UBC Theses and Dissertations

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UBC Theses and Dissertations

Plastic postbuckling behaviour of axially compressed cylindrical shells Tang, Tony K.P. 1974

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P L A S T I C POSTBUCKLING BEHAVIOUR OF AXIALLY COMPRESSED CYLINDRICAL SHELLS by TONY K.P. TANG B.S., U n i v e r s i t y o f M i s s o u r i - R o l l a , 1972. A THESIS SUBMITTED I N PARTIAL FULFILMENT OF MASTER OF APPLIED SCIENCE i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d . THE UNIVERSITY OF B R I T I S H COLUMBIA MAY 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l 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 a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d 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 H e a d o f my D e p a r t m e n t 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 . D e p a r t m e n t 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 8 , C a n a d a i ABSTRACT T h e o r e t i c a l a n a l y s i s i s p r e s e n t e d t o i n v e s t i g a t e t h e p l a s t i c p o s t - b u c k l i n g b e h a v i o u r o f a t h i n c y l i n d r i c a l s h e l l r e s t r a i n e d by a c e n t r a l m a n d r e l u n d e r a x i a l c o m p r e s s i v e l o a d i n g . A x i s y m m e t r i c b u l g e s a r e f o r m e d . W o r k - h a r d e n i n g i s n e g l e c t e d . The T r e s c a s a n d w i c h - s h e l l y i e l d c o n d i t i o n i s u s e d , and i t i s f o u n d t h a t b u l g i n g i s most s u i t a b l y d e s c r i b e d by a t h r e e - h i n g e m o d e l . The b u l g e l e n g t h o f t h e c o l l a p s e mode c a n be p r e d i c t e d a p p r o x i m a t e l y by u s i n g t h e u p p e r - b o u n d t h e o r e m . A g r e e m e n t w i t h t h e t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s i s s a t i s f a c t o r y . i i TABLE OF CONTENTS CHAPTER PAGE I INTRODUCTION 1 I I GENERAL THEORY 1 2 . 1 T h i n - S h e l l T h e o r y 4 2 . 2 T r e s c a S a n d w i c h - S h e l l y i e l d C r i t e r i o n 6 2 . 3 U p p e r - B o u n d Theorem o f P l a s t i c i t y 6 I I I THEORETICAL SOLUTION 8 3 . 1 T h r e e - H i n g e M o d e l 10 3 . 2 F u r t h e r A n a l y s i s f o r T h r e e - H i n g e Model 1 5 3 . 3 F i v e - H i n g e M o d e l and S e v e n - H i n g e M o d e l 17 I V EXPERIMENTAL RESULTS AND DISCUSSION 24 V REMARKS AND CONCLUSIONS 2 7 BIBLIOGRAPHY 2 8 APPENDIX 2 9 i i i L I S T OF FIGURES FIGURE PAGE 1 A x i s y m m e t r i c B u l g e s f o r m e d w i t h C e n t r a l M a n d r e l 3 7 2 S h e l l E l e m e n t u n d e r S y m m e t r i c a l L o a d i n g and B e n d i n g 3 8 3 C o m p a r i s o n o f t h e M i d d l e - S u r f a c e G e o m e t r y f o r C y l i n d r i c a l C o n i c a l S h e l l s 3 9 4 S a n d w i c h - S h e l l E l e m e n t 4 0 5 S t r e s s - S t r a i n D i a g r a m f o r R i g i d , P e r f e c t l y P l a s t i c M a t e r i a l 4 1 6 T r e s c a Y i e l d L o c u s f o r P l a n e S t r e s s 4 2 7 T h r e e - H i n g e M o d e l 4 3 8 D i m e n s i o n l e s s L o a d (P/cr 0) v e r s u s D i m e n s i o n l e s s D i s p l a c e m e n t o f T h r e e - H i n g e M o d e l 4 4 9 F a m i l y o f C u r v e s f o r D i m e n s i o n l e s s Load (P/o^) v e r s u s D i m e n s i o n l e s s D i s p l a c e m e n t (A) f c r T h r e e - H i n g e M o d e l 4 5 1 0 D i m e n s i o n l e s s L o a d ( P / c r 0 ) v e r s u s M e r i d i o n a l E x t e n s i o n a l S t r a i n R a t e a t t h e M i d d l e S u r f a c e tp\ ) f o r T h r e e - H i n g e M o d e l 4 6 1 1 F i v e - H i n g e M o d e l 4 7 1 2 D i m e n s i o n l e s s L o a d (P/cr0) v e r s u s f o r F i v e -H i n g e M o d e l 2 4 8 1 3 S e v e n - H i n g e M o d e l 4 9 1 4 D i m e n s i o n l e s s L e a d ( P / aQ) v e r s u s and QS f o r S e v e n - H i n g e M o d e l 3 5 0 • • 1 5 D i m e n s i o n l e s s l o a d (P/cr 0) v e r s u s <f> and </> f o r S e v e n - H i n g e M o d e l 5 1 • • 1 6 D i m e n s i o n l e s s l o a d ( P / c r 0 ) v e r s u s <£ and <P 3 f c r S e v e n - H i n g e M o d e l . 5 2 1 7 D i m e n s i o n s f o r a T y p i c a l U n i f o r m - W a l l S p e c i m e n 5 3 i v 18 E x p e r i m e n t a l A p p a r a t u s f o r C o m p r e s s i o n T e s t 54 19 U n i f o r m - W a l l S p e c i m e n s a f t e r C o m p r e s s i v e L o a d i n g w i t h C e n t r a l M a n d r e l i n p l a c e 5 5 20 A p p l i e d Load v e r s u s V e r t i c a l D i s p l a c e m e n t ( U n i f o r m - W a l l S p e c i m e n ) 5 6 21 C o m p a r i s o n b e t w e e n T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s 5 7 22 D i m e n s i o n s o f a T y p i c a l N o n - D n i f o r ro-Wa11 S p e c i m e n 5 8 23 N o n - U n i f o r m - W a l l S p e c i m e n s a f t e r C o m p r e s s i v e L o a d i n g w i t h o u t C e n t r a l M a n d r e l 5 9 24 A p p l i e d L o a d v e r s u s V e r t i c a l D i s p l a c e m e n t (Non-U n i f o r m - W a l l S p e c i m e n w i t h o u t C e n t r a l M a n d r e l ) 60 25 R e l a t i v e C o n t r i b u t i o n o f B e n d i n g and S t r e t c h i n g o f M i d d l e S u r f a c e t o t h e T o t a l I n t e r n a l Power D i s s i p a t i o n a s a F u n c t i o n o f t h e D i m e n s i o n l e s s D i s p l a c e m e n t ( A ) f o r T h r e e - H i n g e M o d e l 61 A.1 R e f e r e n c e F r a m e s o f S e v e n - H i n g e M o d e l 62 V N O M E N C L A T U R E r v r = d i s t a n c e from the a x i s to a p o i n t at the middle 0 s u r f a c e o f a s h e l l of r e v o l u t i o n r ( = r a d i u s of c u r v a t u r e of a s h e l l of r e v o l u t i o n 2 = l e n g t h measured on a normal to the meridian between the i n t e r s e c t i o n with the a x i s of r o t a t i o n and the middle s u r f a c e y = d i s t a n c e from vertex measured along cone generator 2 = d i s t a n c e from s h e l l middle s u r f a c e , p o s i t i v e inward 8 - angular c o o r d i n a t e around c i r c u m f e r e n c e = middle s u r f a c e displacement i n the d i r e c t i o n tangent to the meridian of the s h e l l w = middle s u r f a c e displacement i n the d i r e c t i o n normal to the meridian of the s h e l l , p o s i t i v e inward 2h = t h i c k n e s s of sandwich s h e l l 2H = t h i c k n e s s of uniform s h e l l t = t h i c k n e s s of cover sheets i n sandwich s h e l l a = r a d i u s o f c y l i n d r i c a l s h e l l , from a x i s to middle s u r f a c e p = t y p i c a l p o i n t a t the middle s u r f a c e = angle between the normal to the meridian and the a x i s of r o t a t i o n M s , My = bending moments per u n i t l e n g t h N e , N y = membrane f o r c e s per u n i t l e n g t h E e f £y = s t r a i n components cr e , c r y = s t r e s s e s P = e x t e r n a l s t r e s s due to a p p l i e d lead M 0 = bending moment a t the y i e l d hinge = y i e l d s t r e s s A = d i m e n s i o n l e s s displacement d = power d i s s i p a t i o n per u n i t volume D = power d i s s i p a t i o n per u n i t area of middle s u r f a c e W = power of d i s s i p a t i o n L = h a l f - b u l g e l e n g t h L , , L 2 = le n g t h s o f l i n k s of f i v e - h i n g e model 1 = l e n g t h of l i n k s of seven-hinge model V( j = v e l o c i t y of p o i n t { ) (') = -©( ) A>t # where t i s time ( )' '= a ( ) / » Y ( ) L = s u p e r s c r i p t i i d e n t i f i e s q u a n t i t y a s s o c i a t e d with i n t e r i o r sheet of sandwich s h e l l ( f = s u p e r s c r i p t e i d e n t i f i e s q u a n t i t y a s s o c i a t e d with e x t e r i o r sheet of sandwich s h e l l V i ACKNOWLEDGEMENTS The a u t h o r would l i k e t o t h a n k D r . H. Ramsey 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 a c o u r s e o f t h i s r e s e a r c h . The c o m p u t e r f a c i l i t i e s o f t h e C o m p u t i n g C e n t e r o f 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 were u s e d t o do t h e c a l c u l a t i o n s c o n t a i n e d h e r e i n . T h i s r e s e a r c h was s u p p o r t e d by 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 and 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. 