Features of the Buckling of Ribbed Shells

7/28/2019 Features of the Buckling of Ribbed Shells

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International Applied Mechanics. Vol. 36, No . 5. 2000

F E A T U R E S O F T H E B U C K L I N G O F R I BB E D S H E L L S

T o th e B e g i n n i n g o f t h e T h ir d M i l l e n n i u m

V . A . Z a r u t s k i i U D C 5 3 9 . 3

T h e r e s u l ts o f s t a b il i ty a n a l y s i s o f r i b b e d s h e l ls p e r f o r m e d o n t h e b a s i s o f t h e t e c h n i q u e s d e v e l o p e d b yI . Y a . A m i r o a n d h i s f o l l o w e r s a r e r e v ie w e d . A t t e n t i o n is fo c u s e d o n t h e f e a t u r e s o f t h e i r b u c k l i n g d u e t ot h e d i s c r e t e a r r a n g e m e n t o f ri b s.

I n t r o d u c t i o n . T h i n - w a l l e d s t r u c tu r a l e l e m e n t s c a n b e s u b j e c t to c o m p r e s s i v e l o a d s . I n t h is c a s e , t h e ir l o a d - c a r r y i n gc a p a c i ty i s d e t e r m i n e d b y t h e c a p a b i l i t y o f t a k in g u p t h e s e l o a d s w i th o u t b u c k l i n g . T h e r e f o r e , t h e d e v e l o p m e n t o f t e c h n i q u e sf o r d e s ig n i n g s u c h s t r uc tu r al e l e m e n t s h a s b e e n b e i n g o n e o f t h e m o s t u r g e n t p r o b l e m s o f t h e m e c h a n i c s o f s he l ls . T h ed i f f i c u lt i e s a r i s i n g i n th e d e v e l o p m e n t o f su c h t e c h n i q u e s l ie in t h e fa c t ( r e v e a l e d i n c o m p a r i n g t h e r e s u l t s o f e x p e r i m e n t a la n d t h e o r e t i c a l i n v e s t i g a t i o n s ) t h a t e x p e r i m e n t a l l y d e t e r m i n e d c r i t i c a l s t r e s s e s i n 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 a x i a lc o m p r e s s i v e f o r c e s a n d i n s p h e r i c a l s h e l ls s u b j e c t t o a n e x t e r n a l p re s s u r e a r e c o n s i d e r a b l y ( a l m o s t t h r e e t im e s ) l e ss t h a n t h o s e( u p p e r c r i t ic a l s t r e s s e s ) f o u n d b a s e d o n l i n e a r t h e o r y .

H e r e a f t e r , w e w i ll c o n s i d e r , b y a n e x a m p l e o f c y l i n d r ic a l s h e ll s u n d e r a x i a l c o m p r e s s i v e f o r c e s , h o w t h i s f a c t h a sa f f e c t e d th e d e v e l o p m e n t o f t e c h n i q u e s f o r s ta b i l it y a n a l y s i s o f s h e ll s .

U n t i l 1 9 6 3 , i t w a s r e c o m m e n d e d t o u s e a s t h e o r e t i c a l v a l u e s t h e s o - c a l l e d l o w e r c r i t i c a l s tr e s s e s fo u n d i n a p p r o x i m a t es o l u t io n o f t h e n o n l i n e a r e q u i l i b r i u m e q u a t i o n s f o r s h e l l s, s i n c e th e v a l u e s o f th e s t r e s s e s t h u s o b t a i n e d t u r n e d o u t t o b e c l o s et o t h e m i n i m u m e x p e r i m e n t a l v a l u e s [ 4 8 ].

I n 1 9 6 3 , B . A l m r o t h s h o w e d i n [ 99 ] t h a t w h e n t h e f u n c t io n a p p r o x i m a t i n g t h e b u c k l i n g o f a s h e ll is a d ju s t e d , t h el o w e r c r i ti c a l s t r e s s d e c r e a s e s a n d b e c o m e s m u c h l e s s th a n t h e c o r r e s p o n d i n g e x p e r i m e n t a l v a l u e s . I n t h i s c o n n e c t i o n , i t w a sn e c e s s a r y t o d o a w a y w i t h t h e s e s t r e s s e s u s e d a s t h e o r e t ic a l v a l u e s .

