p l i f i c a t i o n s n o t r e q u i r e d by t h e f i n i t e e l ement a n a l y s i s , t h e r e s u l t b e i n g t h a t two v e r y s i m i l a r b u t n o t i d e n t i c a l p r ob l ems a r e compared i n t h e i r s o l u t i o n s . S e c o n d l y , t h e i n c r e m e n t a l L a g r a n g i a n a n a l y s i s p r e s e n -t e d i n t h i s t h e s i s u se s a l i n e a r c o n s t i t u t i v e r e l a t i o n s h i p be tween K i r c h h o f f s t r e s s e s and G r e e n ' s s t r a i n s , whereas i n a c l o s e d f o r m s o l u t i o n t h e same c o n s t i t u t i v e l a w may be u sed t o r e l a t e d i f f e r e n t l y d e f i n e d s t r e s s e s and s t r a i n s . T h i r d l y , e a c h s o l u t i o n o b t a i n e d a t t h e end o f an i n c r e m e n t i s o n l y a l i n e a r a p p r o x i m a t i o n t o t h e t r u e n o n l i n e a r r e s p o n s e d u r i n g t h a t i n c r e m e n t . The f a c t o r s a f f e c t i n g t h e a c c u r a c y o f t h e s e l i n e a r i n c r e m e n t a l s o l u t i o n s were d i s c u s s e d i n C h a p t e r 4 . F i n a l l y , t h e f i n i t e e l emen t p r o c e d u r e i t s e l f o n l y p r o v i d e s an a p p r o x -i m a t e s o l u t i o n o f t h e L a g r a n g i a n v i r t u a l work e x p r e s s i o n s , t h e a c c u r a c y o f w h i c h i s l i m i t e d by t h e c a p a b i l i t i e s o f t h e e l ement and t h e r e f i n e m e n t o f t h e mesh emp loyed . 7.2 E l a s t i c I n f i n i t e P l a t e S t r i p 7.2.1 G e n e r a l The e l a s t i c f i n i t e d e f o r m a t i o n o f an i n f i n i t e p l a t e s t r i p i s a c l a s s i c a l p r o b l e m w h i c h has a c l o s e d f o r m s o l u t i o n g i v e n by T imoshenko and W o i n o w s k y - K r i e g e r [34] . T h i s s o l u t i o n i s d e r i v e d f o r a u n i f o r m l y l o a d e d p l a t e h a v i n g an i n f i n i t e l e n g t h and a f i n i t e b r e a d t h . The s o l u t i o n s f o r b o t h s i m p l y s u p p o r t e d and f i x e d l o n g i t u d i n a l edges have been o b t a i n e d . The s o l u t i o n o f t h e i n f i n i t e p l a t e s t r i p i s o f p r a c t i c a l v a l u e s i n c e T imoshenko shows t h a t t h e s o l u t i o n f o r maximum d e f l e c t i o n s , s t r e s s e s , and b e n d i n g moments i n a p l a t e o f f i n i t e l e n g t h r a p i d l y a p p r o a c h t h e v a l u e s 114 g i v e n by t h e i n f i n i t e p l a t e s t r i p s o l u t i o n when t h e l e n g t h t o b r e a d t h r a t i o i s g r e a t e r t h a n about t h r e e [35] . The two p l a t e s t o be a n a l y s e d , one w i t h s i m p l y s u p p o r t e d l o n g i t u d i n a l edges and t h e o t h e r w i t h f i x e d l o n g i t u d i n a l edge s , a r e shown i n F i g . 16, i n t h e i r de fo rmed and unde fo rmed c o n f i g u r a t i o n s . The l e n g t h o f t h e p l a t e , ou t o f t h e p l a n e o f t h e d i a g r a m , i s t a k e n as b e i n g i n f i n i t e . These two p r o b l e m s t e s t t h e a b i l i t y o f t h e i n c r e m e n t a l L a g r a n g i a n v i r t u a l wo rk e q u a t i o n s , t o g e t h e r w i t h t h e f i n i t e e l e m e n t method and t h e s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e , t o a c c u r a t e l y f o l l o w t h e n o n l i n e a r r e s p o n s e o f t h e p l a t e s , w h i c h i s p r i m a r i l y due t o t h e deve l opment o f l a r g e membrane s t r e s s e s . These l a r g e s t r e s s e s d e v e l o p even though t h e d i s p l a c e m e n t s o f t h e p l a t e r e m a i n r e l a t i v e l y s m a l l w i t h r e s p e c t t o t h e t h i c k n e s s o f t h e p l a t e . The s i m p l y s u p p o r t e d p l a t e w i l l be a n a l y s e d t o a maximum d e f l e c t i o n o f j u s t s l i g h t l y l e s s t h a n t h e t h i c k n e s s o f t h e p l a t e , and t h e f i x e d edged p l a t e t o abou t h a l f t h a t v a l u e . 7 .2 .2 I n f i n i t e P l a t e S t r i p : C l o s e d Form S o l u t i o n The c l o s e d f o r m s o l u t i o n t o t h i s p r o b l e m i s o b t a i n e d by c o n s i d e r i n g an e l e m e n t a l s t r i p o f t h e p l a t e . The g o v e r n i n g d i f f e r e n t i a l e q u a t i o n f o r t h e p l a t e i s g i v e n by D d j * =• -M (7 .1 ) d x 2 where D i s t h e f l e x u r a l r i g i d i t y o f t h e p l a t e and i s g i v e n as E h 3 115 FIXED EDGED INFINITE PLATE STRIP FIG. 16(b) 116 I n t h e two e q u a t i o n s a b o v e , M i s t h e b e n d i n g moment, E i s Y o u n g ' s m o d u l u s , v i s P o i s s o n ' s r a t i o , and w i s t h e d e f l e c t i o n o f t h e p l a t e n o r m a l t o i t s o r i g i n a l p l a n e . E q . 7.1 i s an a p p r o x i m a t i o n o f t h e E u l e r -B e r n o u l l i l a w o f b e n d i n g , be cau se t h e second d e r i v a t i v e o f t h e d e f l e c t i o n i s u sed t o a p p r o x i m a t e t h e c u r v a t u r e o f t h e p l a t e . T h i s a s s u m p t i o n i s v a l i d where t h e s l o p e s (dw/dx ) , o f t h e p l a t e r e m a i n s m a l l , w h i c h i s t r u e f o r t h i s problem-; C o n s i d e r i n g f i r s t t h e s i m p l y s u p p o r t e d p l a t e , t h e b e n d i n g moment a t any c r o s s s e c t i o n o f t h e s t r i p i s g i v e n by T 2 qLx qx - • where S i s t h e i n - p l a n e f o r c e p r e v e n t i n g t h e edges o f t h e p l a t e f r o m mov ing t o g e t h e r , and q i s t h e i n t e n s i t y o f t h e u n i f o r m p r e s s u r e l o a d as shown i n F i g . 1 6 ( a ) . T h i s r e l a t i o n s h i p i s d e r i v e d by a s sum ing t h a t t h e u n i f o r m p r e s s u r e l o a d i n g does n o t change i t s l i n e o f a c t i o n d u r i n g d e f o r m a t i o n , and a l s o t h a t t h e m a g n i t u d e o f any l o a d i n g on a segment i s r e l a t e d t o t h e o r i g i n a l undefo rmed s u r f a c e a r e a . I n r e a l i t y a p r e s s u r e l o a d i n g w o u l d a c t on t h e de fo rmed s u r f a c e a r e a and w o u l d be d i r e c t e d n o r m a l l y t o t h e d e f l e c t e d s u r f a c e a t e v e r y p o i n t . The t y p e o f s u r f a c e t r a c t i o n p r e s c r i b e d f o r t h e c l o s e d f o r m s o l u t i o n i s d e s c r i b e d c o r r e c t l y b y t h e L a g r a n g i a n r u l e o f c o r r e s p o n d e n c e , so a L a g r a n g i a n s u r f a c e t r a c t i o n s h o u l d be u t i l i z e d i n t h e f i n i t e e l e m e n t s o l u t i o n . S h a r i f i and Y a t e s 36 a n a l y s e d t h i s p r o b l e m , b u t s p e c i f i e d a K i r c h h o f f s u r f a c e t r a c t i o n v e c t o r . Now, s u b s t i t u t i n g E q . 7.3 i n t o E q . 7 . 1 , t h e g o v e r n i n g d i f f e r -e n t i a l e q u a t i o n becomes 2 2 d w -::Sw = qLx + cp£ d x 2 J) 2D 2D C7.4) 117 A d o p t i n g t h e n o t a t i o n 2 2 u = S L 4D (7 .5 ) t h e n t h e g e n e r a l s o l u t i o n o f E q . 7.4 i s g i v e n by w = C^inhC 2 ^.) + C ^ o s h f e + ^ y - C L x - x 2 -' 8u D 2u (7 .6 ) A p p l y i n g t h e bounda r y c o n d i t i o n s f o r t h e s i m p l y s u p p o r t e d edges w(0) = 0 w(L ) = 0 (7.7) and s i m p l i f y i n g , t h e s o l u t i o n f o r t h e d e f l e c t i o n w, i s g i v e n as qL w = — 16u 4 D c o s h u ( l - ~ ) Li - 1 c o s h u - . T 2 :' >qL x , T v + y-C L - x ) 8u D (7 .8 ) As can be s e e n , t h e d e f l e c t i o n w i s dependent on t h e p a r a m e t e r u , w h i c h i s a f u n c t i o n o f t h e i n - p l a n e f o r c e S, a s g i v e n i n E q . 