1 • CHAPTER I The o b j e c t i v e o f t h i s i n v e s t i g a t i o n i s t o a n a l y s e t h e p e r f e c t l y p l a s t i c b e h a v i o u r o f t h i n c y l i n d r i c a l s h e l l s r e s t r a i n e d by a c e n t r a l m a n d r e l u n d e r a x i a l c o m p r e s s i o n i n t h e p o s t - b u c k l i n g r a n g e , and t o f o r m u l a t e a m a t h e m a t i c a l m o d e l t o d e s c r i b e t h e power d i s s i p a t i o n d u r i n g t h i s p l a s t i c d e f o r m a t i o n . The p l a s t i c b u c k l i n g b e h a v i o u r o f c y l i n d r i c a l s h e l l s s u b j e c t t o an a x i a l c o m p r e s s i v e l o a d h a s been e x t e n s i v e l y d i s c u s s e d i n t h e p a s t . T h e o r y a nd e x p e r i m e n t a l r e s u l t s a r e a v a i l a b l e i n r e f e r e n c e s (1) and (2), f o r e x a m p l e . R e c e n t l y , Shaw(3) b r i e f l y c o n s i d e r e d t h e p l a s t i c p o s t - b u c k l i n g b e h a v i o u r o f a t h i n c y l i n d r i c a l s h e l l r e s t r a i n e d by a c e n t r a l m a n d r e l u n d e r a x i a l c o m p r e s s i o n . U n d er s u c h l o a d i n g , a x i s y m m e t r i c b u l g e s a r e f o r m e d ( F i g u r e 1 ) . W i t h o u t t h e s t a b i l i z i n g m a n d r e l , t h i n c y l i n d r i c a l s h e l l s b u c k l e a s y m m e t r i c a l l y ( 4 ) , and t h e r e a r e s e v e r a l p o s s i b l e b u c k l i n g modes d i f f e r e n t f r o m t h e p a r t i c u l a r mode s t u d i e d h e r e . Shaw(3) s u g g e s t e d u s i n g t h e e f f e c t o f a x i s y r a m e t r i c b u c k l i n g f o r e n e r g y - a b s o r b i n g d e v i c e s . I n h i s a n a l y s i s , he v i s u a l i z e d t h e a x i s y m m e t r i c b u l g i n g o f t h e s h e l l a s t h e b u c k l i n g o f a s l e n d e r c o l u m n w i t h s i m p l y s u p p o r t e d ends w h i c h a r e c o n s t r a i n e d t o r e m a i n v e r t i c a l l y a l i g n e d . The b u c k l i n g of t h e s h e l l w h i c h i s 2 s l o w l y l o a d e d on a m a n d r e l i n v o l v e s E u l e r i n s t a b i l i t y , t h e f o r m a t i o n and c o l l a p s e o f a p l a s t i c h i n g e , and s t r e t c h i n g o f t h e m i d d l e s u r f a c e u n d e r hoop s t r e s s . F u r t h e r m o r e , Shaw (3) d e v e l o p e d a n e q u a t i o n t o p r e d i c t t h e a p p l i e d l o a d by e q u a t i n g t h e work done by t h e v a r i a b l e a p p l i e d l o a d t o t h e i n t e r n a l work done clue t o b e n d i n g a t t h e h i n g e s and t h e s t r e t c h i n g o f m a t e r i a l due t o t h e c h a n g e i n d i a m e t e r . H o w e v e r , t h e d e r i v a t i o n o f t h i s e q u a t i o n c o n t a i n s a number o f a p p r o x i m a t i o n s . I n o r d e r t o make t h e a n a l y s i s more e x a c t and t o e x t e n d t h e a n a l y s i s , a d i f f e r e n t a p p r o a c h t o t h e a x i s y m m e t r i c b u l g i n g o f t h e s h e l l h a s b e e n t a k e n . The c o n c e p t o f t h e f o r m a t i o n o f p l a s t i c h i n g e s i s a d o p t e d . The s t u d y i s b a s e d on t h i n - s h e l l t h e o r y , t h e T r e s c a s a n d w i c h - s h e l l y i e l d c r i t e r i o n , t h e u p p e r -bound t h e o r e m o f p l a s t i c i t y , and t a k e s a c c o u n t o f g e o m e t r y c h a n g e s ( 5 ) . A s i m i l i a r a p p r o a c h was t a k e n i n t h e a n a l y s i s o f . t h e b e h a v i o u r o f t h e a x i s y m m e t r i c d e f o r m a t i o n o f a s p h e r i c a l s h e l l p r e s e n t e d b y U p d i k e ( 6 ) . H i s a n a l y s i s w h i c h i n v o l v e s d e t e r m i n i n g t h e v e l o c i t y f i e l d a c c o u n t s f o r l a r g e d e f l e c t i o n s by a p p r o x i m a t i n g t h e a c t u a l d e f o r m e d s h a p e o f t h e s h e l l by a s i m p l e g e o m e t r i c s h a p e . The T r e s c a y i e l d c o n d i t i o n i s e m p l o y e d . I n t h e p r e s e n t c a s e , a m o d el w i t h t h r e e p l a s t i c h i n g e s i s c o n s i d e r e d f i r s t . The a n a l y s i s i s t h e n e x t e n d e d t o i n c l u d e f i v e and s e v e n -h i n g e m o d e l s . A l s o , f o r t h e t h r e e - h i n g e m o d e l , t h e b u l g e l e n g t h i s p r e d i c t e d u s i n g t h e u p p e r - b o u n d t h e o r e m ; and t h e p r e s e n c e o f e x t e n s i o n a l s t r a i n r a t e i n t h e m e r i d i o n a l d i r e c t i o n a t t h e m i d d l e s u r f a c e o f t h e b u l g e i s e x a m i n e d . S e v e r a l d i f f e r e n t 3 a s s u m p t i o n s have been c o n s i d e r e d i n c o n s t r u c t i n g t h e m a t h e m a t i c m o d e l , and t h e i r i m p o r t a n c e c o m p a r e d . Some s i m p l e e x p e r i m e n t s were made t o c h e c k t h e v a l i d i t y o f t h i s m o d e l . T h e r e i s s a t i s f a c t o r y a g r e e m e n t . "4 CHAPTER I I GENEfiAL THEORY T h e r e a r e t h r e e b a s i c t h e o r e t i c a l c o n s i d e r a t i o n s e m p l o y e d i n t h i s r e s e a r c h , n a m e l y , t h i n - s h e l l t h e o r y , t h e T r e s c a s a n d w i c h - s h e l l y i e l d c r i t e r i o n and t h e u p p e r - b o u n d t h e o r e m o f p l a s t i c i t y . A number o f a s s u m p t i o n s a r e a l s o i n v o l v e d ; some o f t h e s e a r e i n c l u d e d i n t h e d e v e l o p m e n t o f t h e t h e o r i e s u s e d , a nd some a r e i n t r o d u c e d f o r 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 c o n s i d e r e d . 2.1 T h i n - S h e l l T h e o r y (7) I n t h e c a s e o f a x i s y m m e t r i c a l d e f o r m a t i o n o f a t h i n s h e l l h a v i n g t h e f o r m o f a s u r f a c e r e v o l u t i o n , a s m a l l d i s p l a c e m e n t o f a p o i n t a t t h e m i d d l e s u r f a c e c a n be r e s o l v e d i n t o two c o m p o n e n t s : v . i n t h e d i r e c t i o n o f t h e t a n g e n t t o t h e m e r i d i a n a n d w i n t h e d i r e c t i o n o f t h e n o r m a l t o t h e m i d d l e s u r f a c e ( F i g u r e 2 ) . The c i r c u m f e r e n t i a l s t r a i n EQ and m e r i d i o n a l s t r a i n £y p r o d u c e d by membrane s t r e s s e s and b e n d i n g moments c a n be w r i t t e n i n t e r m s o f t h e d i s p l a c e m e n t s (v,w) a s (2.1.1) 5 e = _ L d v _ _ w _ _ z _ A ( i d w + J i L ) . (2.1.2) y r 1 do> r 1 dcf> d<p A n a x i s y m m e t r i c b u l g e i s now c o n s i d e r e d w h i c h i s f o r m e d u n d e r a x i a l c o m p r e s s i o n i n a t h i n c y l i n d r i c a l s h e l l . T h e u p p e r p o r t i o n o f t h e b u l g e c a n b e r e p r e s e n t e d b y a c o n i c a l s h e l l ( 8 ) b e c a u s e o f t h e i r g e o m e t r i c a l s i m i l a r i t y ( F i g u r e 3 ) . T h e s t r a i n -d i s p l a c e m e n t r e l a t i o n s f o r a c o n i c a l s h e l l c a n b e o b t a i n e d f r o m (2.1.1) a n d ( 2 . 1 . 2 ) b y t a k i n g t h e l i m i t s °°» $ constant , dcj) 0 , a n d s u b s t i t u t i n g r 1 dcj> = dy, c o s j ) = 1 C o t < j > = — • ( 2 . 1 . 3 ) r y' y T h e s t r a i n - d i s p l a c e m e n t r e l a t i o n s o b t a i n e d a r e e 0 = (v - w tan <J> - z ^ ) / y , ( 2 , 1 . 