A f t e r 1 9 6 3 , t h e o re t i c a l in v e s t i g a t i o n s w e r e c a r r i e d o u t t o c l a r i f y w h y t h e e x p e r i m e n t a l a n d t h e o r e ti c a l v a l u e s o fc r i ti c a l s t r e s s e s d i f f e r a n d t o w o r k o u t r e c o m m e n d a t i o n s a s f o r th e s t a b i l it y a n a l y s i s o f s h e l l s, r e s e a r c h e r s r e v e r t i n g t o t h eu p p e r c r i t i c a l s t r e s s e s u s e d a s t h e b a s i c c h a r a c t e r i s t ic o f s h e l l b u c k l i n g . T h e f o l l o w i n g t h r e e p r i m a r y f a c t o r s a f f e c t i n g t h ea b o v e - m e n t i o n e d d i s t in c t i o n b e t w e e n e x p e r i m e n t a l a n d th e o r e t ic a l d a ta w e r e s i n g l e d o u t a n d d e t a i le d :

I . T h e n o n c o m p l i a n c e o f t h e s h e l l - g r i p c o n d i t i o n s r e a l i z e d i n e x p e r i m e n t s a n d a c c e p t e d i n d e s i g n p r o c e d u r e s .2 . T h e h e t e r o g e n e i t y o f t h e s u b c r i ti c a l s t a t e i n t e s t sh e l ls .3 . M a n u f a c t u r i n g i m p e r f e c t i o n s i n t e s t s h e l l s ,T h e f i r st o f t h e a b o v e f a c t o r s w a s a n a l y z e d b y V . K o z h e v n i k o v in [ 7 2] a n d N . A . K i i ' c h e v s k i i a n d S . N . N i k u l i n s k a y a

i n [ 6 8 ]. I t w a s s h o w n t h a t i f t h e e d g e o f a s h el l c a n f r e e l y d i s p la c e 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 , t h e n t h e u p p e r c r i t i c a ls t r e s s e s d e c r e a s e t w o t i m e s in c o m p a r i s o n w i t h t h e c a s e w h e r e s u c h d i s p l a c e m e n t s a r e i m p o s s i b l e . H o w e v e r , s u c h a s h e l l - g r i pc o n d i t i o n w a s n o t r e a l iz e d in t h e k n o w n e x p e r i m e n t s . F o r o t h e r v a r i a n t s o f f i x a t io n o f n o t v e r y s h o r t s h e l ls , t he e f f e c t o fc h a n g i n g t h e g r i p c o n d i t i o n s o n t h e c r i ti c a l s tr e s s e s tu r n s o u t t o b e in s i g n i fi c a n t . T h e i n v e s t i g a t i o n s r e s u l t e d i n t h e c o n c l u s i o nt h a t t h e d i f f e r e n c e b e t w e e n t h e s h e l l - g r i p c o n d i t i o n s r e a l i z e d in t he e x p e r i m e n t a n d t h o s e u s e d i n th e d e v e l o p m e n t o f d e s i g np r o c e d u r e s c a n n o t b e c o n s i d e r e d a s a f a c t o r a c c o u n t i n g f o r th e a b o v e - m e n t i o n e d d i f f e r e n c e b e t w e e n t h e c ri t ic a l s t r e s s e s .

T h e e f f e c t o f t h e s e c o n d f a c t o r , n a m e l y , th e i n h o m o g e n e i t y o f t h e s u b c r i t i c a l s t a te , w a s t h o r o u g h l y a n a l y z e d b yV . I . M y a c h e n k o v ( [6 7 ] a n d t h e r e f e r e n c e s t h e r e i n ). I t w a s s h o w n t h a t w h e n t h e h e t e r o g e n e i t y o f t h e s u b c r i ti c a l s t at e is t a k e ni n t o a c c o u n t , t h e u p p e r c ri t ic a l s t r e s s e s , a s a ru l e , d e c r e a s e . H o w e v e r , t h e c ri t ic a l s t r e s s e s t h u s a d j u s t e d a r e m u c h h i g h e r t h a n