7 . 5 . To d e t e r m i n e t h e f o r c e S, i t i s n e c e s s a r y t o f i n d t h e f o r c e r e q u i r e d t o p r e v e n t t h e ends o f t h e p l a t e f r o m mov ing t owa rd s each o t h e r a s t h e p l a t e unde r goe s l a t e r a l d e f o r m a t i o n . To do t h i s , t h e e x t e n s i o n o f t h e p l a t e s t r i p (A ) , d e f i n e d as t h e d i f f e r e n c e be tween t h e a r c l e n g t h o f t h e m i d d l e s u r f a c e o f t h e p l a t e and t h e c h o r d l e n g t h ( L ) , i s t a k e n t o be p r o d u c e d by t h e f o r c e S. The e x a c t e x p r e s s i o n f o r d e t e r m i n i n g X i s X = 1 + dw 2 dx - L (7 . 9 ) 118 where t he i n t e g r a l g i v e s t h e a r c l e n g t h o f t h e m i d d l e s u r f a c e o f t h e p l a t e . The i n t e g r a l i n E q . 7 .9 i s d i f f i c u l t t o e v a l u a t e however , and so by-a s suming t h a t dw , d x " ^ 1 ( 7 .10 ) t h e e x p r e s s i o n i n E q . 7 .9 f o r A may be a p p r o x i m a t e d t o an a c c e p t a b l e d e g r e e o f a c c u r a c y by , 1 ( L vdw,2 , " l \ fe> d x ( 7 -11 ) T h i s a s s u m p t i o n was a l s o u sed i n a p p r o x i m a t i n g t h e E u l e r - B e r n o u l l i l a w o f b e n d i n g so i s c o n s i s t e n t w i t h t h e a c c u r a c y assumed so f a r . Now t o f i n d t h e f o r c e S, a s sum ing l a t e r a l s t r a i n i n t h e i n f i n i t e d i m e n s i o n o f t h e p l a t e t o be r e s t r a i n e d , t h e s t r e s s and s t r a i n may be r e l a t e d by (1 - V ) and t h e a p p r o x i m a t i o n s a r e made t h a t S = a h ( 7 .13 ) e = A X L " U s i n g E q s . 7 . 1 1 , 7 . 1 2 , and 7.13 t h e i n - p l a n e f o r c e S i s f i n a l l y g i v e n by „ _ Eh f d w / , S ~ " I 2~ ( dx " } d x ( 7 -14 ) 2 ( l - v Z ) L J d x 119 F i n a l l y , s u b s t i t u t i n g t h e e x p r e s s i o n f o r w f r o m E q . 7.8 i n t o Eq . 7 .14 , and s u b s t i t u t i n g f o r S i n E q . 7.14 u s i n g t h e r e l a t i o n s h i p i n E q . 7 . 5 , a t r a n s c e n d e n t a l e q u a t i o n o f t h e p a r a m e t e r u i s o b t a i n e d as 135 16u ' 2 '• 3 t a n h u + u t a n h u - u + 2u 5 1 5 " 2 8 = E h 2.2 2 8 ( 1 - v ) q L ( 7 .15 ) F o r g i v e n m a t e r i a l p r o p e r t i e s , u n i f o r m l o a d i n g q , and h/L r a t i o , t he r i g h t hand s i d e o f E q . 7.15 i s d e f i n e d . The p a r a m e t e r u may t h e n be o b t a i n e d f r o m t h i s t r a n s c e n d e n t a l e q u a t i o n t h r o u g h t h e u se o f any o f s e v e r a l i t e r a t i v e n u m e r i c a l methods . Once h a v i n g e v a l u a t e d t h e p a r a m e t e r u , t h e n t h e d e f l e c t i o n w may be o b t a i n e d f o r any c r o s s s e c t i o n g i v e n by t h e c o o r d i n a t e x , by s u b s t i t u t i n g f o r u i n E q . 7 .8 . By a rguement s o f symmetry , i t i s o b v i o u s t h a t t h e maximum d e f l e c t i o n w i l l o c c u r a t x" = L/2 , t h e r e f o r e s u b s t i t u t i n g t h i s i n t o E q . 7 . 8 , t he maximum d e f l e c t i o n o f t h e p l a t e i s g i v e n by W MAX = |§L-384D s e c h u - 1 +.2~ 17 24 (7 .16 ) F o r t h e f i x e d edge p l a t e , t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n i s s t i l l E q . 7 . 1 , b u t t h e now t h e b e n d i n g moment a t any c r o s s s e c t i o n o f t h e s t r i p i s g i v e n by M = £ L x - ^ x -FSw + M 0 2 2 C7.17) where M 0 i s t h e moment r e q u i r e d a t t h e s u p p o r t s t o p r e v e n t r o t a t i o n o f 120 t h e p l a t e a t t h e s u p p o r t s . The e l e m e n t s o f E q . 7.17 a r e shown i n F i g . 1 6 ( b ) . T h i s e q u a t i o n ha s been d e r i v e d u s i n g t h e same a s s u m p t i o n s abou t t h e s u r f a c e t r a c t i o n as i n t h e s i m p l y s u p p o r t e d c a s e , t h e r e f o r e t h e same comments a p p l y a g a i n i n t h i s c a s e . F o l l o w i n g t h e same deve l opment as f o r t h e s i m p l y s u p p o r t e d p l a t e , and u s i n g t h e b o u d a r y c o n d i t i o n s b e l o w a l o n g w i t h t h e o b s e r v a t i o n t h a t t h e d e f l e c t e d s u r f a c e i s s y m m e t r i c a l w i t h r e s p e c t t o t h e c e n t r e o f t h e s t r i p (x = L / 2 ) , dw(O) = dw(L/2) = 0 dx dx w(0) = 0 ( 7 .18 ) t h e n t h e f o l l o w i n g t r a n s c e n d e n t a l e q u a t i o n f o r t h e p a r a m e t e r u i s f o u n d , •135 16u S 2 u 3 -15 u + 4u - 3u 5 s i n h u 5 t a n h u = -E~h 2,2 2 8 ( 1 - v ) q L ( 7 .19 ) H a v i n g f ound t h e p a r a m e t e r u t h r o u g h t h e u se o f t h e t r a n s c e n -d e n t a l e q u a t i o n , t h e maximum d e f l e c t i o n o f t h e f i x e d edge p l a t e , a t x = L/2, i s e v a l u a t e d by w. MAX 5 k 384D 2M 3 _u + c s c h u 2 - c o t h u (7 .20) I n summary, t h e r e a r e s e v e r a l o b s e r v a t i o n s t o be made abou t t h e c l o s e d f o r m s o l u t i o n s p r e s e n t e d by T imoshenko . The f i r s t o b s e r v a t i o n i s t h a t t h e s u r f a c e t r a c t i o n s as employed i n t h e d i f f e r e n t i a l e q u a t i o n s s h o u l d be d e s c r i b e d by a L a g r a n g i a n s u r f a c e t r a c t i o n i n t h e f i n i t e e l ement a n a l y s i s , so t h a t t h e c l o s e d f o r m p r o b l e m and t h e f i n i t e e l ement p r o b l e m c o r r e s p o n d i n t h e l o a d i n g a p p l i e d . S e c o n d l y , t h e d e r i v a t i o n o f t h e c l o s e d 121 f o r m s o l u t i o n s r e q u i r e s t h a t t h e s l o p e s o f t h e p l a t e ( dw/dx ) , r e m a i n s m a l l . T h i s i s n o t a s e r i o u s r e s t r i c t i o n s i n c e l a r g e d i s p l a c e m e n t s w o u l d be r e q u i r e d t o g i v e t h e s l o p e m a g n i t u d e s n e c e s s a r y t o i n v a l i d a t e t h e s o l u t i o n . T h i s w o u l d i n t u r n r e q u i r e a m a t e r i a l w i t h an u n u s u a l l y l a r g e e l a s t i c d e f o r m a t i o n l i m i t . T h i r d l y , c a r e must be t a k e n i n c h o o s i n g a p p r o p r i a t e bounda ry c o n d i t i o n s f o r t h e f i n i t e e l ement a n a l y s i s i n o r d e r t h a t t h e y be c o n s i s t e n t w i t h t h e bounda r y c o n d i t i o n s imposed i n t h e c l o s e d f o r m s o l u t i o n s . T h e r e f o r e , i n t h e c a s e o f t h e s i m p l y s u p p o r t e d edge p l a t e , i n - p l a n e and ou t o f p l a n e d i s p l a c e m e n t s o f t h e m i d d l e s u r f a c e a t t h e s u p p o r t s must be p r e v e n t e d , b u t t h e edges must b e f r e e t o r o t a t e . F o r t h e p l a t e w i t h f i x e d edge s , i n a d d i t i o n t o t h e c o n s t r a i n t s on t h e s i m p l y s u p p o r t e d p l a t e r o t a t i o n o f t h e p l a t e a t t h e s u p p o r t s must be p r e v e n t e d . I t s h o u l d be n o t e d howeve r , t h a t i n t h e c l o s e d f o r m s o l u t i o n s t o b o t h p r o b l e m s , t h e r e i s no s p e c i a l r e s t r a i n t a t t h e s u p p o r t s t o p r e v e n t any P o i s s o n e f f e c t s t h a t m i g h t o c c u r i n t h e t h i c k n e s s d i m e n s i o n o f t h e p l a t e s . C a r e s h o u l d t h e r e f o r e be t a k e n i n n o t g r e a t e r r e s t r a i n t i n t h e f i n i t e e l ement a n a l y s i s . F o u r t h l y , t h e c o n s t i t u t i v e r e l a t i o n s h i p g i v e n i n E q . 7.12 i s u t i l i z e d w i t h t h e s t r e s s and s t r a i n d e f i n i t i o n s g i v e n i n E q . 7 . 