4 ) dv d ,dw. ,„ .. c . E = - z — (—) . (2.1.5) y dy dy dy T h e s t r a i n r a t e s a r e o b t a i n e d b y t a k i n g t h e i r d e r i v a t i v e s w i t h r e s p e c t t o t i m e , 6 £ f i = (v - w tan (j) - zw')/y , (2.1.6) (2.1.7) where v 1 = d v / d y and w' = dw/dy. 2.2 T r e s c a S a n d w i c h - S h e l l Y i e l d C r i t e r i o n (9) The T r e s c a s a n d w i c h - s h e l l e l e m e n t c o n s i s t s o f two c o v e r s h e e t s o f e g u a l t h i c k n e s s j o i n e d by a l a y e r t h a t c a r r i e s o n l y s h e a r and no a x i a l s t r e s s e s o r b e n d i n g moments ( F i g u r e 4 ) . The c o v e r s h e e t s a r e t r e a t e d a s b e i n g i n f i n i t e l y t h i n , s o t h e v a r i a t i o n s o f s t r e s s a n d s t r a i n e v e r t h e t h i c k n e s s c a n be n e g l e c t e d . 2.3 U p p e r - B o u n d Theorem o f P l a s t i c i t y The s t r e s s - s t r a i n d i a g r a m f o r a r i g i d , p e r f e c t l y p l a s t i c m a t e r i a l i s shown i n F i g u r e 5. When t h e a b s o l u t e v a l u e o f t h e u n i a x i a l s t r e s s cr i s b e l o w t h e y i e l d s t r e s s cr^, t h e s p e c i m e n i s r i g i d ; when t h e a b s o l u t e v a l u e o f cr e q u a l s cr0f t h e s p e c i m e n i s c a p a b l e o f p l a s t i c f l o w u n d e r c o n s t a n t s t r e s s . 7 To e s t a b l i s h a n u p p e r - b o u n d f o r t h e a p p l i e d l e a d , i t i s n e c e s s a r y t o u s e a • K i n e m a t i c a l l y A d m i s s i b l e V e l o c i t y F i e l d * d e f i n e d by P r a g e r ( 9 ) . S u c h a f i e l d i n a r i g i d , p e r f e c t l y p l a s t i c c o n t i n u u m i s one w h i c h o b e y s a l l t h e c r i t e r i a o f p l a s t i c f l o w , t h e c o n d i t i o n o f i n c o m p r e s s i b i l i t y and t h e b o u n d a r y c o n d i t i o n s on t h e v e l o c i t i e s . The power o f t h e e x t e r n a l l o a d We i s s m a l l e r o r e q u a l t o t h e power d i s s i p a t e d i n t e r n a l l y N-L : w < w, e - 1 (2.3.1) 8 CHAPTER I I I THEORETICAL SOLUTION D u r i n g b e n d i n g , most o f t h e work c f t h e e x t e r n a l l o a d i s d i s s i p a t e d i n p l a s t i c d e f o r m a t i o n , r a t h e r t h a n s t o r e d a s e l a s t i c s t r a i n e n e r g y . I n o r d e r t o s i m p l i f y t h e a n a l y s i s s t i l l f u r t h e r , t h e e f f e c t o f work h a r d e n i n g i s n e g l e c t e d . T h e s e a s s u m p t i o n s c a n The a c t u a l y i e l d c o n d i t i o n f o r most d u c t i l e m a t e r i a l s i s g e n e r a l l y c o n s i d e r e d t o be w e l l a p p r o x i m a t e d by t h e Von M i s e s y i e l d c o n d i t i o n o f maximum o c t a h e d r a l s t r e s s . H o w e v e r , t h e T r e s c a y i e l d c o n d i t i o n b a s e d on maximum s h e a r s t r e s s i s a r e a s o n a b l y good a p p r o x i m a t i o n t o r e a l i t y and f r e q u e n t l y l e a d s t o somewhat s i m p l e r m a t h e m a t i c a l c a l c u l a t i o n s b e c a u s e i t i s p i e c e -w i s e l i n e a r . A p p l i c a t i o n o f t h e T r e s c a y i e l d c o n d i t i o n i s p a r t i c u l a r l y s i m p l e when t h e p r i n c i p a l d i r e c t i o n s ( c i r c u m f e r e n t i a l and m e r i d i o n a l ) a r e f i x e d a n d known a s t h e y a r e f o r r o t a t i o n a l l y s y m m e t r i c s h e l l s . I n t h e T r e s c a s a n d w i c h - s h e l l , t h e r e s u l t a n t membrane f o r c e s ( N er Nij ) and b e n d i n g moments ( M 3 , M y ) a r e r e l a t e d t o t h e s t r e s s e s i n t h e c o v e r s h e e t s by be m o d e l l e d by a r i g i d . p e r f e c t l y p l a s t i c m a t e r i a l . (3.0.1) (3.0.2) 9 (3.0.3) [ = (a 1 - a e ) t h . y y y (3.0.4) The c o n d i t i o n t h a t an element of the sandwich s h e l l have t h e same c a p a c i t y as an u n i f o r m s h e l l of t h i c k n e s s 2H ( F i g u r e 4 ) , w i t h r e s p e c t t o membrane f o r c e and bending moment, can be s a t i s f i e d by p u t t i n g t=H and h=H/2. Thus the T r e s c a s a n d w i c h - s h e l l y i e l d c o n d i t i o n i s e q u i v a l e n t t o the assumption t h a t , i n an u n i f o r m s h e l l o b e y i n g t h e T r e s c a y i e l d c o n d i t i o n , t h e r e i s a p i e c e - w i s e u n i f o r m s t r e s s v a r i a t i o n t h r o u g h the t h i c k n e s s w i t h a d i s c o n t i n u i t y a t the middle s u r f a c e . The f l o w r u l e i n the u n i f o r m s h e l l i s s a t i s f i e d a t z=+ H/2, but v i o l a t e d e l s e w h e r e . The T r e s c a s a n d w i c h - s h e l l y i e l d c o n d i t i o n can be e x p r e s s e d more c o n v e n i e n t l y i n terms of the a c t u a l s t r e s s e s i n th e two c o v e r s h e e t s ( cr~g , cryL , crj , a* ) , r a t h e r t h a n i n terms of t h e r e s u l t a n t membrane f o r c e s and bending moments(5). From (2.1.6) and (2.1.7), i t f o l l o w s t h a t the c o r r e s p o n d i n g g e n e r a l i z e d s t r a i n r a t e s [ G# t Gy r Gg * Gy ) a r e r e l a t e d to the m i d d l e - s u r f a c e v e l o c i t i e s v, w by & a X - (-w tan <j> + v hw')/y, (3.0.5) i = v' - hw • I I (3.0.6) y 10 e a e = (-w t a n <J) + v + hw')/y, (3.0.7) G y 6 = v'+ hw". (3.0.8) A s h e l l e l e m e n t i s i n t h e p l a s t i c s t a t e when a t l e a s t one c o v e r s h e e t i s i n t h e p l a s t i c s t a t e . W i t h r e f e r e n c e t o t h e T r e s c a y i e l d l o c u s f o r p l a n e s t r e s s shown on cr 6 , cr^ a x e s a s t h e h e x a g o n ABCDEF i n F i g u r e 6 , e i t h e r t h e s t r e s s p o i n t i , t h a t i s t h e p o i n t ( °e , ) , o r t h e s t r e s s p o i n t e, t h e p o i n t ( aQ , e ) , must l i e on t h e y i e l d l o c u s . However, e x a m i n a t i o n o f ( 3 . 0 . 5 ) - ( 3 . 0 . 8 ) r e a d i l y r e v e a l s t h a t i f one c o v e r s h e e t i s r i g i d , no p l a s t i c d e f o r m a t i o n o f t h e s h e l l i s p o s s i b l e ( 3 ) . H e n c e , f o r p l a s t i c d e f o r m a t i o n o f t h e s h e l l t o o c c u r , b o t h s t r e s s p o i n t s i and e must l i e s i m u l t a n e o u s l y cn t h e y i e l d l o c u s . I t i s assumed t h a t t h e s h e l l m a t e r i a l i s i s o t r o p i c , t h e r e f o r e , t h e y i e l d s t r e s s e s i n t h e m e r i d i o n a l and c i r c u m f e r e n t i a l d i r e c t i o n s a r e t h e same. As shown i n F i g u r e 6 , •o~D i s t h e y i e l d s t r e s s i n b o t h t h e Y and Q d i r e c t i o n s . 3.1 T h r e e - H i n g e M o d e l The s i m p l e s t m a t h e m a t i c a l m o d e l w h i c h d e s c r i b e s t h e power 11 d i s s i p a t i o n i n t h e b u l g e i s a t h r e e - h i n g e m o d e l , shown i n F i g u r e 7. I t i s assumed t h a t t h e r e i s no e x t e n s i o n a l s t r a i n i n t h e m e r i d i o n a l d i r e c t i o n a t t h e m i d d l e s u r f a c e d u r i n g b u c k l i n g , t h a t i s , £ylz=o = 0 . T h u s , t h e m o t i o n o f t h e m i d d l e s u r f a c e o f t h i s m o d e l i s t r e a t e d as a r i g i d - b a r l i n k a g e . I n c o m p r e s s i o n l o a d i n g , p o i n t s A and C a r e assumed move i n v e r t i c a l b u t o p p o s i t e d i r e c t i o n s , and p o i n t B i n t h e h o r i z o n t a l d i r e c t i o n . The downward v e l o c i t y VA i s g i v e n by V A = - L cos 4> o> (sin <J> j + cos <j> k) . (3.1.1) The p a r t i c l e v e l o c i t y V p i n t h e m i d d l e s u r f a c e h a s two v e l o c i t y c o m p o n e n t s (v,w) : yp = vj + wk , where v , = cb x (y - a sec cb)j , ~p/A Z then v = -L cos <j) sin cb cb , (3.1.2) 2 * w = - (L cos cb - y + a sec cb)cb . (3.1.3) 12 The v e l o c i t y w v a r i e s l i n e a r l y i n y as l o n g as AE r e m a i n s s t r a i g h t . The n e g a t i v e s i g n s i n ( 3 . 