S . P . T i m o s h e n k o I n s t i t u te o f M e c h a n ic s , N a t i o n a l A c a d e m y o f S c i e n c e s o f U k r a i n e , K i e v . T r a n s l a t e d f r o m P r i k l a d n a y aM e k h a n i k a , V o l . 3 6 , N o . 5 , p p . 3 - 3 1 , M a y , 2 0 0 0 . O r i g i n a l a rt ic l e su b m i t t e d N o v e m b e r 1 2 , 1 9 9 9 .

1 0 6 3 - 7 0 9 5 / 0 0 / 3 6 0 5 - 0 5 5 9 5 2 5 . 0 0 @ 2 0 0 0 K l u w e r A c a d e m i c / P l e n u m P u b l i s h e r s 5 5 9


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A r e v i e w o f t h e o r e t i c a l s tu d i e s o f th e s t a b i li t y o f r i b b e d s h e l ls i s p r e s e n t e d b e l o w . T h e s e s t u d ie s w e r e c a r r i e d o u t a tt h e S. P . T i m o s h e n k o I n s t i t u te o f M e c h a n i c s o f t h e N a t i o n a l A c a d e m y o f S c i e n c e s o f U k r a i n e b y I . Y a . A m i r o a n d t h ee m p l o y e e s o f t h e d e p a r t m e n t o f b u i l d i n g m e c h a n i c s o f t h i n - w a l l e d s t ru c t u r e s c r e a t e d b y h i m i n 1 9 6 2 .

T h e s e s t u d i e s w e r e t h e f i r s t t o p r o p o s e t e c h n i q u e s f o r s t a b il i ty a n a l y s i s o f c y l i n d r i c a l s h e l l s t h a t a r e r e i n f o r c e d w i t hc r o s s r ib s a n d 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 . T h e s e t e c h n i q u e s h a v e p e r m i t t e d o n e t o c l a s s i f y t h e w a v e m o d e s i n t h eloca l buck l ing o f she l l s due to the d i s c re te a r rangem ent o f the r ibs [1, 2 , 22 , 23] .

T h e e f f e c t o f t h e n u m b e r a n d r i g i d i ty o f th e r i b s o n t h e c r it i c al s tr e s se s w a s t h o r o u g h l y s t u d i e d, I t w a s s h o w n t h a tt h e c ri t ic a l s t r e s se s o f o n e o f t h e l o c a l - b u c k l i n g m o d e s a r e m i n i m u m o v e r a w i d e r a n g e o f t h e p a r a m e t e r s o f r i b b e d s h e l ls .T h i s p r o v e d t h e f a c t t h a t v a r i a n t s o f t h e t h e o r y o f f i b b e d s h e l l s c o n s t r u c te d w i t h o u t r e g a r d f o r t h e p r e s e n c e o f a l l p o s s i b l el o c a l - b u c k l i n g w a v e m o d e s c a n l e a d t o a n o v e r e s t im a t i o n o f t h e c r it i c a l s t re s s e s a n d , w h e n b e i n g u s e d f o r o p t i m a l d e s i g n o fshe l l s , to cons ide rab le d i s to r t ion o f the pa ram ete rs o f op t im a l s t ruc tu res [30 , 31 , 32 , e tc . ] .

T h e f o l l o w i n g t w o v a r i a n t s o f t h e a b o v e - m e n t i o n e d te c h n i q u e s w e r e d e v e l o p e d : s i m p l i f i e d ( m o n o m i a l a p p r o x i m a t i o no f r e s o l v i n g f u n c t i o n s ) [ 1 , 2] a n d r e f i n e d [ 25 , 3 3, 4 4 ] ( p o l y n o m i a l a p p r o x i m a t i o n o f r e s o l v i n g f u n c t i o n s ) . B o t h v a r i a n t sp r o d u c e c l o s e r e s u l t s i f t h e s h e l l i s r e i n f o r c e d w i t h m o d e r a t e l y r i g i d r ib s [ 2 2 , 2 3 , 4 4 ] .