13 . The e f f e c t o f t h e a p p r o x i m a t i o n s t o t h e s t r e s s and s t r a i n i s somewhat d i f f i c u l t t o p r e d i c t q u a l i t a t i v e l y . The s t r e s s a a s d e s c r i b e d i s a L a g r a n g i a n t y p e , b u t t h e s t r a i n d e f i n i t i o n i s o f . a s m a l l s t r a i n t y p e . S i n c e t h e f i n i t e e l emen t a n a l y s i s w i l l r e l a t e K i r c h h o f f s t r e s s e s and L a g r a n g i a n s t r a i n s w i t h e s s e n t i a l l y t h e same c o n s t i t u t i v e r e l a t i o n s h i p , t h e . c losedr; f -orm mode l and t h e f i n i t e e l e m e n t a n a l y s i s m o d e l d i f f e r s l i g h t l y . F i n a l l y , i n t h e c l o s e d f o r m s o l u t i o n s no a c c o u n t i s made o f t h e s h e a r d e f o r m a t i o n s o f t h e p l a t e s unde r t h e u n i f o r m l o a d i n g , wherea s t h e f i n i t e e l emen t w i l l a u t o -m a t i c a l l y i n c l u d e s h e a r d e f o r m a t i o n . Thus , i f e v e r y t h i n g e l s e was i n e x a c t c o r r e s p o n d e n c e , t h e d e f l e c t i o n f o u n d by t h e f i n i t e e l emen t a n a l y s i s 122 s h o u l d be g r e a t e r t h a n t h a t p r e d i c t e d by t h e c l o s e d f o r s o l u t i o n , by t h e amount o f s h e a r d e f o r m a t i o n o f t h e p l a t e . The t r a n s c e n d e n t a l e q u a t i o n s f o r b o t h t h e s i m p l y s u p p o r t e d edge p l a t e , E q . 7 . 15 , and t h e f i x e d edge p l a t e , E q . 7 . 19 , w i t h t h e d i m e n s i o n s and m a t e r i a l p r o p e r t i e s g i v e n b e l o w were s o l v e d . E = 1 0 7 p s i ( 6.895 x 1 0 1 0 kPa ) h = 0.2 i n . ( 0.508 cm ) L = 10.0 i n . ( 25 .4 cm ) v = 0 .25 The n u m e r i c a l method employed t o s o l v e t h e t r a n s c e n d e n t a l e q u a t i o n s was N e w t o n ' s Method o f T a n g e n t s . H a v i n g t h e n o b t a i n e d t he p a r a m e t e r u , f o r a r a n ge o f l o a d i n t e n s i t i e s , t h e maximum d e f l e c t i o n o f t h e p l a t e a t t h e c e n t r e o f t h e span d i v i d e d by t h e t h i c k n e s s o f t h e p l a t e ( w j ^ j j / h ) J W A S e v a l u a t e d . The n u m e r i c a l p r o c e d u r e r e q u i r e d t h e p a r a m e t e r ( w ^ ^ / h ) t o be o b t a i n e d t o s i x s i g n i f i c a n t f i g u r e a c c u r a c y . The n u m e r i c a l r e s u l t s f o r b o t h bounda r y c o n d i t i o n s a r e g i v e n i n T a b l e I I I . The f i n i t e e l ement a n a l y s i s r e s u l t s w i l l be compared a g a i n s t t h e s e c l o s e d f o r m r e s u l t s . 7 . 2 . 3 I n f i n i t e P l a t e S t r i p : F i n i t e E l ement A n a l y s i s The f i n i t e e l emen t u s i n g t h e n o n u n i f o r m i n t e g r a t i o n scheme d e v e l o p e d i n ' t h e p r e v i o u s c h a p t e r , i s u t i l i z e d t o o b t a i n a n u m e r i c a l s o l u t i o n t o t h e i n f i n i t e p l a t e s t r i p p r o b l e m . The p l a t e i s a n a l y s e d by e m p l o y i n g a g r i d o f e l e m e n t s t h r o u g h t h e t h i c k n e s s o f t h e p l a t e . The e l e m e n t s a r e t h e n u sed w i t h a p l a i n s t r a i n c o n s t i t u t i v e r e l a t i o n s h i p t o r e p r e s e n t t h e c o n d i t i o n o f r e s t r a i n e d s t r a i n i n t h e i n f i n i t e d i m e n s i o n . On l y h a l f o f t h e p l a t e span need be a n a l y s e d by a rguements o f symmetry about t h e m i d - s p a n o f t h e p l a t e . I n f i n i t e p l a t e s t r i p s TABLE I I I CLOSED FORM SOLUTION RESULTS FOR TWO INF IN ITE PLATE STRIPS p s i q (kPa) SIMPLY SUPPORTED EDGES w MAX h FIXED EDGES w MAX h 0 0 0 2 ( 13 .79 ) 0.168562 0.036585 4 (27 .58 ) 0.291145 0.072958 6 (41 .37 ) 0.380915 0.108918 8 (55 .16 ) 0.451510 0.144288 10 (68 .95 ) 0.510024 0.178917 12 (82 ,74 ) 0.560287- 0.212692 14 (96 .53 ) 0.604556 0.245527 16 (110 .32 ) 0.644265 0.277371 18 (124 .11 ) 0.680381 0.308195 20 (137 .90 ) 0.713587 0.337995 22 (151.68) 0.744382 0.366781 24 (165.47) 0.773146 0.394577 26 (179 .26 ) 0 .800170 0.421413 28 (193 .05 ) 0.825687 0.447328 30 (206 .84 ) 0.849884 0.472362 32 (220 .63 ) 0.872914 0.496556 34 (234.42) 0.894903 0.519954 36 (248.21) 0.915958 0.542598 38 (262.00) 0 .936170 0.564526 40 (275.79) 0.955615 0.585779 124 w i t h b o t h s i m p l y s u p p o r t e d and w i t h f i x e d s u p p o r t s a r e a n a l y s e d , u s i n g t h e same d i m e n s i o n s and m a t e r i a l p r o p e r t i e s as u s ed i n t h e p r e v i o u s s e c t i o n i n o b t a i n i n g t h e c l o s e d f o r m s o l u t i o n s . F o r t h e p l a t e w i t h s i m p l y s u p p o r t e d edge s , t h e f i n i t e e l ement a n a l y s i s was p e r f o r m e d u s i n g t h e g r i d o f e l e m e n t s shown i n F i g . 1 7 ( a ) . The i n c r e m e n t a l v i r t u a l work e q u a t i o n s and t h e s e l f - c o r r e c t i n g p r o c e d u r e fo rmed t h e b a s i s o f t h e method o f a n a l y s i s . The p r o b l e m was a n a l y s e d t w i c e , once u s i n g l o a d i n g i n c r e m e n t s o f 4 p s i ( 27.58 k P a ) , up t o a maximum l o a d i n g o f 40 p s i ( 275.8 k P a ) , t h e n a second a n a l y s i s was made u s i n g a l a r g e r l o a d i n g i n c r e m e n t o f 8 p s i ( 55 .16 k P a ) , up t o t h e same maximum l o a d i n g . The r e s u l t s o b t a i n e d f r o m t h e s e two a n a l y s e s , f o r t h e d e f l e c t i o n p a r a m e t e r wj^x/ n> a r e g i v e n n u m e r i c a l l y i n T a b l e IV and shown g r a p h i c a l l y i n F i g . 18, where t h e y a r e compared w i t h t h e c l o s e d f o r m s o l u t i o n . I t c an be seen f r o m t h e r e s u l t s o b t a i n e d t h a t t h e f i n i t e e l emen t a n a l y s i s o g i v e s e x c e l l e n t r e s u l t s even f o r t h e l a r g e r l o a d i n g i n c r e m e n t a n a l y s i s . I n t h e 4 p s i l o a d i n g i n c r e m e n t a n a l y s i s i t can be seen t h a t t h e l o a d c o r r e c t i o n a p p l i e d t o t h e s o l u t i o n has c au sed some o s c i l l a t i o n abou t t h e c l o s e d f o r m s o l u t i o n . The u se o f t h e s m a l l e r l o a d i n g i n c r e m e n t c an a l s o be seen t o g i v e a s o l u t i o n t h a t more c l o s e l y f o l l o w s t h e c l o s e d f o r m s o l u t i o n a t e v e r y i n c r e m e n t . I t s h o u l d be n o t e d however , t h a t t h e 8 p s i i n c r e m e n t a n a l y s i s , e x c e p t f o r t h e f i r s t l o a d s t e p w h i c h i s r e a l l y o n l y a l i n e a r a n a l y s i s , f o l l o w s t h e n o n l i n e a r d e f o r m a t i o n p a t h v e r y w e l l . N e x t , t h e p l a t e h a v i n g f i x e d l o n g i t u d i n a l edges was a n a l y s e d u s i n g t h e same e l ement g r i d as i n t h e p r e v i o u s p r o b l e m , b u t t h e bounda ry c o n d i t i o n s a t t h e s u p p o r t have been a p p r o p r i a t e l y m o d i f i e d , a s shown i n F i g . 1 7 ( b ) . The n u m e r i c a l r e s u l t s o b t a i n e d f o r w M A y / h a r e g i v e n i n 125 l3£ I* INF IN ITE PLATE STRIP WITH SIMPLY SUPPORTED EDGES F IN ITE ELEMENT GRID F I G . 17 (a ) INF IN ITE PLATE STRIP WITH FIXED EDGES F IN ITE ELEMENT GRID F I G . 1 7 ( b ) 126 TABLE IV F IN ITE ELEMENT RESULTS FOR THE INF IN ITE PLATE STRIP WITH SIMPLY SUPPORTED LONGITUDINAL EDGES q w MAX h w MAX h p s i C l o s e d Form S o l u t i o n F i n i t e E l ement A n a l y s i s ( U s i n g M i d -•depth Node) 4 p s i s t e p s 8 p s i s t e p s 0 0 0 0 4 0.291145 0.353943 (+21.6) 8 0.451510 0.422102 ( - 6 . 