1 . 2 ) and ( 3 . 1 . 3 ) i n d i c a t e t h a t t h e a n g u l a r v e l o c i t y has a n e g a t i v e v a l u e f o r c o m p r e s s i v e l o a d i ng. From ( 3 . 0 . 5 ) - (3.0.8) , ( 3 . 1 . 2 ) and ( 3 . 1 . 3 ) , t h e m e r i d i o n a l a n d c i r c u m f e r e n t i a l s t r a i n r a t e s i n t h e two c o v e r s h e e t s become e 6 = 0 , (3.1.6) v i 1 = 0 , (3.1.7) y . e (y tan <b - a sec A tan d) + h) * / r i , O N e = * x — * (p (3.1.8) 0 y ' . i _ _ (y tan cb - a sec <j) tan 4> - h) ' (3 1 9 ) e 0 y The power d i s s i p a t i o n p e r u n i t v o lume f o r p l a n e s t r e s s i n a t h i n s h e e t o b e y i n g t h e T r e s c a y i e l d c o n d i t i o n i s g i v e n by P r a g e r (4) , and i s d = °-f (|e Q| + \iy\ •+ \tQ + ty\) . (3.1.10) The e x p r e s s i o n f o r power d i s s i p a t i o n p e r u n i t v o l u m e d i n t h e o u t e r and i n n e r s h e e t s c a n be w r i t t e n a s d 6 = f (IVI + I V + K + ty^ ( 3 ' 1 - 1 1 ) d" = ^ ( l ^ 1 ! + + i ^ 9 i + V ! ) 0.1.12) where d = d e +dl . (3.1.13) 13 The d i s s i p a t e d power D per u n i t a r e a o f mi d d l e s u r f a c e can be found by e v a l u a t i n g (3.1. 6) - (3. 1.9) and (3. 1. 11) - (3. 1.1 3) . T h i s g i v e s D _ t a |"j y t a n $ ~ a s e c ^ t a n $ n ^ | o Ll y _l_ | y tan (fa - a sec cfa tan cb - h ^ j (3.1.14) When D i s i n t e g r a t e d o ver the a r e a of t h e middle s u r f a c e AB, the d i s s i p a t e d power WAB i s found t o be L + a sec cb W B^ = 2Tfta Q J [| y s i n cb - a tan <b + h cos cb | a sec cb + |y s i n (b - a tan cb - h cos cb | ] ] c b | dy (3.1.15) S i n c e the r e g i o n BC i s t h e s y m m e t r i c a l c o u n t e r p a r t o f AB, t h e t o t a l power d i s s i p a t e d ' i n the w a l l of the b u l g e i s 2W t e . The t o t a l power d i s s i p a t e d i n t e r n a l l y , Wj_ , i s o b t a i n e d by a d d i n g the q u a n t i t y 2W^Q and the power d i s s i p a t i o n ( WA, WB, Wc ) a s s o c i a t e d w i t h the y i e l d h i n g e s (A ,B , C ) , where W = 2TT a M <b| , (3.1.16) A o Wg = | 4TT M Q ( a + L cos cb) cb | , (3.1.17) W = |2TT a M o>| • ' (3.1.18) C o 14 A s shown i n F i g u r e 7, A and C a r e t h e y i e l d h i n g e s a t t h e r i g i d - p l a s t i c b o u n d a r i e s . The s t r a i n r a t e s i n t h e c o v e r s h e e t s f o r b o t h o f t h e h i n g e s a r e Ge = °/ £ o = ° , Gy-*--co and ^y-^+co. T h u s , t h e s t r e s s p o i n t e l i e s on t h e s i d e AB and i on t h e s i d e DE o f t h e y i e l d l o c u s i n F i g u r e 6. H i n g e E c o n n e c t s t h e two r e g i o n s o f p l a s t i c d e f o r m a t i o n AB and BC, F i g u r e 7. The s t r a i n r a t e s i n t h e c o v e r s h e e t s f o r h i n g e B a r e e e > 0 , G e > 0 , 6^-*+co an d £y-*-oo. I t i s e x p e c t e d t h a t t h e s t r e s s p o i n t s e, i f o r h i n g e B l i e a t t h e c o r n e r s D and B o f t h e y i e l d l o c u s f o r t h e o u t e r a n d i n n e r s h e e t s r e s p e c t i v e l y . T h e r e t h e n o r m a l i t y c o n d i t i o n a l l o w s t h e s t r a i n - r a t e v e c t o r s t o ha v e an i n f i n i t e y co m p o n e n t and a f i n i t e , n o n - z e r o Q component. H e n c e , a t t h e L e , y i e l d h i n g e s , o"g , °\j = -°~o , and t h e b e n d i n g moment i s , f r o m ( 3 . 0 . 4 ) , Mo = 2 t h a 0 ' (3.1.19) S u b s t i t u t i n g t h e a b o v e v a l u e o f M 6 and p u t t i n g t=H and h=H/2 i n t o W i, y i e l d s L + a sec 4> 2 • • /• W = 4TTH (2a + L cos <J>) | (J> | + AtrH I (|y s i n <f> - a tan <}> a sec $ + 1 L C | S _ 1 | + | y s . n ^ _ a fcan . _ H_co i LJ>| ) d y ( 3 1 2 Q ) T h e ' e x t e r n a l power o f t h e a p p l i e d l o a d i s new c a l c u l a t e d . 15 T h e e x t e r n a l power e q u a l s t h e a b s o l u t e v a l u e o f t h e p r o d u c t o f t h e a p p l i e d l o a d and t h e r e l a t i v e a p p r o a c h v e l o c i t y o f t h e e n d s o f t h e c y l i n d e r . H e n c e , We = | 8TT a H L cos cb P <j> | . (3.1.21) E q u a t i n g t h e e x t e r n a l power and t h e power d i s s i p a t e d i n t e r n a l l y , t h e u p p e r - b o u n d P/c^, i s f o u n d t o be L + a sec cb P T 2 2 f , i . , cos <b | — = 2aH + LH cos cb + H / (| y s i n cb - a tan cb + ^ — 1 o i a sec cp i H cos (bi , N + |y s m cb - a tan cb - ^ *"| °y) /2aH L cos cb (3.1.22) A t y p i c a l p l o t o f d i m e n s i o n l e s s l o a d (P/o^ ) v e r s u s d i m e n s i o n l e s s d i s p l a c e m e n t ( A ) f o r t h e n u m e r i c a l v a l u e s : L=0.35 i n c h e s , H=0.035 i n c h e s , a=0.7875 i n c h e s , i s g i v e n i n F i g u r e 8, where t h e d i m e n s i o n l e s s d i s p l a c e m e n t A i s g i v e n by A = I - s i n 4> (3.1.23) 3.2 F u r t h e r A n a l y s i s f o r T h r e e - H i n g e M o d e l The c h a r a c t e r i s t i c b u c k l i n g l e n g t h (2L) i n a t h i n c y l i n d r i c a l s h e l l u n d e r a x i a l c o m p r e s s i o n w h i c h p r e v a i l s a f t e r t h e s t a r t o f b u c k l i n g ( p o s t - b u c k l i n g ) c a n be p r e d i c t e d 16 a p p r o x i m a t e l y f r o m t h e u p p e r bound ( 3 . 1 . 2 2 ) . By u s i n g t h e same n u m e r i c a l v a l u e s f o r t h e d i m e n s i o n s o f t h e t h i n c y l i n d r i c a l s h e l l a s i n t h e p r e v i o u s a n a l y s i s and a s e r i e s o f d i f f e r e n t v a l u e s f o r t h e b u l g e l e n g t h (21) , t h e f a m i l y o f c u r v e s f o r P/CT^ v e r s u s A o f t h e t h r e e - h i n g e m o d el c a n be f o u n d and i s shown i n F i g u r e 9. S i n c e t h e b e s t u p p e r bound o c c u r s f o r t h e c u r v e w i t h t h e l o w e s t P / c r 0 v a l u e , t h e c u r v e w i t h L=0.25 i n c h e s , w h i c h h a s t h e l o w e s t v a l u e o f P/ ej0 t o r a w i d e r a n g e o f A , i s s e l e c t e d . T h e r e f o r e , t h e h a l f - b u l g e l e n g t h i s p r e d i c t e d t o be 0.25 i n c h e s . H o wever, i t s h o u l d be n o t e d t h a t b e y o n d A = 0 . 3 5 , t h e c u r v e w i t h L=0.35 i n c h e s h a s t h e l o w e s t P / e r 0 v a l u e . I n o r d e r t o i n v e s t i g a t e t h e v a l i d i t y o f t h e a s s u m p t i o n t h a t t h e e x t e n s i o n a l s t r a i n i n t h e m e r i d i o n a l (y) d i r e c t i o n a t t h e m i d d l e s u r f a c e i s z e r o f o r t h e t h r e e - h i n g e m o d e l , i t i s c o n s i d e r e d t h a t t h i s m e r i d i o n a l e x t e n s i o n a l s t r a i n r a t e i s now u n i f o r m i n y, t h a t i s , e l _ = £ ( c o n s t a n t ) . (3.2.1) y|z=0 W i t h t h e same c a l c u l a t i o n a s i n t h e p r e v i o u s a n a l y s i s f o r t h e t h r e a - h i n g e m o d e l , e x c e p t t h a t t h e p a r t i c l e v e l o c i t y v p i s g i v e n by V = V A + V + V • ( 3 . 2 . 