T h e t e c h n i q u e s o r i g i n a l l y p r o p o s e d f o r r i b b e d 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 a x i a l c o m p r e s s i v e f o r c e s w e r e t h e ng e n e r a l i z e d t o s h e l l s u n d e r e c c e n t r i c c o m p r e s s i o n [4 , 8 9 ], e x t e r n a l p re s s u r e [ 2 0 , 4 4 , 9 1 - 9 4 ] , t o r q u e s [ 8 5 ] , a n d c o m b i n a t i o n sof these and o the r loads [25 , 31 , 80 , 88 , e tc .] .

A f t e r w a r d s , t e c h n i q u e s w e r e d e v e l o p e d f o r s ta b i l i ty a n a l y si s o f r i b b e d c y l i n d r i c a l s h e l l s w i t h l a rg e r e c t a n g u l a rn o t c h e s u n d e r d i f f e r e n t l o ad s , a n d t h e e f f e c t o f d i f f e r e n t t y p e s o f r e i n f o r c e m e n t o f th e n o t c h e s o n t h e c r i t ic a l lo a d s w a s s t u d i e d[20 , 86 , 87 , e tc . ] .T e c h n i q u e s f o r s t a b i l it y a n a l y s i s o f r i b b e d c o n i c a n d s p h e r ic a l s h e ll s a n d r i b b e d s h e l ls o f a p o s i t i v e G a u s s i a nc u r v a t u r e w i t h a n a r b i t r a ry s h a p e o f t h e m e r i d ia n w e r e d e v e l o p e d [ 2 0, 5 3 , 5 4 , 9 1 - 9 4 ] .

T o d e s c r i b e t h e d e f o r m a t i o n o f s h e ll s, u se w a s m a d e o f b o t h t h e c l a s si c a l th e o r y o f s h e l l s b a s e d o n t h e K i r c h h o f f - L o v eh y p o t h e s e s f o r a s h e l l a n d t h e K i r c h h o f f - C l e b s c h h y p o t h e s e s f o r r i b s [2 0 , 2 5 , 31 , e t c .] a n d a r e f i n e d t h e o r y o f s he l ls b a s e do n t h e T i m o s h e n k o m o d e l f o r b o t h t h e s h e l l a n d t h e f ib s [ 5 6 , 6 3 , e t c. ] .

I n p a r t i c u l a r , i t w a s s h o w n f o r t h e f ir s t t i m e t h a t t h e e f f e c t o f t h e d i s c r e t e a r r a n g e m e n t o f r ib s i s p o t e n t i a t e d w h e nt h e b u c k l i n g o f a s h e l l is d e s c r i b e d u s i n g t h e r e f in e d t h e o r y o f s h e l ls [ 5 6 , 6 3 ] .

A s a f i n a l r e s u l t o f a l l t h e p r o b l e m s c o n s i d e r e d , r e c o m m e n d a t i o n s o n h o w t o c h o o s e t h e r a t i o n a l p a r a m e t e r s o fr e i n f o r c e m e n t w e r e f o r m u l a t e d [ 2 0 , 2 5 , 3 1, e tc . ].

F u r t h e r i n v e s t i g a t i o n s w e r e b a s e d o n t h e n o n l i n e a r e q u i l i b r i u m e q u a t i o n s f o r e v a l u a t i o n o f t h e c r it i c al s t r e ss e s i nshe l l s wi th in i t i a l de f lec t ions . In pa r t i cu la r , i t was show n tha t c r i t i ca l s t re s ses c lose to expe r im enta l va lues cou ld be ob ta ine di f th e d e s i g n p r o c e d u r e u s e s a s i n i t i a l d e f l e c t io n s t h o s e m e a s u r e d u n d e r n e a r l y c r i t ic a l l o a d s [ 2 5 , 3 8 , 3 9 , 7 5 , 7 6 , 8 2 ] .