5 ) 0.707885 (+57.) 12 0.560287 0.586580 ( +4.7) 16 0.644265 0.644570 (+0.05) 0 .673315(+4 .5 ) 20 0.713587 0.720350 (+0.95) 24 0.773146 0.777995 (+0.63) 0 .787600(+1.9 ) 28 0.825687 0.830960 (+0.64) 32 0.872914 0.878320 (+0.62) 0 .883325(+1.2 ) 36 0.915958 0.921635 (+0.62) 40 0.955615 0.961590 (+0.62) 0 .965385(+1.0 ) N o t e : The numbers i n d i c a t e d i n p a r e n t h e s e s a r e t h e p e r c e n t a g e a b s o l u t e e r r o r o f t h e f i n i t e e l emen t r e s u l t s when compared t o t h e c l o s e d f o r m s o l u t i o n . 127 128 T a b l e V where t h e y a r e compared t o t h e c l o s e d f o r m s o l u t i o n , and a l s o shown g r a p h i c a l l y i n F i g . 19 where t h e c l o s e d f o r m s o l u t i o n i s a g a i n shown. The f i x e d edge p l a t e was a n a l y s e d u s i n g t h e 4 p s i and 8 p s i i n c r e m e n t s o f l o a d i n g as was u sed f o r t h e s i m p l y s u p p o r t e d p l a t e . A g a i n , e x c e l l e n t r e s u l t s were o b t a i n e d u s i n g b o t h i n c r e m e n t s o f l o a d i n g . Tha t t h e 4 p s i i n c r e m e n t a n a l y s i s i s n o t s i g n i f i c a n t l y b e t t e r t h a n t h e 8 p s i i n c r e m e n t o n e , i s p r i m a r i l y due t o t h e n a t u r e o f t h e n o n l i n e a r p r o b l e m . T h i s p a r t i c u l a r p r o b l e m i s n o t as h i g h l y n o n l i n e a r a s i s t h e s i m p l y s u p p o r t e d c a s e . I n b o t h t h e s i m p l y s u p p o r t e d and f i x e d bounda ry c o n d i t i o n p l a t e s ' a n a l y s i s , i t s h o u l d be n o t e d t h a t t h e r e s u l t s were o b t a i n e d „us ing a r e a s o n a b l y c o a r s e mesh o f f i n i t e e l e m e n t s . The u se o f two e l e m e n t s t h r o u g h t h e t h i c k n e s s was r e q u i r e d a t l e a s t i n t h e c a s e o f t h e s i m p l y s u p p o r t e d p l a t e t o c o r r e c t l y m o d e l t h e bounda r y c o n d i t i o n s . I t i s e x -p e c t e d t h a t t h e u se o f e i t h e r more s o p h i s t i c a t e d e l e m e n t s w i t h h i g h e r o r d e r shape f u n c t i o n s , o r t h e u se o f a g r e a t e r number o f t h e n o n u n i f o r m r e c t a n g u l a r e l e m e n t s d e v e l o p e d i n t h i s t h e s i s , w o u l d r e s u l t i n even more a c c u r a t e s o l u t i o n s t o t h e two p r o b l e m s p o s e d . 7.3 The E l a s t i c a 7.3.1 G e n e r a l The i n f i n i t e p l a t e p r o b l e o f t h e p r e v i o u s s e c t i o n ha s a c l o s e d f o r m s o l u t i o n t h a t i s o b t a i n e d by s o l v i n g t h e a p p r o x i m a t e d i f f e r e n t i a l e q u a t i o n g i v e n as E q . 7 . 1 . The a p p r o x i m a t i o n i n h e r e n t i n t h a t e q u a t i o n i s s a t i s f a c t o r y when t h e s l o p e s , and hence d e f l e c t i o n s , r e m a i n r e a s o n a b l y s m a l l . When t h e s l o p e s and d e f l e c t i o n s become l a r g e howeve r , t h e n t h e e x a c t d i f f e r e n t i a l e q u a t i o n o f t h e d e f l e c t i o n c u r v e must be u s e d . T h i s 129 TABLE V F IN ITE ELEMENT RESULTS FOR THE INF IN ITE PLATE STRIP WITH FIXED LONGITUDINAL EDGES q p s i w MAX h C l o s e d Form S o l u t i o n w MAX h F i n i t e E l ement A n a l y s i s ( U s i n g M i d - d e p t h Node) 4 p s i s t e p s 8 p s i s t e p s 0 0 0 0 4 0.072958 0.070619 ( -3 .2 ) 8 0.144288 0.139505 ( -3 .3 ) 0.141237 (• -2.1) 12 0.212692 0.206531 ( -2 .9) 16 0.277371 0.270013 ( -2 .7 ) 0.269333 (--2.9) 20 0.337995 0.329659 ( -2 .5 ) 24 0.394577 0.385444 ( -2 .3 ) 0.387621 (--1.8) 28 0.447328 0.437562 ( -2 .2 ) 32 0.496556 0 .486300 ( -2 .1 ) 0.488911 (-•1.5) 36 0.542598 0 .531970 ( -2 .0) 40 0.585779 0.574880 ( -1 .9 ) 0.577590 (- 1.4) N o t e : The numbers i n d i c a t e d i n p a r e n t h e s e s a r e t h e p e r c e n t a g e a b s o l u t e e r r o r o f t h e f i n i t e e l emen t r e s u l t s when compared t o t h e c l o s e d f o r m s o l u t i o n . 130 MAX h 4 o az lb 20 24 28 32 36 40 <t Cpsi) F IN ITE ELEMENT RESULTS FOR THE INF IN ITE PLATE STRIP (J IXED EDGES) F I G . 19 131 i s t h e E u l e r - B e r n o u l l i l a w o f b e n d i n g e x p r e s s e d by D 4^" = " M (7.21) ds where D i s a g a i n t h e f l e x u r a l r i g i d i t y , M i s t h e moment a t t h e p a r t i c u l a r c r o s s s e c t i o n , and d6/ds r e p r e s e n t s t h e c u r v a t u r e o f t h e d e f l e c t i o n c u r v e . The t e r m dO/ds i s t h e r a t e o f change o f 6, t h e a n g l e o f r o t a t i o n o f . t h e d e f l e c t i o n c u r v e , w i t h r e s p e c t t o s w h i c h i s a r u n n i n g c o o r d i n a t e a l o n g t h e d e f l e c t i o n c u r v e . Now, a s suming t h a t t h e c o n s t i t u t i v e r e l a t i o n -s h i p i s t h a t o f a l i n e a r e l a s t i c m a t e r i a l , t h e e x a c t shape o f t h e e l a s t i c c u r v e s a t i s f y i n g E q . 7.21 i s c a l l e d t h e e l a s t i c a . The m a t h e m a t i c a l s o l u t i o n o f t h e e l a s t i c a was f i r s t i n v e s t i g a t e d by B e r n o u l l i , E u l e r , L a g r a n g e , and P l a n a . Many d i f f e r e n t t y p e s o f beams and l o a d i n g c o n d i t i o n s have s i n c e been p o s t u l a t e d and s o l v e d f o r , b u t o n l y two w i l l be u sed h e r e t o e v a l u a t e t h e a b i l i t y o f t h e i n c r e m e n t a l v i r t u a l wo rk e q u a t i o n s w i t h t h e n o n u n i f o r m l y i n t e g r a t e d f i n i t e e l emen t and t he s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e , i n f o l l o w i n g l a r g e d i s p l a c e m e n t s . The two e l a s t i c a s o l u t i o n s a v a i l a b l e t h a t were cho sen f o r c o m p a r i s o n p r u p o s e s a r e , a c a n t i l e v e r w i t h a v e r t i c a l t i p l o a d , and a c a n t i l e v e r w i t h a u n i f o r m l y d i s t r i b u t e d l o a d . The method o f o b t a i n i n g t h e c l o s e d f o r m s o l u t i o n s w i l l be b r i e f l y d i s c u s s e d , t h e n t h e f i n i t e e l ement s o l u t i o n s w i l l be o b t a i n e d and compared t o t h e c l o s e d f o r m s o l u t i o n s . 7.3.2 C a n t i l e v e r W i t h A V e r t i c a l T i p L o a d : C l o s e d Form S o l u t i o n C o n s i d e r t h e c a n t i l e v e r AB shown i n F i g . 20, h a v i n g a v e r t i c a l t i p l o a d P wh i ch , i s assumed t o p r o d u c e l a r g e d e f l e c t i o n s and t o d e f o r m t h e beam t o a c o n f i g u r a t i o n A B ' . The v e r t i c a l and h o r i z o n t a l d i s p l a c e -132 CANTILEVER WITH VERTICAL TIP LOAD.UNDERGOING LARGE DISPLACEMENTS FIG. 20 133 merits o f t h e f r e e end o f t h e beam a r e d e n o t e d by 6^ and 6^ r e s p e c t i v e l y , and t h e a n g l e o f r o t a t i o n a t t h e f r e e end o f t h e c a n t i l e v e r i s d e n o t e d by 6^. N o t e t h a t t h r o u g h o u t t h e l a r g e d e f o r m a t i o n s e x p e r i e n c e d , t h a t t h e t i p l o a d P does n o t change i t s l i n e o f a c t i o n and i t s m a g n i t u d e i s i n d e p e n d e n t o f t h e d e f o r m a t i o n , and i s t h u s p r o p e r l y d e s c r i b e d as a L a g r a n g i a n t r a c t i o n v e c t o r . A s sum ing t h a t t h e l e n g t h o f t h e d e f l e c t i o n c u r v e a t A B ' i s e q u a l t o t h e i n i t i a l l e n g t h L, t h a t i s no a x i a l e x t e n s i o n , t h e n t h e e x p r e s s i o n f o r t h e b e n d i n g moment can be d e r i v e d and s u b s t i t u t e d i n t o E q . 