2 ) ~p ~ A ~p/A ~y where V y= £(y-a s e e d } ] i s t h e v e l o c i t y due t o s t r a i n i n t h e m e r i d i o n a l d i r e c t i o n . E q u a t i n g t h e power o f t h e e x t e r n a l l o a d t o t h e t o t a l i n t e r n a l d i s s i p a t e d power y i e l d s t he u p p e r - b o u n d 17 L + a sec cb i . P a (4aH 2 + 2HL cos cb) | ( b | + H j { 2y cos cb |e| a sec cb H + | (y s i n cb - a t a n cb + cos <b)cb - (y cos cb - a)e| H " * + | (y s i n cb - a tan cb + — cos (b)<b + a e| H + | (y s i n cb - a tan cb - — cos cb)cb - (y cos cb - a)e| H * " i 1 + | (y s i n cb - a tan cb - — cos cb)<b + a£ | | dy /4aH L cos ci | | | . (3.2.3) o F i c j u r e 10 shows t h e c u r v e s o f v e r s u s J z _ Q f o r <^> =89 and ^ 1 0 . The minimum v a l u e o f P/o^ o c c u r s a t 6y| z- o=0 i n b o t h c a s e s . a c c o r d i n g t o t h e u p p e r - b o u n d t h e o r e m o f p l a s t i c i t y , t h e r e s u l t i n d i c a t e s t h a t t h e a s s u m p t i o n o f f , =0 i s t h e b e s t one f o r t h i s r 1-y,z = o m o d e l . T h i s r e s u l t i n d i c a t e s t h a t c i r c u m f e r e n t i a l and b e n d i n g d e f o r m a t i o n s d o m i n a t e t h e power d i s s i p a t i o n . I n t h e e a r l y s t a g e s o f b u l g i n g , power i s d i s s i p a t e d m a i n l y i n s t r e t c h i n g i n t h e c i r c u m f e r e n t i a l d i r e c t i o n . As t h e b u l g i n g p r o c e s s c o n t i n u e s , t h e power d i s s i p a t e d i n t h e h i n g e s i n c r e a s e s r e l a t i v e t o t h e c o n t r i b u t i o n o f s t r e t c h i n g . A f t e r t h e b u l g e i s h a l f - f o r m e d , t h e b e n d i n g d e f o r m a t i o n b e g i n s t o d o m i n a t e t h e power d i s s i p a t i o n ( F i g u r e 2 5 ) . 3.3 F i v e - H i n g e M o d e l and S e v e n - H i n g e M o d e l I n v i e w o f t h e g e o m e t r i c a l s h a p e o f t h e a x i s y m m e t r i c b u l g e , 18 a f i v e - h i n g e m o d e l would a p p e a r t o o f f e r a c l o s e r f i t t o t h e d e f o r m a t i o n t h a n a t h r e e - h i n g e one. T h i s m o d e l i s c o n s t r u c t e d by u s i n g two p a i r s o f l i n k s o f d i f f e r e n t l e n g t h s and i s shown i n F i g u r e 11. common f a c t o r i n t h e e x p r e s s i o n f o r i n t e r n a l power d i s s i p a t i o n a n d e x t e r n a l power, and h e n c e c a n c e l s when t h e two a r e e q u a t e d ( 3 . 1 . 2 2 ) . T h u s , t h e v a l u e o f P/cr0 a t e a c h i n s t a n t o f p l a s t i c d e f o r m a t i o n i s e a s i l y c o m p u t e d . H o w e v e r , a s t h e m o d e l becomes more c o m p l i c a t e d , a g r e a t e r number o f v a r i a b l e s i s r e q u i r e d t o s p e c i f y t h e m o t i o n . I n t h e f i v e - h i n g e m o d e l , t h e r e a r e two i n d e p e n d e n t a n g u l a r v e l o c i t i e s , one o f w h i c h c a n be a s s i g n e d a r b i t r a r i l y . By a s i m i l a r method o f c a l c u l a t i o n a s f o r t h e t h r e e - h i n g e m o d e l , t h e u p p e r - b o u n d f u n c t i o n P/cr 0 o f t h e f i v e -h i n g e model h a s t h e f o r m I n t h e t h r e e - h i n g e m o d e l , t h e a n g u l a r v e l o c i t y i s a o a sec 4". + L a sec <J> + (3.3.1) where: 3. = y s i n (j), - a tan <J> B_ = y s i n cj> - (a + L cos $ ) tan A. = (a + L. cos <{> ) sec <$> 19 The s e a r c h f o r t h e minimum v a l u e o f P / c r 0 by m a t h e m a t i c a l a n a l y s i s i n a c o m p l i c a t e d m o d e l l e a d s t o v e r y cumbersome c a l c u l a t i o n s . However, a n u m e r i c a l t r i a l - a n d - e r r o r a p p r o a c h l e a d s r a p i d l y t o an a n s w e r . T h e r e i s no a s s u r a n c e , h o w e v e r , t h a t t h e a b s o l u t e minimum has been o b t a i n e d , and n o t j u s t a l o c a l minimum. I n c a r r y i n g o u t a n u m e r i c a l t r i a l - a n d - e r r c r p r o c e d u r e t o f i n d t h e minimum v a l u e o f P/cr^, , t h e v a r i a b l e s w h i c h a r e most l i k e l y t o g o v e r n t h e p l a s t i c b u c k l i n g a r e s e l e c t e d . T e n t a t i v e v a l u e s a r e a d o p t e d f o r a l l b u t one o f t h e s e v a r i a b l e s w h i c h a r e t h e n h e l d c o n s t a n t w h i l e t h e f i r s t one i s v a r i e d . Then t h e f i r s t v a r i a b l e i s h e l d c o n s t a n t a t t h e v a l u e f c r w h i c h P/cr 0 i s a minimum, and t h e p r o c e s s i s r e p e a t e d i n t u r n w i t h t h e o t h e r v a r i a b l e s u n t i l t h e minimum v a l u e o f P/cr 0 f o r a l l t h e v a r i a b l e s i s f o u n d . The c h a r a c t e r i s t i c v a r i a b l e s i n (3.3.1) f o r t h e f i v e - h i n g e m o d e l a r e <p, , <fiz , and cpz. C o n s t a n t v a l u e s a r e a s s i g n e d t o t h e a n g l e s . c/> and cpz • qS, i s p u t e g u a l t o m i n u s one. The same n u m e r i c a l v a l u e s f o r t h e h a l f - w a l l t h i c k n e s s H (=0.035 i n c h e s ) , t h e s h e l l r a d i u s a ( = 0 . 7 8 7 5 i n c h e s ) and t h e h a l f - b u l g e l e n g t h L ( = 0 . 3 5 i n c h e s ) u s e d i n S e c t i o n 3.1 f o r t h e t h r e e - h i n g e m o d e l a r e u s e d h e r e . The l e n g t h s L, and Lz ( F i g u r e 11) a r e a r b i t r a r i l y c h o s e n a s f o l l o w s : L=L,+L,2. and L ; = 2 L 2 . The r e s u l t s a r e p l o t t e d i n F i g u r e 12 w h i c h shows t h a t t h e b e s t u p p e r - b o u n d o c c u r s when = ~ 1 * D i f f e r e n t v a l u e s f o r t h e a n g l e s cp( , <£2 a n d o f t h e r a t i o L , : L Z a r e s e l e c t e d t o v e r t i f y t h e r e s u l t s . I t i s f o u n d t h a t t h e r e s u l t s a r e i n s e n s t i t i v e t o t h e s e g e o m e t r i c p a r a m e t e r s . 2 0 S i n c e i t i s f o u n d t h a t db=-. q^ = - 1 , t h e l i n k s AB and BC a r e r o t a t i n g w i t h t h e same a n g u l a r v e l o c i t y , and b e c a u s e o f s y m m e t r y , l i k e w i s e , l i n k s CD and DE. H i n g e s B artd E a r e n o t f u n c t i o n i n g and t h e r e f o r e o n l y t h r e e h i n g e s a r e o p e r a t i n g i n t h e f i v e - h i n g e m o d e l . I n o r d e r t o a v o i d t h e need o f a r b i t r a r i l y s e l e c t i n g t h e r a t i o o f t h e l e n g t h s L ( :L^ f o r t h e f i v e - h i n g e model a n d t o h a v e a more r e a l i s t i c model o f t h e b u l g e , a s e v e n - h i n g e model i s n e x t c o n s i d e r e d . F i g u r e 13 shows t h e s e v e n - h i n g e m o d e l , w h i c h h a s s i x r i g i d l i n k s . E a c h l i n k h a s t h e same l e n g t h X , where L=3i.. The m o d e l i s d e s c r i b e d by s i x c h a r a c t e r i s t i c v a r i a b l e s , n a m e l y <£v , ^ , <fi^ , * • * c/> , cpz , and . By a s i m i l a r method o f c a l c u l a t i o n a s f o r t h e t h r e e - h i n g e m o d e l , t h e e x p r e s s i o n f o r P/ <x0 o f t h e s e v e n - h i n g e m o d e l i s _P_ a = { H ( a [ ^ l + B x - 4»2| + B 2 | o> 2 - 4>3| + B 3 | ^ | ) ( a s e c (f>^ + I) l^l / \r\x + | cos cjhj + In-L " f cos ' f l j dy ( a s e c cj)^) B 2 s e c cb2 + B.^ s e c <J>2 J y 2 + 4>2(n2 + | c o s o> 2) | i • • H I + 1*4^ u 2 + 4>2(n2 - 2 c o s 4>2) I ay 21 B 3 sec d»3 B 2 sec <J> /i • • • H \H1P3 + £<J>2 y 3 + <f> 3(ri 3 + - c o s <|>3) dy} + | 4 x P 3 + 4 2 P 3 + * 3 ( n 3 - f cos 4>3) | 2a l\ ^sln(<P1 - <J>2)<J>2 + tan(<J>3 - 4> 2H 3 sin 4>1 + £ ^  + cosC^ - <}>2) <J>2 / + sec((J>3 - tj>2) s inC^ - 4>2>c]>3 + sec((})3 - <J>2) cos(4^ - <J>2)4)3 cos 4^ 1 J , (3.3,2) where: B^ = a + £ cos <j>^ , B 2 = a + H cos (J)^  + H cos 4>2» B 3 = a + A cos <{>^  + £ cos <J>2 + £. cos 4>3> = y sin ^ - a tan cj^ , D 2 = y sin <j>2 ~ (a + £ cos 4>^ ) tan <J>2j n 3 = y sin 4>3 - (a + 2, cos <\>^ + 2, cos f}>2) tan 4>3> y 2 = cos (4> - 4>2) sin <J>2 + sin - 4>2) cos 4>2, y 3 = c o s ^ - 4>2) c o s ^ - <J>3) sin (J>3 + sin(4) 1 - <$>2) s±n(<$>1 - <jr ) sin 4> + cos (4> - <{)2) sinCc^ - c?3) cos (}>3 - s i n ^ - <{>2) cos((()1 - $ ) cos 4>3> P 3 = c o s ^ - 4>3) sin <j>3 + sin(d>1 - <J>3) cos (J)3. 2 2 The c o m p l e t e c a l c u l a t i o n i s shown i n t h e A p p e n d i x . The n u m e r i c a l t r i a l - a n d - e r r o r p r o c e d u r e u s e d p r e v i o u s l y i s a l s o a d o p t e d h e r e t o f i n d t h e v a l u e s o f «r\ , <fe and ^ 3 c o r r e s p o n d i n g t o t h e minimum P/ cr 0 d u r i n g p l a s t i c b u c k l i n g . The r e s u l t s f o r t h e n u m e r i c a l v a l u e s : L=0.35 i n c h e s , H=0.035 i n c h e s a n d a=0.7875 i n c h e s u s e d i n S e c t i o n 3.1 f o r t h e t h r e e - h i n g e m o d e l , a r e shown i n F i g u r e 14 i n a t h r e e - d i m e n s i o n a l g r a p h o f P / c r 0 v e r s u s c/>2 a n d <£3 , f o r t h e a n g l e s ^ = 8 9 . 