M u c h a t t e n t i o n w a s g i v e n t o t h e l o a d - c a r r y i n g c a p a c i t y o f s h e l l s . I n p a r t i c u l a r , t e c h n i q u e s w e r e d e v e l o p e d f o rd e t e r m i n a t i o n o f t h e l i m i t l o a d s f o r c y l i n d r i c a l sh e l ls 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 a n d i n t e r n a l p r e s s u r e [ 3 1 ] a n d t h ec o l l a p s e l o a d s f o r l o c a l l y b u c k l e d s h e l ls t h a t a re c a p a b l e o f t a k i n g u p f u r t h e r i n c r e a s e i n t h e l o a d [ 3 8 ].

T h e c o n d i t i o n s f o r a p p l i c a b i l i t y o f t e c h n i q u e s b a s e d o n t h e t h e o r y o f s t r u c t u r a ll y o r t h o t r o p i c s h e l l s w e r e d e f i n e du s i n g t h e a s y m p t o t i c m e t h o d [ 2 5 , 5 8 ] ,

T h u s , I . Y a . A m i r o a n d h i s c o l l e a g u e s w e r e t h e f i r s t to f u l l y s t u d y a ll t h e f e a t u r e s o f t h e d e f o r m a t i o n o f f i b b e d s h e l l sb u c k l i n g d u e t o t h e d i s c r e te a r r a n g e m e n t o f ri b s.

A r e v i e w o f e x p e r i m e n t a l i n v e s t ig a t i o n s o f th e s t a b i l it y o f r i b b e d s h el l s is c o n t a i n e d i n [ 2 8 ] . A r e v i e w o f t h e o r e t i c a li n v e s t i g a t i o n s o f t h e s t r e s s - s t r a i n s t a t e o f r ib b e d s h e ll s u n d e r s t at ic a n d d y n a m i c l o a d i n g a n d s o m e g e n e r a l c o n s i d e r a t i o n sr e g a r d i n g t h e e f f e c t o f t he d i s c r e t e a r r a n g e m e n t o f r i b s o n t h e i r s ta b i li t y a r e p r e s e n t e d i n [ 1 0 0 ] . B e l o w , t h e s e c o n s i d e r a t i o n sa r e e x t e n d e d t o t h e a n a l y s is o f t h e b u c k l i n g o f r i b b e d s h e l l s d u e t o t h e d i s c r e te a r r a n g e m e n t o f f i b s . S t u d i e s w i t h s i m i l a r r e s u lt sa r e d e s c r i b e d i n [ 1 0 1 , 1 0 3 ] .

1 . S t a b i l i t y o f I s o t r o p i e S h e l l s u n d e r S t a ti c L o a d i n g .1 . 1 . D e s i g n P r o c e d u r e s . A t t h e f i r s t s t ag e o f t h e s t a b il i ty a n a l y s is o f r i b b e d s h e l l s , c i r c u l a r c l o s e d c y l in d r i c a l s h e l l 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 w e r e c o n s i d e r e d ( Fi g . 1 ). T h e m i x e d e n e r g y m e t h o d w a s u s e d . I t w a s a s s u m e d t h a t:- t h e c l a s si c a l i h e o r y o f s h e ll s b a s e d o n t h e K i r c h h o f f - L o v e h y p o t h e s e s c a n b e u s e d f o r d e s ig n o f t h e c a si n g a n d t h e

t h e o r y o f r o d s b a s e d o n t h e K i r c h h o f f - C l e b s c h h y p o t h e s e s f o r d e si g n o f th e f i b s ;- t h e s u b c r i t i c a l st a te i s h o m o g e n e o u s ;

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- t he ri bs and t he ca s i ng have i den t ica l def l ec t i ons and a ng l es o f ro t a t ion a l ong t he con t ac t l i nes ;- t h e rib s a n d t h e c a s i n g d e f o r m i n d e p e n d e n t l y a lo n g t h e r i bs ; t h e r e f o r e , t h e i r n o n s y m m e t r i c a r r a n g e m e n t a b o u t t h e

m e d i a n s u r fa c e o f t h e c a s i n g c a n n o t b e t a k e n i n t o a c c o u n t ; a n d- t he she l l i s r e i n fo rced wi t h an o r t hogonal ne t o f r i b s a r r anged a l ong i t s genera t r i ces and d i r ec t r i ces . Such a

fo rm ul a t i o n o f t he p rob l em w as f i r st p roposed by I . Ya . A m i ro [ 1 , 2 ] and was use d i n ma ny s t ud i es i n t h is sub j ec t a r ea . Int h i s connec t i on , fo l l owi ng t he m onograph [31 ] . we wi l l b r i e f l y s ta t e t he bas i c r e l a t ions u sed i n t h i s fo rmul a t i on .