7 . 2 1 , t h e E u l e r - B e r n o u l l i l a w o f b e n d i n g . The f l e x u r a l r i g i d i t y i n t h i s c a s e i s g i v e n by D = E I ( 7 .22 ) where I i s t h e moment o f i n e r t i a o f t h e c r o s s s e c t i o n . A f t e r a p p l y i n g bounda r y c o n d i t i o n s and c o n s i d e r a b l e m a n i p u l a t i o n o f t h e g o v e r n i n g e q u a t i o n , t h e s o l u t i o n i s o b t a i n e d i n te rms o f e l l i p t i c i n t e g r a l s . The t r a n s c e n d e n t a l e q u a t i o n t o be s o l v e d i s F ( k ) - F(k,<j>) =\/ | | i ( 7 .23 ) where , 1 + s i n 0, k =\ b <j) = a r c s i n 2 1 ~k?2~ (7 .24 ) and F ( k ) i s a c o m p l e t e e l l i p t i c i n t e g r a l of. t h e f i r s t k i n d , w h i l e F(k,<j>) i s an e l l i p t i c i n t e g r a l o f t h e f i r s t k i n d . They a r e d e f i n e d by and 134 TT/2 F ( k ) = f d t ( 7 .25 ) '0 / I . 2 . 2 1 - k s i n t F(k s<f») = j ^ ( 7 .26 ) ' 0 , 2 . 2 1 - k s i n t Knowing t h e l e n g t h o f t h e c a n t i l e v e r L, and f l e x u r a l r i g i d i t y E I , t h e t r a n s c e n d e n t a l e q u a t i o n g i v e n i n E q . 7.23 i s s o l v e d by a s sum ing a v a l u e o f 6 , and t h e n f i n d i n g t h e c o r r e s p o n d i n g v a l u e o f t h e l o a d P. The e q u a t i o n f o r t h e v e r t i c a l d e f l e c t i o n o f t h e end o f t h e c a n t i l e v e r i s g i v e n as V 4E I P L 2 E ( k ) - E(k,cf>) (7 .27) where E ( k ) i s a c o m p l e t e e l l i p t i c i n t e g r a l o f t h e second k i n d and E(k,cj>) i s an e l l i p t i c i n t e g r a l o f t h e second k i n d . F i n a l l y , t h e h o r i z o n t a l d e f l e c t i o n o f t h e c a n t i l e v e r i s f o und f r o m Sfi , 1 - / 2 E I s i n 9 b ( 7 . 28 ) V P L 2 E l l i p t i c i n t e g r a l s o f b o t h k i n d s a r e w e l l t a b u l a t e d , and Ro jahn[36 ] . ,has c a l c u l a t e d d e f l e c t i o n s and r o t a t i o n s o f t h i s c a n t i l e v e r 2 i n t e rms o f t h e n o n d i m e n s i o n a l p a r a m e t e r PL /EI . The d e f l e c t i o n s and r o t a t i o n s a r e g i v e n a l s o i n t h e i r n o n d i m e n s i o n a l f o r m i n T a b l e V I , and t h e s e v a l u e s w i l l be u sed t o compare t h e f i n i t e e l emen t r e s u l t s w i t h . 7 .3 .3 C a n t i l e v e r W i t h A V e r t i c a l T i p L o a d : F i n i t e E l ement A n a l y s i s The f i n i t e e l ement a n a l y s i s was p e r f o r m e d f o r a c a n t i l e v e r TABLE V I ANGLE OF ROTATION AND DEFLECTIONS OF A CANTILEVER BEAM WITH A T IP LOAD P L 2 e b 6 V E I TT/2 L L 0 0 0 0 0.25 0.079 0.083 0.004 0 .50 0.156 0.162 0.016 0.75 0.228 0.235 0.034 1.0 0.294 0.302 0.056 2 .0 0.498 0.494 0.160 3.0 0.628 0.603 0.255 4 . 0 0.714 0.670 0.329 5.0 0.774 . 0.714 0.388 6.0 0.817 0.744 0.434 7.0 0.849 0.767 0.472 8.0 0.874 0.785 0.504 9.0 - 0.894 0.799 0.531 1 0 . 0 0.911 0.81.1 0.555 oo - 1.000 1.000 1 .000 whose d i m e n s i o n s and m a t e r i a l p r o p e r t i e s a r e g i v e n i n F i g . 2 1 ( a ) . The f i n i t e e l emen t mesh u s e d , and t h e bounda r y c o n d i t i o n s imposed a r e shown i n F i g . 2 1 ( b ) . The p r o b l e m was a n a l y s e d u s i n g t h e i n c r e m e n t a l v i r t u a l work e q u a t i o n s and t h e s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e , a l o n g w i t h t h e n o n u n i f o r m l y i n t e g r a t e d r e c t a n g u l a r f i n i t e e l e m e n t . Two s o l u t i o n s were o b t a i n e d u s i n g two d i f f e r e n t l o a d i n g i n c r e m e n t mag-n i t u d e s . L e t t i n g .2 E I K = i T ( 7 . 2 9 ) Then , t h e two a n a l y s e s were p e r f o r m e d u s i n g i n c r e m e n t s o f t h e l o a d P c o r r e s p o n d i n g t o i n c r e m e n t s i n t h e p a r a m e t e r K o f 0.25 and 0 . 5 . The r e s u l t s o f t h e two a n a l y s e s u s i n g t h e d i f f e r e n t i n c r e m e n t s o f K, f o r t h e r o t a t i o n and t h e v e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t s a t t h e c a n t i l e v e r ' s f r e e end a r e g i v e n i n T a b l e s V I I , V I I I , and I X , f o r a r ange o f K v a l u e s f r o m 0.25 t o 10.0 . The h o r i z o n t a l and v e r t i c a l d e f l e c t i o n s g i v e n a r e t h o s e o f t h e m i d - d e p t h node a t t h e f r e e end o f t h e c a n t i l e v e r . The r o t a t i o n of, t h e f r e e e n d , 6^, was e v a l u a t e d by f i n d i n g t h e a n g l e t h a t t h e l i n e segment j o i n i n g t h e two m i d - d e p t h nodes n e a r e s t t h e f r e e end , fo rms w i t h t h e h o r i z o n t a l . The f i n i t e e l ement s o l u t i o n v a l u e s f o r 6^ a r e t h e r e f o r e an a v e r a g e o f t h e r o t a t i o n o f t h e d e f l e c t e d m i d d l e s u r f a c e o v e r t h e segment f r o m 9L/10 t o L w i t h t h e mesh u s e d . The r e s u l t s f o r 6 , 6 , and 6^ a r e a l s o g i v e n g r a p h i c a l l y i n F i g s . 2 2 , 23 and 24 r e s p e c t i v e l y , a l o n g w i t h t h e c l o s e d f o r m s o l u t i o n . I t c an be seen f r o m t h e T a b l e s and F i g u r e s g i v e n t h a t t h e f i n i t e e l ement s o l u t i o n r e s u l t s c o r r e s p o n d f a i r l y w e l l w i t h t h e c l o s e d f o r m s o l u t i o n , e s p e c i a l l y when a s m a l l e r l o a d i n g i n c r e m e n t (0 .25 ) i s u s e d . The re i s an o s c i l l a t i o n o f t h e f i n i t e e l ement s o l u t i o n a t l ow — b — E = 1.2 x 1 0 4 psi j L t v = 0.2 r — h 1 L = 10.0 in . 6 h - 1.0 i n . cross section b = 1.0 i n . CANTILEVER WITH A VERTICAL T IP LOAD F I G . 21 (a ) CANTILEVER WITH A VERTICAL T IP LOAD: F IN ITE ELEMENT GRID F I G . 21 (b ) TABLE V I I F IN ITE ELEMENT RESULTS FOR CANTILEVER END ROTATION K E I 6 b T T / 2 C l o s e d Form S o l u t i o n 9 b H B . i t e E l emen t R e s u l t s K I n c rement 0.5 K I n c rement 0.25 0 0 0 0 0.25 0.079 . 6.-'. 0.0783 0.50 0.156 0.1543 0.0862 0.75 0.228 0.2058 1.0 0.294 0.1581 0.2097 1.5 0.2867 0.3149 2 .0 0.498 0.2981 0.4175 2.5 0.4733 0.4975 3.0 0.628 0.4781 0.5632 3.5 0.5626 0.6169 4 . 0 0.714 0.5842 0.6617 4 .5 0.6839 0.6991 5.0 0.774 0.6918 0.7302 5.5 0.7548 0.7564 6.0 0.817 0.7636 0.7788 6.5 0.7989 0.7984 7.0 0.849 0.8103 0.8156 7.5 0 0.8307 0.8308 8.0 0.874 0.8429 0.8443 8.5 0.8561 0.8564 9.0 0.894 0.8669 0.8672 9.5 0.8768 0.8769 1 0 . 0 0.911 0.8857 0.8857 1 1 1 1 1 1 1 1 1-2 3 4 5 6 7 8 9 10 K F IN ITE ELEMENT RESULTS FOR CANTILEVER END ROTATION F I G . 22 TABLE V I I I F IN ITE ELEMENT RESULTS FOR CANTILEVER VERTICAL END DEFLECTIONS K E I u -C l o s e d Form S o l u t i o n 6 . —£ F i n i t e E l ement R e s u l t s L K I n c rement 0.5 K I n c rement 0.25 0 • 0 0 0 0.25 0.083 0.0828 0 .50 0.162 0.1657 0.0948 0.75 0.235 0.2190 1.0 0.302 0.1710 0.2238 1.5 0.3167 0.3328 2 .0 0.494 0.3259 0.4319 2.5 0.5025 0.5058 3 .0 0.603 0.4973 0.5631 3.5 0.5777 0.6079 4 . 0 0.670 0.5926 0.6435 4.5 0.6695 0.6722 5.0 0.714 0.6733 0.6955 5.5 0.7166 0.7148 6.0 0.744 0.7233 0.7312 6.5 0.7463 0.7453 7.0 0.767 0.7553 0.7577 7.5 0.7689 0.7687 8.0 0.785 0.7780 0.7784 8.5 0.7872 0.7872 9.0 0.799 0.7951 0.7.952 9.5 0.8025 0.8058 1 0 . 0 0.811 0.8091 0.8090 141 6 F IN ITE ELEMENT RESULTS FOR CANTILEVER VERTICAL END DISPLACEMENTS F I G . 