5 , <^> =89° ana $ 3 = 8 8 . 5 ° . The b e s t u p p e r - b o u n d o c c u r s when <£2 = <£3 and b o t h a p p r o a c h m i n u s i n f i n i t y . T h u s , l i n k s BC and CC, and a l s o DE and EF a r e r o t a t i n g w i t h t h e same a n g u l a r v e l o c i t y w h i l e l i n k s AB an d FG a r e s t a t i o n a r y . T h e r e f o r e , t h e s e v e n - h i n g e m o d el i s o p e r a t i n g as t h e t h r e e - h i n g e m o d e l w i t h a b u l g e l e n g t h o f 41 i n s t e a d o f 6J*. H o w e v e r , f r o m t h e a n a l y s i s i n S e c t i o n 3.2 ( F i g u r e 9 ) , t h e h a l f - b u l g e l e n g t h i s p r e d i c t e d t o t e L=0.25 i n c h e s , w h i c h i s s h o r t e r t h a n t h e a v e r a g e v a l u e o f L=0.35 i n c h e s m e a s u r e d f r o m t h e e x p e r i m e n t s . T a k i n g L=0.25 i n c h e s f r o m F i g u r e 9, and u s i n g t h e same n u m e r i c a l v a l u e s f o r t h e h a l f - w a l l t h i c k n e s s (H) , t h e s h e l l r a d i u s (a) and t h e a n g l e s ( <f> t ^, <£3) u s e d i n F i g u r e 14, t h e t h r e e - d i m e n s i o n a l g r a p h o f P/ cr 0 v e r s u s <f>z a n d <£3 (<^=-1) i s p l o t t e d i n F i g u r e 15. The b e s t u p p e r - b o u n d o c c u r s when cf>z = c/^  =-1. The a n a l y s i s o f t h e s e v e n - h i n g e model i s c o n t i n u e d by a s s i g n i n g a new s e t o f a n g l e s : $,=22°, <£z = 38 6 and <£3 =16°. U s i n g t h e same n u m e r i c a l v a l u e s f o r H and a u s e d i n F i g u r e 14, t h e t e s t • * • upper-bound occurs when = qS = d> =-1 for the half-bulge length L=0.25 inches (Figure 16) . These results imply that a three-hinge motion i s preferred. The seven-hinge model is operating as a three-hinge model with a preferable half-bulge length L=0.25 inches. The search for the minimum values of the upper-bound on P/ cr Q was also conducted with the optimization subroutine (10) in the computer l i b r a r y . These results agree with those using the numerical tr ial -and-error method. 24 CHAPTER I V EXPERIMENTAL RESULTS AND DISCUSSION C y l i n d r i c a l s h e l l s made f r o m 65 ST a l u m i n u m were t e s t e d i n o r d e r t o v e r i f y t h e t h e o r e t i c a l p r e d i c t i o n s . E a c h s h e l l was 4 i n c h e s i n h e i g h t and r e i n f o r c e d a t b o t h e n d s . T h e s e s p e c i m e n s w e re 1.645 i n c h e s i n o u t s i d e d i a m e t e r and had a u n i f o r m w a l l t h i c k n e s s o f 0.07 i n c h e s ( F i g u r e 1 7 ) . The s p e c i m e n s were n o t h e a t - t r e a t e d i n any way a f t e r f a b r i c a t i o n . The m a t e r i a l had an i n i t i a l y i e l d p o i n t o f a p p r o x i m a t e l y 40,000 p s i and a t a n g e n t m o d u l u s o f a p p r o x i m a t e l y 200,000 p s i p a s t t h e y i e l d p o i n t . The s p e c i m e n s , w h i c h were r e s t r a i n e d by a c e n t r a l m a n d r e l , were l o a d e d i n a p o s i t i v e d i s p l a c e m e n t t e s t i n g m a c h i n e . The a p p l i e d l o a d was t r a n s f e r r e d by a l o a d i n g c o l l a r f r o m t h e l o a d i n g h e a d ( F i g u r e 1 8 ) . The s u r f a c e s b e t w e e n t h e s p e c i m e n s , t h e m a n d r e l , and t h e l o a d i n g c o l l a r were l u b r i c a t e d w i t h g r a p h i t e g r e a s e , The f i r s t s p e c i m e n was l o a d e d u n t i l t h e b u l g e was c o m p l e t e l y c o l l a p s e d . The r e m a i n d e r were c o m p r e s s e d by l e s s e r a mounts as i l l u s t r a t e d i n F i g u r e 19. The b u l g e s d i d n o t f o r m a t t h e same p l a c e i n a l l t h e t h e s p e c i m e n s . The a p p l i e d l o a d v e r s u s v e r t i c a l d i s p l a c e m e n t f o r t h e c a s e o f c o m p l e t e c o l l a p s e o f t h e b u l g e i s p l o t t e d i n F i g u r e 20. A c o m p a r i s o n b e t w e e n t h e e x p e r i m e n t a l and t h e t h e o r e t i c a l ( t h r e e - h i n g e m o d e l ) P/cr 0 ( d i m e n s i o n l e s s l o a d ) v e r s u s A ( d i m e n s i o n l e s s 25 d i s p l a c e m e n t ) c u r v e i s i l l u s t r a t e d i n F i g u r e 2 1 . I n t h e t h e o r e t i c a l a n a l y s i s , a c o m p l e t e s o l u t i o n must s a t i s f y t h r e e s e t s o f e q u a t i o n s : t h e y i e l d c o n d i t i o n , t h e f l o w r u l e , and t h e e q u i l i b r i u m e q u a t i o n s . A s o l u t i o n b a s e d on a k i n e m a t i c a l l y a d m i s s i b l e v e l o c i t y f i e l d w h i c h s a t i s f i e s t h e y i e l d c o n d i t i o n and t h e f l o w r u l e p r o v i d e s an u p p e r - b o u n d on t h e l o a d s o f t h e c o m p l e t e s o l u t i o n . However, t h e u p p e r - b o u n d l o a d d e t e r m i n e d from p e r f e c t l y - p l a s t i c a n a l y s i s i s n o t n e c e s s a r i l y g r e a t e r t h a n t h e l o a d o b s e r v e d e x p e r i m e n t a l l y . S i n c e t h e y i e l d s t r e s s i s n o t a p r e c i s e p h y s i c a l q u a n t i t y , a n d s i n c e t h e a l u m i n u m a l l o y u s e d i n t h e s p e c i m e n s d o e s w o r k - h a r d e n , s e l e c t i n g an a p p r o p r i a t e v a l u e f o r t h e y i e l d s t r e s s i s a m a t t e r o f some u n c e r t a i n t y . The v a l u e o f 40,000 p s i f o r t h e y i e l d s t r e s s i s t o be r e g a r d e d as an i n i t i a l y i e l d s t r e s s . When t h i s v a l u e i s u s e d t o o b t a i n t h e d i m e n s i o n l e s s l o a d P/cr0 , t h e e x p e r i m e n t a l l o a d -d e f l e c t i o n c u r v e , F i g u r e 2 0 , l i e s c o n s i d e r a b l y a b o v e t h e one o b t a i n e d f r o m t h e u p p e r - b o u n d c a l c u l a t i o n s . C h o o s i n g a l a r g e r v a l u e f o r t h e y i e l d s t r e s s , w h i c h a c c o u n t s f o r w o r k - h a r d e n i n g , l o w e r s t h e e x p e r i m e n t a l l e a d - d e f l e c t i o n c u r v e and b r i n g s i t i n t o c l o s e r a g r e e m e n t w i t h t h e u p p e r - b o u n d a n a l y s i s . F c r t h i s p r o b l e m , p e r f e c t l y - p l a s t i c a n a l y s i s d e e s n o t c l o s e l y p r e d i c t t h e l o a d , b u t i t d o e s g i v e a c l o s e p r e d i c t i o n o f t h e c o l l a p s e mode. F o u r a d d i t i o n a l s p e c i m e n s were made f o r t e s t i n g w i t h o u t a r e s t r a i n i n g c e n t r a l m a n d r e l . T h e s e s p e c i m e n s , made f r o m t h e same m a t e r i a l , 65 ST a l u m i n u m , were m a c h i n e d t o t h e same d i m e n s i o n s 26 a s t h e s p e c i m e n s o f t h e p r e v i o u s t e s t s . I n o r d e r t c f o r m an a x i s y m m e t r i c b u l g e a t a p r e d e t e r m i n e d p o s i t i o n , t h e w a l l t h i c k n e s s was r e d u c e d f r o m 0.07 i n c h e s t o a minimum o f 0.065 i n c h e s , s y m m e t r i c a l l y o v e r a l e n g t h o f 1.0 i n c h ( F i g u r e 2 2 ) . The t e s t i n g p r o c e d u r e was s i m i l a r t o t h a t u s e d p r e v i o u s l y . A x i s y m m e t r i c b u l g e s f o r m e d a s e x p e c t e d i n the s e c t i o n o f r e d u c e d w a l l t h i c k n e s s . F u r t h e r m o r e , i t was f o u n d t h a t t h e s e s p e c i m e n s h a v i n g s l i g h t l y n o n - u n i f o r m w a l l t h i c k n e s s a n d t e s t e d w i t h o u t a m a n d r e l b u c k l e d t h e same as t h e u n i f o r m - w a l l s p e c i m e n s t e s t e d w i t h t h e m a n d r e l . F i g u r e 24 shows t h e p l o t o f a p p l i e d l o a d v e r s u s v e r t i c a l d i s p l a c e m e n t f o r t h e c a s e o f c o m p l e t e c o l l a p s e o f t h e b u l g e . T h i s c u r v e shows v e r y c l o s e a g r e e m e n t w i t h t h e c u r v e i n F i g u r e 20 f o r t h e u n i f o r m - w a l l s p e c i m e n . I t i s n o t e d t h a t t h e m e a s u r e d v a l u e s o f t h e h a l f - b u l g e l e n g t h i n t h e s e t e s t s a v e r a g e 0.32 i n c h e s . The m e a s u r e d v a l u e s o f t h e h a l f - b u l g e l e n g t h i n t h e p r e v i o u s t e s t s f o r t h e u n i f o r m - w a l l s p e c i m e n s a v e r a g e 0.35 i n c h e s . The a g r e e m e n t b e t w e e n t h e s e two h a l f - b u l g e l e n g t h s i s v e r y c l o s e . 