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' i l . t 2 w i , t i FOj ~c1+ ~ Z E F j lZ j -+ + d xj = lk ] 2r~r[

+Bj~ d v 2- + ,/--~2 + Kj , ~ dr , ( 2 )

/ 2 = r [ ~A = p h f S i 1 [ 0 _ ~ _

o oL c - ) y - v J d x d v~gx 2J 2 t G3x)

562


5/27

awj dx.j= l 0

Here , x and y a re the Ca r te s ian co ord ina te s o f a po in t o n th e m edia n sur face o f the cas ing , w and ~p a re the d e f lec t ion o f the cas ingand the s tre s s func t ion , E an d v a re the e la s t ic m odu lus and P o i s son s ra t io o f the m a te r ia l o f the cas ing and the ribs ( i t wasas sum ed tha t the r ibs and the cas ing w ere m ade f rom the s am e m a te r ia l ), 17, r, and l a re the th icknes s and the rad ius o f the m ediansur face and the l eng th o f the she l l , E i, B j , / ~ , and k a re th e c ro s s -sec t iona l a rea o f the j th long i tud ina l rib , i t s bend ing r ig id i ty inthe rad ia l p lane , to rs iona l r ig id i ty , and the i r num ber , F jd B j , K j . , and k I are the characteris t ics (s im ilar to F j , B j , K j , and k) o f thej l th r ing r ib . e j and e j j a re the s t ra ins o f the axes o f the in d ica ted r ibs , 0 j and 0 j~ a re the ang les o f twis t o f the i r c ros s s ect ions , pis the axial com pre ss iv e s t ress in the cas ing, e j = - ~ , and ~j~ = 0, an d

E h 3D - 12 (1 - 0 2) "T h e c o n d i t i o n s o f r i g i d c o n t a c t o f t h e ca s i n g a n d t h e r i b s u n d e r t h e a b o v e a s s u m p t io n s a r e

O w Ij = w l y = y j . O j = -~ - y y = y ]

I ~ V j l = W I X = . ~ 1 ' 0 j l = ~ - ~ X m . , j l " ( 4 )w h e r e y j an d x j~ ar e t h e a x ia l c o o r d i n a t e s o f t h e j t h l o n g i t u d i n a l a n d t h e j l t h r i n g r i b s.

T o e v a l u a t e t h e s t r e s s f u n c t io n , t h e f o l l o w i n g e q u a t i o n w a s u s e d :b 4 a 4 . . . . 0 20 4 q ) + 2 q) , + (p E w ( 5 )O x 4 c 3 x Z O y - c3 y 4 r O x 2 "

A s f o l l o w s f r o m [ 2 5 ] , it w a s d i s c o v e r e d t h a t E q . ( 5 ) c o u l d n o t b e w r i t t e n w i t h a l l o w a n c e f o r t h e r i g i d c o n t a c t o f t h ec a s in g a n d t h e r i b s a l o n g t h e r i b a x e s . T h e r e f o r e , a c c o u n t i n g f o r t h e n o n s y m m e t r i c a r r a n g e m e n t o f t h e r i b s w i t h t he u s e o ft h e m i x e d e n e r g y m e t h o d c a l l s f o r th e i n t r o d u c t io n o f a d d i t i o n a l a s s u m p t i o n s a n d s u b s t an t i at i o n o f t h e p o s s ib i l it y o f u s i n gthem .