23 142 TABLE I X F IN ITE ELEMENT RESULTS FOR CANTILEVER HORIZONTAL END DEFLECTIONS *-s2 E I \ L C l o s e d Form S o l u t i o n (5, n F i n i t e E l ement R e s u l t s L K I n c rement 0.5 K I n c rement 0.25 0 0 0 0 0.25 0.004 0 0.50 0.016 0 0.0051 0.75 0.034 0.0193 1.0 0.056 0.0166 0.0298 1.5 0.0468 0.0676 2.0 0.160 0.0632 0.1175 2.5 0.1365 0.1660 3.0 0.255 0.1568 0.2116 3.5 0.2141 0.2528 4.0 0.329 0.2339 0.2899 4.5 0.3081 0.3228 5.0 0.388 0.3210 0.3518 5.5 0.3752 0.3776 6.0 0.434 0.3875 0.4007 6.5 0.4216 0.4217 7.0 0.472' 0.4360 0.4408 7.5 0.4579 0.4583 8.0 0.504. 0.4730 0.4744 8.5 0.4889 0.4893 9.0 0.531 0.5027 0.5030 9.5 0.5157 0.5158 10.0 0.555 . 0.5277 0.5278 143 1 2 3 4 5 6 7 8 9 1 0 K F IN ITE ELEMENT RESULTS FOR CANTILEVER HORIZONTAL END DISPLACEMENTS F I G . 24 K v a l u e s c au sed by t h e s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e . The o s c i l l a t i o n may be a t t r i b u t e d t o t h e e f f e c t o f l i n e a r i z i n g t h e v i r t u a l work e q u a t i o n s , w h i c h c au sed p a r t o f t h e v i r t u a l work e x p r e s s i o n s t o be n e g l e c t e d . The ab sence o f t h e n e g l e c t e d p o r t i o n a p p e a r s t o have cau sed t h e i n c r e m e n t a l s t i f f n e s s m a t r i c e s t o be t oo s t i f f . I t i s o n l y t h e a c t i o n o f t h e l o a d c o r r e c t i o n t e r m i n t h e s e l f - c o r r e c t i n g s o l u t i o n p r o c e d u r e t h a t p r e v e n t s t h e s o l u t i o n o b t a i n e d f r o m d i v e r g i n g . The m a g n i t u d e o f t h i s o s c i l l a t i o n c o u l d be r e d u c e d e i t h e r by u s i n g s m a l l e r i n c r e m e n t s o f l o a d i n g , as can be seen f r o m t h e r e s u l t s shown h e r e , o r by i t e r a t i n g a t each l o a d i n g i n c r e m e n t s t e p t o s a t i s f y a c r i t e r i a o f c o n v e r g e n c e . 7.3 .4 C a n t i l e v e r W i t h U n i f o r m L o a d i n g : C l o s e d Form S o l u t i o n , The c l o s e d f o r m s o l u t i o n f o r t h i s p r o b l e m shown i n F i g . 2 5 ( a ) , was o b t a i n e d by Holden^37J . The s o l u t i o n a g a i n s t a r t s f r o m t h e E u l e r -B e r n o u l l i l a w o f b e n d i n g , g i v e n i n E q . 7 . 21 , f r o m w h i c h , a- suming a a u n i f o r m l o a d i n g i n t e n s i t y q , t h e f o l l o w i n g i s o b t a i n e d , .2 ^ = - J s c o s e (7.30) ds The bounda ry c o n d i t i o n s a p p l i e d a r e 7 ^ = 0 a t s = 0 ds (7.31) 0 = 0 a t s = L H o l d e n t h e n s e p a r a t e s t h e second o r d e r e q u a t i o n , E q . 7 .30 , i n t o a s y s t e m o f two f i r s t o r d e r e q u a t i o n s and i n t e g r a t e s u s i n g n u m e r i c a l methods t o g e t e a s a f u n c t i o n o f s f o r p r e s c r i b e d q . The d e f l e c t i o n s a r e o b t a i n e d 145 CANTILEVER WITH UNIFORM LOADING: F IN ITE ELEMENT GRID F I G . 25Cb) 146 by i n t e g r a t i n g a l o n g t h e l e n g t h o f t h e c a n t i l e v e r u s i n g t h e known v a l u e s o f 6, t h a t were d e t e r m i n e d f i r s t . H o l d e n u s e d a f o u r t h o r d e r R u n g e - K u t t a p r o c e d u r e f o r t h e d i f f e r e n t i a l e q u a t i o n s and a S i m p s o n ' s r u l e f o r t h e i n t e g r a t i o n s r e q u i r e d . H o l d e n assumes t h e c a n t i l e v e r t o be i n e x t e n s i o n a l . N u m e r i c a l r e s u l t s were n o t g i v e n by H o l d e n i n h i s p a p e r , t h e r e f o r e h i s s o l u t i o n was o b t a i n e d as a c c u r a t e l y a s p o s s i b l e by g r a p h i c a l means f r o m a r e p o r t by B a t h e , e t a l [38] , who a l s o u se t h i s s o l u t i o n f o r c o m p a r i s o n p u r p o s e s . 7 .3 .5 C a n t i l e v e r W i t h U n i f o r m L o a d i n g : F i n i t e E l e m e n t A n a l y s i s The f i n i t e e l ement s o l u t i o n t o t h i s p r o b l e m was o b t a i n e d u s i n g t h e g r i d o f n o n u n i f o r m l y i n t e g r a t e d r e c t a n g u l a r f i n i t e e l e m e n t s a s shown i n F i g . 2 5 ( b ) . The t i p d e f l e c t i o n r e s u l t s were o b t a i n e d u s i n g two d i f f e r e n t l o a d i n g i n c r e m e n t m a g n i t u d e s , and t h e r e s u l t s a r e g i v e n i n T a b l e X. The r e s u l t s a r e g i v e n f o r a l o a d p a r a m e t e r K ' , where K ' i s d e f i n e d a s T 3 f f - ( 7 -32 ) Two a n a l y s e s w e r e p e r f o r m e d u s i n g i n c r e m e n t s o f t h e u n i f o r m l o a d q c o r r e s p o n d i n g t o i n c r e m e n t s o f K ' o f 0 .5 and 1.0 . B o t h a n a l y s e s i n c l u d e d t h e r ange o f K ' f r o m K '= 0 t o K '= 10.0 . The t i p d e f l e c t i o n v a l u e s g i v e n a r e t h o s e o f t h e m i d - d e p t h node . A n o t h e r a n a l y s i s was a l s o p e r f o r m e d , i n w h i c h t h e l o a d c o r r e c t i o n a s p e c t o f t h e o t h e r two a n a l y s e s was n o t u s e d . The r e s u l t s o f t h i s t r i a l a r e a l s o g i v e n i n T a b l e X . These t h r e e s o l u t i o n s a r e shown g r a p h i c a l l y i n F i g . 26 , where H o l d e n ' s s o l u t i o n i s g i v e n f o r c o m p a r i s o n p u r p o s e s . As c a n be s een f r o m F i g . 26 , t h e i n c r e m e n t a l f i n i t e e l e m e n t 147 TABLE X T IP DEFLECTIONS OF A UNIFORMLY LOADED CANTILEVER:  F IN ITE ELEMENT RESULTS K ' 6 V L K ' I n c r emen t = 0.5 K ' I n c r emen t = 1 . 0 K ' I n c r emen t = 1.0 (No l o a d c o r r e c t i o n ) 0 0 0 0 0.5 0.062479 1.0 0.079183 0.124957 0.124957 1.5 0.170092 2 .0 0.181301 0.137298 0.138484 2.5 0.286930 3.0 0.293628 0.312856 0.151783 3.5 0.387183 4 . 0 0.393764 0.316576 0.164853 4 .5 0.480691 5.0 0.485628 0.446118 0.177861 5.5 0.561300 6.0 0.561300 0.456431 0.190308 6.5 0.633140 7.0 0.635942 0.637573 0.202696 7.5 0.695887 8.0 0.697049 0.625872 0.214860 8.5 0.750017 9.0 0.749188 0.697172 . 0.226804 9.5 0.7946041 10.0 0.791740 0.713169 0.238529 148 TIP"DEFLECTIONS OF A UNIFORMLY LOADED CANTILEVER: F IN ITE ELEMENT RESULTS F I G . 26 149 solutions are somewhat e r r a t i c i n the i r incremetal steps. The o s c i l l a -t ions are due to the act ion of the load correct ion term i n t ry ing to prevent the so lu t ion from diverging from equi l ibr ium states. The analysis of th i s problem using increments i n R' of 0 . 5 , appears to be s l i g h t l y less e r r a t i c than that using increments of 1.0 , as should be expected. The rapid divergence of the f i n i t e element analysis using a solut ion procedure without equi l ibr ium checks i s c l e a r l y shown i n F i g . 26, and thus shows the value of using the se l f -cor rec t ing solut ion procedure i n the analysis of nonlinear problems. The f i n i t e element analysis using increments of K ' of 0.5 shows s l i g h t l y greater t i p deflect ions than Holden's closed form solut ion predic t s . In actual fact , an e l a s t i c i t y solut ion i f i t could be found, should predict greater t i p deflect ions due to shear deformation and extension of the can t i l ever , which Holden's solut ion does not consider. For t h i s type of large def lec t ion problem then, the incremental v i r t u a l work equations, the nonuniformly integrated rectangular f i n i t e element, and the se l f -cor rec t ing so lu t ion procedure combine to give an effect ive analysis technique. 