27 CHAPTER V REMARKS AND CONCLUSIONS The f a m i l y o f c u r v e s i n F i g u r e 9 i n d i c a t e s t h a t , when t h e b u l g e i s a l m o s t c o l l a p s e d , t h e d i s s i p a t e d power due t o b u l g i n g i s n o t s t r o n g l y r e l a t e d t o t h e b u l g e l e n g t h o v e r a l a r g e r a n g e o f l e n g t h s . The h a l f - b u l g e l e n g t h L ( = 0 . 3 5 i n c h e s ) o b t a i n e d f r o m m e a s u r e m e n t s i s s l i g h t l y d i f f e r e n t f r o m t h e h a l f - b u l g e l e n g t h L ( = 0 . 2 5 i n c h e s ) o b t a i n e d f r o m t h e u p p e r - b o u n d a n a l y s i s . One o f t h e r e a s o n s i s t h a t t h e measurement i s t a k e n a t t h e m i d d l e s u r f a c e a l o n g t h e a r c w h i l e t h e t h e o r y g i v e s t h e l e n g t h o f t h e mo d e l i n a s t r a i g h t l i n e . I n g e n e r a l , t h e a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l r e s u l t s a n d t h e t h e o r y i s s a t i s f a c t o r y . I n one c a s e i n t h e a n a l y s i s , t h e e x t e n s i o n a l s t r a i n r a t e a t t h e m i d d l e s u r f a c e ( £ j z _ ^ f ° r t h e t h r e e - h i n g e m o d el h a s been c o n s i d e r e d u n i f o r m . T h i s y i e l d s t h e same r e s u l t a s i f z e r o e x t e n s i o n a l s t r a i n r a t e i s assumed. I t was i n i t i a l l y b e l i e v e d t h a t i n c r e a s i n g t h e number o f s e g m e n t s i n t h e model w o u l d l o w e r t h e u p p e r - b o u n d . H o w e v e r , o f t h e m o d e l s c o n s i d e r e d i n t h e p r e s e n t p e r f e c t l y - p l a s t i c a n a l y s i s , i t i s f o u n d t h a t t h e t h r e e - h i n g e m odel y i e l d s t h e l o w e s t u p p e r - b o u n d . However, t h e c a l c u l a t i o n o f t h e u p p e r - b o u n d d e p e n d s s t r o n g l y on t h e y i e l d c o n d i t i o n , and t h e q u e s t i o n r e m a i n s o p e n a s t o w h e t h e r o t h e r y i e l d c o n d i t i o n s l e a d t o t h e same r e s u l t . 28 BIBLIOGRAPHY 1 . B a t t e r m a n , S.C., " P l a s t i c B u c k l i n g o f A x i a l l y C o m p r e s s e d C y l i n d r i c a l S h e l l s '«, AIAA J o u r n a l , Volume 3, Number 2, 1965. 2. F l o r e n c e , A.L. and G o o d i e r , J . N . , " Dynamic P l a s t i c B u c k l i n g o f C y l i n d r i c a l S h e l l s i n S u s t a i n e d A x i a l C o m p r e s s i v e Flow ", J o u r n a l o f A p p l i e d M e c h a n i c s , T r a n s . ASME, S e r i e s E, March 1968. 3 . Shaw, M i l t o n C., " D e s i g n s f o r S a f e t y : t h e M e c h a n i c a l Fuse M e c h a n i c a l E n g i n e e r i n g , A p r i l 1973. A l l a n , T., 5 . Ramsey, H., " E x p e r i m e n t a l and A n a l y t i c a l I n v e s t i g a t i o n o f t h e B e h a v i o u r o f C y l i n d r i c a l T u b e s S u b j e c t t o A x i a l C o m p r e s s i v e F o r c e s ", M e c h a n i c a l E n g i n e e r i n g S c i e n c e , Volume 10, Number 2, 1968. " P l a s t i c , C o l l a p s e o f S t e e p C o n i c a l S h e l l s u n d e r A x i a l C o m p r e s s i o n ", CSME J o u r n a l , Volume 1, November 1972. 6. U p d i k e , D.P., " On t h e L a r g e D e f o r m a t i o n o f a R i g i d - P l a s t i c S p h e r i c a l S h e l l C o m p r e s s e d by a R i g i d P l a t e ", J o u r n a l o f E n g i n e e r i n g f o r I n d u s t r y , A u g u s t 1972. 7. T i m o s h e n k o , S. and W o i n o w s k y - K r i e g e r , 5., T h e o r y of P l a t e s and S h e l l s ", McGraw H i l l , New Y o r k , N.Y., 1959. 8 . Hodge, P.G., J r . , " L i m i t A n a l y s i s o f R o t a t i o n a l l y S y m m e t r i c P l a t e s and S h e l l s P r e n t i c e - H a l l , Mew Y o r k , N . J . , 1963. 9. P r a g e r , W., " An I n t r o d u c t i o n t o P l a s t i c i t y " , A d d i s o n - W e s l e y , I n c . , R e a d i n g , Mass., 1959. 10. B i r d , C. , L e e , C M . and O ' R e i l l y , D., " UBC N I P : N o n - L i n e a r F u n c t i o n O p t i m i z a t i o n ", C o m p u t i n g C e n t e r , 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 , O c t o b e r 1973. 29 A P P E N D I X C a l c u l a t i o n s f o r S e v e n - H i n g e M o d e l ( F i g u r e A . 1 ) I n t h e s e v e n - h i n g e model s e c t i o n s ABCD and DEFG u n d e r g o s y m m e t r i c d e f o r m a t i o n ; t h e r e f o r e , o n l y s e c t i o n AECD need be c o n s i d e r e d . H i n g e A i s assumed s t a t i o n a r y , h i n g e s B, C and D t r a n s l a t e d u r i n g b u l g i n g . The v e l o c i t i e s o f B, C and D a r e f o u n d by a s s u m i n g t h a t AB, BC, CD a r e r i g i d l i n k s c o n s t r a i n e d t o move i n a m e r i d i o n a l p l a n e ( t h e e x t e n s i o n a l s t r a i n r a t e ELJ i s assumed t o be z e r o ) . Hence, By e x a m i n i n g t h e v e l o c i t y o f C r e l a t i v e t o B, t h e a b s o l u t e v e l o c i t y o f C i s f o u n d t o be (1) y c = I z<p1 + i c o s ( o > 1 - <j) 2) 0 2 k + £ s i n (4>1 - c b 2 ) o> 2 j . ( 2 ) F i n i a l l y , Vp i s o b t a i n e d a s , + £ s e c ( < b 3 - cf> 2) s i n (<b - cb ) cj> (3) 30 The power d i s s i p a t e d i n t h e h i n g e s x s WA = WG = 2TfaM o | ^ | , (4) WB = Wp = 2TT (a + £ cos cb 1) M q | - ^) | , (5) Wc = W£ = 2TT (a + £ cos <|>1 + £ cos o>2) M q | (<j>2 - £ ) | , (6) •| WD = 2TT (a + £ cos c}^ + £ cos cj>2 + £ cos <f>3) M Q |o> | . (7) S u b s t i t u t i n g M 0= 2 t h c r 0 f r o m (3.1.19) i n ( 4 , 5, 6, 7) and t h e n a d d i n g t h e s e e x p r e s s i o n s y i e l d s f o r t h e t o t a l power WH d i s s i p a t e d i n t h e h i n g e s WH = 8rrth OQ £ a I f J + (a + £ cos <j>1) | - <j>2) | + (a + £ cos <b1 + £ cos o>2) | (<f>2 - <j>3) | + (a + £ cos <J> + £ cos Q ) 2 + £ cos (b^) (J^l (8) The power d i s s i p a t e d , due t o c i r c u m f e r e n t i a l s t r a i n i n g o f s e c t i o n AB i s now c a l c u l a t e d . A t y p i c a l p o i n t P has j u s t a t r a n s v e r s e v e l o c i t y c omponent w g i v e n by w = (y - a sec c^) e) . (9) From ( 3. 0 . 5) - (3 . 0. 8) and (10) t h e m e r i d i o n a l and c i r c u m f e r e n t i a l s t r a i n r a t e s i n t h e two c o v e r s h e e t s become (y tan (b - a sec cb tan cb + h) ^e = 1 y ^ , (10) (y tan cb - a sec c L tan A _ h) e j - i - j - * 1 . ( I D = 0 , (12) kX = 0 . (13) S u b s t i t u t i n g f r o m (10) - (13) i n ( 3 . 1. 11) - ( 3 , 1. 