150 CONCLUSIONS The analysis of the f i n i t e deformation of li n e a r e l a s t i c bodies has been approached i n t h i s thesis, through an incremental v i r t u a l work formulation using the Lagrangian description. To obtain numerical solutions to the incremental equations derived, the f i n i t e element method was u t i l i z e d with a self-correcting solution technique. The part i c u l a r f i n i t e element used was a nonuniformly numerically integrated, eight degree of freedom, b i l i n e a r rectangle. The a n a l y t i c a l procedure developed, comprised of the incremental v i r t u a l work equations, the f i n i t e element method and the self-correcting solution technique, has been used to obtain numerical solutions for four nonlinear f i n i t e deformation problems. The results of these analyses were compared against available closed form solutions. In a l l four problems, the a n a l y t i c a l procedure developed gave solutions with excellent agreement to their respective closed form solutions. The value of the self-correcting solution technique i n preventing divergence of the incremental analysis from the correct nonlinear path was shown for the case of a uniformly loaded cantilever. From the resu l t s obtained, and their excellent agreement to the closed form solutions, i t i s concluded that the a n a l y t i c a l procedure as advanced i n t h i s thesis i s an effective technique for obtaining the nonlinear f i n i t e deformation response of e l a s t i c bodies. The computer program that was developed to implement t h i s solution technique, w i l l be made available to the Department of C i v i l Engineering, University of B r i t i s h Columbia. 15.1 BIBLIOGRAPHY ZIENKIEWICZ, O.C., "The F i n i t e Element Method i n St r u c t u r a l and  Continuum Mechanics," McGraw-Hill Ltd. London, England, (1967). COOK, R.D., "Concepts and Applications of F i n i t e Element Analysis," J. Wiley and Sons Inc. (1974). ODEN, J.T., " F i n i t e Elements of Nonlinear Continua," McGraw-Hill (1972) . TURNER, M.J., DILL, E.H., MARTIN, H.C., and MELOSH, R.J., "Large Deflections of Structures Subjected to Heating and External Loads," ... Journal of Aerospace Sciences Vol. 27, No. 2 (1960) pp. 97-106, 127. GALLAGHER, R.H., AND PADLOG, J . , "Discrete Element Approach to Stru c t u r a l I n s t a b i l i t y Analysis," American I n s t i t u t e of Aeronautics and Astronautics Journal, Vol. 1, No. 6 (1963) pp. 1437-1439. MARTIN, H.C., "On the Derivation of S t i f f n e s s Matrices f o r the Analysis of Large Deflection and S t a b i l i t y Problems," Proceedings  of the Conference on Matrix Methods i n Structural Mechanics, Wright-Patterson A i r Force Base, Ohio, Oct. 1965, pp. 697-716. MARCAL, P.V., "The E f f e c t of I n i t i a l Displacements on Problems of Large Deflection and S t a b i l i t y , " Report ARPA E54, Nov. 1967, Brown University, D i v i s i o n of Engineering. HIBBIT, H.D., MARCAL, P.V. and RICE, J.R., "A F i n i t e Element Formulation f o r Problems of Large St r a i n and Large Displacement," Technical Report N00014-0007/2, June 1969, Brown University, D i v i s i o n of Engineering. FELIPPA, C.A., and SHARIFI, P., "Computer Inplementation of Nonlinear F i n i t e Element Analysis," Numerical Solution of Nonlinear S t r u c t u r a l Problems, American Society of Mechanical Engineers, New York (1973), pp. 31-50. • SHARIFI, P., and POPOV, E.P., "Nonlinear F i n i t e Element Analysis of Sandwich Shells of Revolution," American I n s t i t u t e of Aeronautics and Astronautics Journal,. Vol. 11, No. 5 (1973) pp. 715-722. YAGHAMI, S., AND POPOV, E.P., "Incremental Analysis of Large Deformations i n Continuum Mechanics with Applications i n E l a s t i c i t y , " International Journal of Solids and Structures, Vol. 7, No. 10 (1971) pp. 1375-1393. -152. 12. YAGHAMI, S., " I n c r e m e n t a l A n a l y s i s o f La r ge D e f o r m a t i o n s i n M e c h a n i c s o f S o l i d s w i t h A p p l i c a t i o n s t o A x i s y m m e t r i c S h e l l s o f R e v o l u t i o n , " NASA CR-1350 , ( 1969 ) . 13. TILLERSON, J . R . , STR ICKL IN, J . A . , AND HAISLER, W.E. , " N u m e r i c a l Methods f o r t h e S o l u t i o n o f N o n l i n e a r P rob lems i n S t r u c t u r a l A n a l y s i s , " N u m e r i c a l S o l u t i o n o f N o n l i n e a r S t r u c t u r a l P r o b l e m s , A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s , New Yo rk ( 1 9 7 3 ) , p p . 6 7 - 1 0 1 . 14. BATHE, K . J . , OZDEMIR, H . , and WILSON, E.L. " S t a t i c and Dynamic G e o m e t r i c and M a t e r i a l N o n l i n e a r A n a l y s i s , " R e p o r t No. UCSESM 74 -4 , Feb. 1974, S t r u c t u r a l E n g i n e e r i n g L a b o r a t o r y , U n i v e r s i t y o f C a l i f o r n i a ( B e r k e l e y ) . 15. ZUDANS, Z . , REDDI, M.M., and TSA I , H . , "DYPLAS, A F i n i t e E l ement Dynamic E l a s t i c - P l a s t i c La rge D e f o r m a t i o n A n a l y s i s P r o g r a m , " N u c l e a r E n g i n e e r i n g and D e s i g n , V o l . 27 (1974) p p . 398 -412 . 16. FUNG, Y . C . , " F o u n d a t i o n o f S o l i d M e c h a n i c s , " P r e n t i c e - H a l l ( 1 965 ) . 17. IB ID, p p . 4 40 - 441 . 18. STR ICKL IN, J . A . , HAISLER, W.E . , and VON RIESEMANN, W.A., " E v a l u a t i o n o f S o l u t i o n P r o c e d u r e s f o r M a t e r i a l a nd/o r G e o m e t r i c a l l y N o n l i n e a r S t r u c t u r a l A n a l y s i s , " A m e r i c a n I n s t i t u t e o f A e r o n a u t i c s and  A s t r o n a u t i c s J o u r n a l , V o l . 11 , No. 3 (1973) p p . 292 -299 . 19. IB ID. 20. MELOSH, R . J . , " B a s i s f o r D e r i v a t i o n o f M a t r i c e s f o r t h e D i r e c t S t i f f n e s s M e t h o d , " A m e r i c a n I n s t i t u t e o f A e r o n a u t i c s and A s t r o n a u t i c s  J o u r n a l , V o l . 1, No. 7 ( 1 963 ) , p p . 1631-1637. 21. ODEN, J . T . , " A G e n e r a l Theo r y o f F i n i t e E l e m e n t s , I. T o p o l o g i c a l C o n s i d e r a t i o n s , " I n t e r n a t i o n a l J o u r n a l o f N u m e r i c a l Methods i n  E n g i n e e r i n g , V o l . 1 ( 1 969 ) , p p . 205 -221 . 22. OL IVE IRA, E.R. de ARANTES e , " C o m p l e t e n e s s and Conve rgence i n t h e F i n i t e E l ement M e t h o d , " P r o c e e d i n g s o f t h e 2nd C o n f e r e n c e on M a t r i x  Methods i n S t r u c t u r a l M e c h a n i c s , W r i g h t - P a t t e r s o n A i r F o r c e Ba se , O h i o , (1968) p p . 1061-1090. 23. OLIVEIRA, E.R. de ARANTES e, " T h e o r e t i c a l F o u n d a t i o n s o f t h e ^ F i n i t e E l ement M e t h o d , " I n t e r n a t i o n a l J o u r n a l o f S o l i d s and S t r u c t u r e s , V o l . 4 ( 1 9 6 8 ) , p p . 929 -952 . 24. MELOSH, R . J . , " B a s i s f o r D e r i v a t i o n o f M a t r i c e s f o r t h e D i r e c t S t i f f n e s s M e t h o d , " A m e r i c a n I n s t i t u t e o f A e r o n a u t i c s and A s t r o n a u t i c s  J o u r n a l , V o l . 1, No. 7 ( 1 9 6 3 ) , pp. 1631^1637. 153 25. OLSON, M.D., " C o m p a t a b i l i t y ( o f F i n i t e E l emen t s i n S t r u c t u r a l M e c h a n i c s ) , " P a p e r p r e s e n t e d a t t h e W o r l d Cong re s s on F i n i t e E l ement Methods i n S t r u c t u r a l M e c h a n i c s , Bournemouth, D o r s e t , E n g l a n d , O c t o b e r 1 2 - 1 7 , 1975. ( R e p r i n t e d as S t r u c t u r a l R e s e a r c h S e r i e s R e p o r t No. 12, Depar tment o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Co lumbia, O c t o b e r 1975.) 26. IB ID 27. McLay, R.W., " C o m p l e t e n e s s and Conve rgence P r o p e r t i e s o f F i n i t e E l ement D i s p l a c e m e n t F u n c t i o n s - A G e n e r a l T r e a t m e n t , " A m e r i c a n  I n s t i t u t e o f A e r o n a u t i c s and A s t r o n a u t i c s P a p e r N o . ' 6 7 - 1 4 3 , . N e w Yo r k New Y o r k ( 1 967 ) . 28 . STRANG, G . , F IX , G . F . , " A n A n a l y s i s o f t h e F i n i t e E l ement M e t h o d , " P r e n t i c e - H a l l ( 1 973 ) . 