1 3) , t h e power d i s s i p a t i o n d p e r u n i t v o l u m e i s t h e n d. = a o (y tan <b^  - a sec <j>^  tan ^ + h) # (y tan cb.. - a sec cb tan cb - h) + I ~ ^ L — * (14) I n t e g r a t i o n o f d g i v e n by (14) o v e r t h e v o l u m e o f s e c t i o n AB g i v e s , f o r t h e power d i s s i p a t i o n w B^ i n t h a t s e c t i o n . (a sec cb^ + H) W A B = 2 7 r a 0 f c I I j My s i n <t>1 ~ a t a n + h cos cb-J (a sec c b ^ ) + |y s i n cj> - a tan <+> - tlcos (p j ]dy . (15) The c a l c u l a t i o n f o r t h e power d i s s i p a t i o n i n s e c t i o n BC i s c a r r i e d o u t i n t h e same manner a s f o r s e c t i o n AB. I n t h i s 32 s e c t i o n , t h e p a r t i c l e v e l o c i t y Vp w h i c h h a s two c o m p o n e n t s (v,w) i s g i v e n a s V = v j . + wk, , (16) ~p J l I and V = V_ + V , . (17) *jp ~B "p/B The v e l o c i t y V Q i n t h e r e f e r e n c e f r a m e o f t h e s e c t i o n BC i s w r i t t e n a s ZB = ~  1 * i s l n ( * i " V h  + 1 * i c o s ^ i ~ ^ k i ' (18) The v e l o c i t y Vp, i s = ^ y - (a + I cos (J)^ ) sec cf>2 S u b s t i t u t i n g ( 1 6 ) , (18) and (19) i n ( 1 7 ) , y i e l d s v = - £ <f> sin (cS - tj) ) , • • — • w = £ cos (cj^ - (J>2) + y - (a + I cos <$> ) sec <f> j <fi <t>2 kj . (19) (20) A f t e r t h e v e l o c i t y f i e l d o f s e c t i o n EC i s f o u n d , t h e power d i s s i p a t i o n i s c a l c u l a t e d by t h a same p r o c e d u r e l e a d i n g t o (15). 33 The c a l c u l a t i o n y i e l d s (a + £ cos 4> + £ cos 4>2) sec <J)? W = 2vo t BC o £ 4>1 [cos (d^ - (J>2) s i n <J> (a + £ cos 4^) sec 4>2 + s i n - <()2) cos (J>2] + <|, [y s i n 4>2 - (a + £ cos $ ±) tan 4>2 .hcos d) ] I + I £ (j> [cos ( 4 ^ - 4>2) s i n <J>2 + s i n ( 4 ^ - 4>2) cos <f>2] + 4>2 [y s i n 4>2 - (a + £ cos 4^) tan 4>2 - hjcos fy^] \ dy (21) The v e l o c i t y f i e l d i n s e c t i o n CD i s w - £ 4^ s i n (4>1 - 4>3) - £ cos ( 4 ^ - 4>2) ^ s i n ( 4 ^ - 4^) + £ s i n (4)^ - 4>2) 4>2 cos ( 4 ^ - 4>3) £ 4^ cos (4>1 - 4>3) + £ cos (4>1 - 4>2) 4*)2 cos (<$>1 - 4>3) + £ s i n (4>1 - 4>2) 4>2 s i n ( 4 ^ - <jj ) y - (a + £ cos 4>-. + £ cos 4>2) sec 4>3 (22) and t h e power d i s s i p a t e d i n s e c t i o n CD 34 (a + % cos + I cos 4>2 + I cos 4^) sec 4>3 WCD = 2Tra Qt J" \l <$>± [cos - 4>3) sin 4>3 (a + I cos 4^ + 2 cos 4>2) sec 4>3 i '+ sin (<j)1 - <j>3) cos <J>3] + I (j>2[cos - 4>2) cos - 4,3) sin <J) f '.+ sin (<J» - <()2) sin (cf^ - 4>3) sin 4>3 j + cos (§ 1 - <^ 2) sin ( 4 ^ - (|) ) cos 4>3 - sin (4>1 - 4>2)jcos (4>1 - 4>3) cos <p^] + [y sin 4>3 - (a + I cos 4> i » • ' '• . . . . + I cos 4>2) tan 4>3 + hcos 4>3] <p \ +\Z 4^ [cos ( 4 ^ - 4>3> sin 4>. i + . sin (4>1 - 4,3) cos 4,3] + £ 4,2 [cos - 4^) Cos - 4,3) sin 4>3 + sin(4) 1 - 4>2) sin (4>1 - 4)3) sin ^ + cos ' - ^ ) s i n - ^ ) c o s - sin (4>i - <{,2) cos (4>i - 4>3) cos 4>3)] j + [y sin 4>3 _ (a + £ cos 4>^  + & c o s <f > 2) tan d>^  — K cos 4>3] 4>3 I dy The t o t a l power d i s s i p a t e d i n t e r n a l l y i s o b t a i n e d t y a d d i n g t h e power d i s s i p a t e d i n t h e h i n g e s and i n t h e s e c t i o n s , h e n c e , W i = WH + 2 ( WAB + W B C + WCD> ' <24) The e x t e r n a l power We o f t h e a p p l i e d l o a d (4rtaHP) i s t h e p r o d u c t o f t h e a p p l i e d l o a d and t h e r e l a t i v e a p p r o a c h v e l o c i t y o f t h e e n d s o f t h e s h e l l . The e x t e r n a l power i s g i v e n l y 35 We = | 87TaHP | £ s i n (cj^ - cj>2) o>2 + £ tan (c{>3 - cj>2) <J>3 + £ sec (cb - o>2) s i n (cb1 - cb0) <f> 1 r 2 ' r 3 s i n cb^  + £<bn + £ cos (o>1 - <J>2) o>2 + £ sec (cb3 - cb2) cos (cj^ - fl>2) <f>3 cos cb J l -(25) E q u a t i n g t h e e x t e r n a l power We w i t h t h e power d i s s i p a t e d i n t e r n a l l y W-t , and p u t t i n g t=H and h=H/2, y i e l d s t h e f o l l o w i n g u p p e r - b o u n d f o r P/ c r 0 , ^ - = {H(a + 1^  - o>2| + B 2 |o>2 - 4>3| + B 3 |o>3|) (a sec <b^  + £) (a sec cb^) H H 1^ + cos cbj + \r\± - -| cos cbj dy B 2 sec <f>2 B 1 sec cb2 i * * H i + y 2 + cb2(n2 - - cos o)2) j dy B 3 sec <b3 B 2 sec cj>3 f \H1P3 + 4 2 y 3 + $ 3 <n 3 + f cos cj>3) j dy} + |Ao>1p3 + £o>2 y 3 + fl)3(n3 - | cos c^) | / [ 2a £| [ + sec(cb 3 - o>2) sin(cb 1 - <(>2)<J>3 + sec(cb 3 - cb2) c o s ^ - <J>2)<1>3 t o o cj^ + costcb^ - fl>2) cb cos cb. 2 (3.3.2) where: B^ = a + £ cos (J)^ , B 2 = a + £ cos c{>^  + £ cos (f>2, B 3 = a + £ cos cj^  + £ cos 4>2 + £ cos <t>3, = y sin (f)^  - a tan <j»^ , n 2 = y sin <J>2 - (a + £ cos c}^ ) tan 4>2> n 3 = y sin ^ - (a + £ cos <j>1 + £ cos <}>2) tan <j>3 y 2 = cos - (|)2) sin 4>2 + sin - 4>2> cos (J>2 y 3 = c o s ( ( f ) 1 ~ 4>2^ c o s ^ i ~ ^ s i n ^ 3 + s inC^ - <f>2) sin((()1 - c^) sin <j> + cos (<J>1 - <J>2) sin(4>1 - 4>3) cos <J> - sin((f>1 - cj>2) cos((})1 - 4>3) cos 4> , P 3 = c o s ^ - (J>3) sin '<J>3 + sin((j>1 - cos 4> . » 9 g u r e 1 A x f s y m m e t r i c B u l g e s f o r m e d w i t h C e n t r a l M a n d r e l 38 A B : E l e m e n t M e r t d f a n A B Q P : P o s i t i o n o f S h e l l E l e m e n t b e f o r e A x i s y m m e t r i c D e f o r m a t i o n A'B'P'Q ' : P o s i t i o n o f S h e l l E l e m e n t a f t e r A x i s y m m e t r i c D e f o r m a t i o n F i g u r e 2 . S h e l l E l e m e n t u n d e r S y m m e t r i c a l L o a d i n g a n d B e n d i n g 39 0 C o n i c a l - S h e l l M idd le Sur face .(b) Cyi i ndr i ca l Shel 1 , 0 C y l i n d r i c a l - S h e l l M idd le Su r f ace F igure 3 Comparison of the M idd l e-Su r f a ce Geometry f o r C y l i n d r i c a l and Con i c a l S h e l l s F i p;u r e h S a n d w i c h - S h e l l E l e m e n t 0 0 5> F i g u r e 5 S t r e s s - S t r a i n D i a g r a m f o r R i g i d , P e r f e c t l y P l a s t i c M a t e r i a l © £ e > 0 , £ y — » ® £e = o , E y — » F i g u r e 6 T r e s c a Y i e l d L o c u s f o r P l a n e S t r e s s F Igure 7 Three-Hln*e Model 45 Flfiure 9 Family of Curves for Dimensionless Load(P/o;) versus 0 Inens I on I ess Displacement (A) for Three-Hlnse Model F i g u r e 10 D i m e n s i o n l e s s L o a d ( P / c r ) v e r s u s M e r i d i o n a l F x t e n s i o n a l S t r a i n R a t e a t t h e M i d d l e S u r f a c e (e„| ) f o r T h r e e - H i np:e M o d e l y i z=o -p-47 A p p 1 i e d L o a d F i g u r e 11 F i v e - H i n g e M o d e l A p p l i e d L o a d . 1+9 i A p p l i e d Load A p p l i e d L o a d F i g u r e 13 Seven-Hinge Model 50 51 F i g u r e T 5 — • D i m e n s i o n l e s s L o a d ( ^ v e r s u s c^> a n d </> f o r S e v e n ^ H l n g e M o d e l ^ 2 I 52 F i g u r e 1 6 D i m e n s i o n l e s s L o a d ( P / o O ve r s u s < j ,and <j> f o r S e v e n - H i n ^ e M o d e l 5 3 S i d e V i e w 0.37" ,Top V i e w 0.07" F i g u r e 17 D i m e n s i o n s o f a T y p i c a l U n i f o r m - W a l l S p e c i m e n L o a d i n g Head F i g u r e 1 9 U n i f o r m - W a l l S p e c i m e n s a f t e r C o m p r e s s i v e L o a d i n g w i t h C e n t r a l M a n d r e l i n p l a c e ( C o m p r e s s i v e l o a d w a s i n c r e a s e d o n t h e s p e c i m e n s I 56 F igure 2 0 A p p l i e d Load ve rsus V e r t i c a l Disp1acemen t ( Uni form-Wal l Specimen ) 57 o.o o . i o .J o.4 0.5 0.6 a.7 0.8 o.« D i m e n s i o n l e s s D i s p l a c e m e n t ( A ) F i g u r e 21 C o m p a r i s o n b e t w e e n T h e o r e t i c a l a n d E x p e r i m e n t a l R e s u l t s F i g u r e 22 D i m e n s i o n s o f a T y p i c a l N o n - U n i f o r m - W a 1 1 S p e c i m e n 60 P O I N T E D I N U . S . A . V e r t i c a l D i s p l a c e m e n t [INCHES] F i g u r e 2b A p p l i e d Load v e r s u s V e r t i c a l D i s p l a c e m e n t ( Non-Uniform-Wall Specimen wi.thout C e n t r a l Mandrel ) • I -2 -3 -4 . «5, -6 -7 , -8 -9 J-Q . D i m e n s i o n J e s s D i s p l a c e m e n t ( A ' F i g u r e 25 R e l a t i v e C o n t r i b u t i o n o f B e n d i n g and S t r e t c h i n g o f M i d d l e S u r f a c e t o t h e T o t a l I n t e r n a l Power D i s s i p a t i o n as a F u n c t i o n o f t h e D i m e n s i o n l e s s D i s p l a c e m e n t ( A ) f o r T h r e e - H i n g e Model F i g u r e A . l R e f e r e n c e Frames o f S e v e n - H i n g e M o d e l 

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