29. MELOSH, R . J . " B a s i s f o r D e r i v a t i o n o f M a t r i c e s f o r t h e D i r e c t S t i f f n e s s M e t h o d , " A m e r i c a n I n s t i t u t e o f A e r o n a u t i c s and A s t r o n a u t i c s J o u r n a l , V o l . 1, No. 7 ( 1 9 6 3 ) , p p . 1631-1637. 30. COOK, R.D., " C o n c e p t s and A p p l i c a t i o n s o f F i n i t e E l emen t A n a l y s i s , " J . W i l e y and Sons I n c . (1974) p p . 103 -105 . 31. ZIENKIEWICZ, O .C . , " The F i n i t e E l ement Method i n S t r u c t u r a l and  Con t i nuum M e c h a n i c s " M c G r a w - H i l l L t d . , London,. E n g l a n d (1967) p p . 144-152. 32. COOK, R.D., " C o n c e p t s and A p p l i c a t i o n s o f F i n i t e E l ement A n a l y s i s , " J . W i l e y and Sons I n c . (1974) pp . 131 -135 . •3'3.. TIMOSHENKO, S., WOINOWSKY-KRIEGER, S., " T h e o r y o f P l a t e s and S h e l l s , " McGraw - H i l l L t d . , L o n d o n , E n g l a n d (1959) pp . 6 - 12 . 34. IB ID pp . 6 -20 35 . IB ID pp . 118-131 36. ROJAHN, C . , " L a r g e D e f l e c t i o n s o f E l a s t i c Beams , " t h e s i s f o r t h e d e g r e e o f E n g i n e e r , S t a n f o r d U n i v e r s i t y , June 1968. 37. HOLDEN, J . T . , "On t h e F i n i t e D e f l e c t i o n s o f T h i n Beams , " I n t e r n a t i o n a l  J o u r n a l o f S o l i d s and S t r u c t u r e s , V o l . 8, No. 8 (1972) pp . 1051 - 1055 . 38 . BATHE, K - J . , HALUK, 0 . , WILSON, E . L . , . " S t a t i c and Dynamic G e o m e t r i c and M a t e r i a l N o n l i n e a r A n a l y s i s , " R e p o r t No. UC SESM 7 4 - 4 , F e b r u a r y 1974, U n i v e r s i t y o f C a l i f o r n i a ( B e r k e l e y ) , S t r u c t u r a l E n g i n e e r i n g L a b o r a t o r y , p p . 117 -120 . 154 APPENDIX. A - THREE DIMENSIONAL ANALYSIS OPERATOR MATRICES In s e c t i o n 6 . 3 . 2 , use was made o f t h r e e d i f f e r e n t o p e r a t o r m a t r i c e s t h a t b e c a u s e o f t h e i r s i z e were n o t shown t h e r e . These o p e r a t o r m a t r i c e s w i l l be shown i n t h i s a p p e n d i x , and a r e l a t i o n s h i p w i l l be d e v e l o p e d be tween them t o r e d u c e t h e c o m p u t a t i o n a l e f f o r t r e q u i r e d . F i r s t , t h e o p e r a t o r m a t r i x [B^] i s r e q u i r e d t o o p e r a t e on t he v e c t o r ( u ) t o g ive{e. } as shown i n Eq . 6 . 12 , and t h e e l emen t s o f {e } a r e JLi JLi d e f i n e d as i n Eq . 6 . 9 . U s i n g t h e s e r e q u i r e d r e l a t i o n s h i p s t h e n , 8ai 3ai 3a 2 3ac 3ai 3a^ 3ai 3a 2 3a 3 3ac 3 a 2 3a] 3a, da-. '3a. (A. S e c o n d l y , t he o p e r a t o r m a t r i x [ B ^ ( l u ) ] i s r e q u i r e d t o o p e r a t e on t h e v e c t o r {u} t o g i v e { e ^ } as shown i n Eq . 6 . 13 , where t h e e l emen t s o f { e ^ } a r e d e f i n e d by Eq . 6 .10 . T h e r e f o r e , [ B ^ L ( 1 u ) ] i s g i v e n i n m a t r i x f o r m , on t h e f o l l o w i n g page , as 155 ro rt rO co 3 ro d co CM 3 CM rt ro ro r O CM ro 3 Osl CM CO ro CD r O rt CO rt CO rt CD ro CD CO t—v / — \ f—\ f—s r—1 ro i—i CM ** «\ •* CO CM CO ro CO CD rt 3 3 3 r-H ro i—i l—1 / — \ ^ — ' \ J Osl H •n + + + CO I—1 ro r-H CM CD 1—I CD rt CD rt CO rt CD s — ' CO CO CO / — \ CM l V t—% ,—* rt co CO 3 CM 3 ro rt ro CM 3 CM rt ro TM CM ro ro CO rt ro rt ro rt ro rt ro rt CO ro ro ro ro • — \ , — v CO ro —t CM *\ ^ CM CM CM- - CM CM CM 3 3 ro rt 3 3 3 1—1 •—1 ro i—i ,—i 1 — ' * — ' f—\ - / - 1 f + 'CM + + + CM ro i—i 3 ro i—1 CM <ro rt ro rt i—i ro rt ro rt ro rt CO ro v * ro ro ro /—\ ' — » CM ,—> r-rt ro CM 3 CM 3 CM 3 CM 3 CM 3 CM 3 CM 3 ro rt ro ro rt ro CM rt ro CM rt ro ro rt ro CM i-H 3 CM rt ro rt ro CM rt ro rt ro rt ro ro rt CM rt ro I |CN 2 OQ 156 T h i r d l y , t h e o p e r a t o r m a t r i x [ L ] , as d e f i n e d i n E q . 6.35 i s g i v e n by 3 8a ! 3 3 a 2 3 3 a 3 0 0 0 0 0 0 U ] T = 0 0 0 3 3a] 3 3 a 2 3 3 a 3 0 0 0 (A 0 0 0 0 0 0 3 3 a ! 3 3 a 2 3 3 a 3 F i n a l l y , t h e m a t r i x o f i n i t i a l s t r e s s e s as r e q u i r e d by t he r e l a t i o n s h i p i n E q . 6.36 i s g i v e n by [XST] lsl2 'Sis 0 0 0 0 0 0 X S 2 1 ^ 2 2 0 0 0 0 0 0 ^ 3 1 x s s L s 3 3 0 0 0 0 0 0 0 0 0 1 Sn X S l 2 0 0 0 0 0 0 ^ 1 1 s 2 2 1 S 2 3 0 0 0 (A 0 0 0 ^31 ^33 0 0 0 0 0 0 0 0 0 xsn lSu 0 0 0 0 0 0 1 s 2 2 ^ 2 3 0 0 0 0 0 0 ^31 1 S 3 2 XS33 Now r a t h e r t h a n c a l c u l a t e a l l t h e o p e r a t o r m a t r i c e s s e p a r a t e l y , r e l a t i o n s h i p s w i l l be e s t a b l i s h e d be tween them i n o r d e r t o a v o i d a d u p l i c a t i o n o f e f f o r t . B e g i n n i n g w i t h t h e o p e r a t o r [ L ] , t h e o p e r a t o r [B ] may be o b t a i n e d t h r o u g h t h e u se o f a t r a n s f o r m a t i o n m a t r i x [ A ] , t h a t i s M M-MMW 157 (A.5) where the matrix J~AJ is defined as 2 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 • 0 1 0 0 0 0 0 ' -0 0 0 0 2 (A.6) Advantage may be gained by use of the fact that [A] is symmetric about i t s major diagonal, in performing the required matrix mult ipl icat ion. Next, the operator [ B N L ( l u ) ] m a Y be readily obtained from the result of Eq. A.5 , through:':the use of the matrix of i n i t i a l derivatives, [ l u i j] defined in Eq. 6.51 for use in the Kirchhoff surface traction is obtained from the matrix product integral . The operator below, BNL< l u> Vlu>]M- M l>] 1»J (A. 7) Thus, beginning with the operator [ L ] , the other two required operators [ B l ] and [ B ^ U ) ] are obtained in sequence using the transformation matrix [A] and then the matrix of i n i t i a l derivatives 158 [ 1 u . . ] . T h i s a pp r oach s h o u l d r e d u c e t h e c o m p u t a t i o n a l e f f o r t r e q u i r e d i n o b t a i n i n g [B^] and [ B ^ ^ ( 1 u ) ] , by a v o i d i n g an u n n e c e s s a r y amount o f d u p l i c a t i o n o f c a l c u l a t i o n s i n f o r m i n g t h e s e two o p e r a t o r s . 159 APPENDIX B - TWO DIMENSIONAL ANALYSIS OPERATOR MATRICES In s e c t i o n 6 . 4 . 2 , t h e o p e r a t o r m a t r i c e s [L~], [B~, ] and [ B ~ N L ( 1 u ) ] as w e l l as t h e m a t r i x o f i n i t i a l s t r e s s e s [ X ST ] f o r two d i m e n s i o n a l a n a l y s i s were d e f i n e d . They were n o t shown f u l l y i n t he ma in t e x t due t o t h e i r s i z e b u t a r e g i v e n i n f u l l i n t h i s a p p e n d i x . I n a d d i t i o n r e l a t i o n s h i p s w i l l be e s t a b l i s h e d between t h e t h r e e o p e r a t o r m a t r i c e s i n an a n a l o g o u s manner t o t h o s e o f A p p e n d i x A f o r t h r e e d i m e n s i o n a l a n a l y s i s . The o p e r a t o r m a t r i x [B^] i s g i v e n by r ^ 3ai 9 9a2 0 9a 2 9ai (B n e x t [B ^ u ) ] i s g i v e n by [ B ^ u ) ] = ( ^ 2 , 2 ) 9a 2 9 ^ ' + ' ° U 2 ^ 917 (B 160 and f i n a l l y [L] i s d e f i n e d as [L] = 3 a x 3 3 a 2 • 0 0 3 a x 3 3 a 9 (B. 3) P r o c e e d i n g i n an a n a l a g o u s manner t o A p p e n d i x A , i t may be s i m p l y shown t h a t [ B L ] [N ] = [A] [L] [N] (B. 4) where [A] = 1 0 0 0 0 0 0 1 0 1 1 0 (B. 5) S i m i l a r l y [ B J J L ^ U J ] [N] = [B ] [N] l u 2 , l L l u l , 2 ^ 2 , 2 (B. 6) The r e s u l t o f Eq . B. 6 may be a l t e r n a t e l y , and f o r c o m p u t a t i o n a l pu rpose s , more c o n v e n i e n t l y , o o b t a i n e d by 161 [ B J ^ U J ] [N] ^1,1 0 1 u 2 } i 0 0 ^ 1 ^ 2 0 ^ 2 , 2 ^ 1 , 2 ^ 1 , 1 ^ 2 , 2 ^ 2 , 1 [L] [N] (B. 7) Final ly the matrix of i n i t i a l stresses is given by i^ ST] ^ 1 2 0 X S i 2 ^ 2 2 0 0 0 0 0 0 0 lsn ls12 l S i 2 ^ 2 2 (B. 8) 

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