I n a c c o r d a n c e w i t h t h e o r d i n a r y p r o c e d u r e o f a p p l i c a t i o n o f t h e m i x e d e n e r g y m e t h o d t o t h e s o l u t i o n o f s h e ll s t a bi l it yp r o b l e m s , a f u n c t i o n a p p r o x i m a t i n g t h e d e f l e c t i o n o f a s h e l l i s s p e c i f i e d , t h e n qa i s d e t e rm i n e d f r o m ( 5 ) a n d w j , w j ? 0 i , and0 jj f r o m ( 4 ) . A f t e r t h a t , s u b s ti t u t i n g t h e f u n c t i o n a p p r o x i m a t i n g w a n d % w j , w j, O j, and 0j , into U and A , w e d e t e r m i n e t h ee x t r e m u m o f E , a n d f r o m t h e e q u a t i o n ( t h e s y s t e m o f e q u a t i o n s ) o b t a i n e d , w e f i n d t h e d e s ir e d c r i ti c a l s t re s se s .

S i n c e , u n d e r a c t u a l c o n d i t i o n s , t h e e d g e s o f s h e l l s a r e s t i f f e n e d b y s t r o n g r i n g s o r t h e r e a r e d i a p h r a g m s a t t h e m , int h e s tu d i e s b e i n g c o n s i d e r e d , t h e e d g e s o f t h e s h e ll w e r e a s s u m e d h i n g e d . I n t h i s c a s e , it is c o n v e n i e n t t o a p p r o x i m a t e t h ed e f l e c t io n o f t h e c a s i n g b y t h e d o u b l e t r i g o n o m e t r i c s e ri e s

c o s , v + s in " V )'= Z Z s in dm C m , , D i n , ,r rt~t= l 1 1=0

(6 )

7 tm rw h e r e d _ -t / l l

563


6/27

W h e n t h e d e f l e c t i o n i s a p p r o x i m a t e d i n s u c h a w a y , th e d e t e r m i n a t i o n o f t h e c r it i c al s t re s s e s i s r e d u c e d t o t h ec a l c u l a t i o n o f t h e m i n i m u m r o o t s o f i n fi n it e d e t e r m i n a n t s . I t i s p o s s i b l e t o s h o w t h a t t h e s e d e t e r m i n a n t s a r e o f a n o r m a l t y p ea n d , i n p r a c ti c e , t he r o o t s o f f i n i t e - o r d e r d e t e r m i n a n t s a r e ca l c u la t e d . S u c h a s o l u t i o n o f th e p r o b l e m i s u s e d i n [ 3 5 , 4 4 ] . I nt h e f ir s t s t u d i e s in t hi s s u b j e c t a r e a , f o r e s e e a b l e a n a l y t i c a l re s u l ts w e r e d e r i v e d t h r o u g h t h e m o n o m i a l a p p r o x i m a t i o n o f t h ed e f l e c t i o n [ 1 - 3 , 3 2 , 3 6 , 3 7 ] ( o n e t e r m w i t h i n d i c e s m a n d n w a s r e t a i n e d i n s e r i e s (6 ) ). I n t h i s c a s e , o n t h e a s s u m p t i o n t h a t a l lt h e r ib s o f o n e d i r e c t i o n h a v e t h e s a m e r i g i d i t y c h a r a c t e r i s ti c s ( t h e i n d i c e s j a n d Jl a r e o m i t t e d a n d B j = B , K j = K ,B jl = B 1, K it = K ] , a n d F j = F ) a n d a r e e q u i d i s ta n t , t h e f o l l o w i n g s i m p l e t h e o r e t i ca l f o r m u l a s f o r th e p a r a m e t e r o f c ri t ic a ls t re s s e s w e r e o b t a i n e d ( h e r e , a s o p p o s e d t o [ 2 5 ] , th e t w i s t i n g o f r i n g ri b s i n t h e a x i s y m m e t r i c b u c k l i n g m o d e i s n o t t a k e n i n t oa c c o u n t ) :

1 + "h- 2) + ( ~ k d ' 2 + - - '" q m O - l + y k L h d m 1 2 ( l - v [ , d ,~ ) j

I ~ 1 1 S 1 1 1 - - 1 + 2 I r e ; i n1 (2 ) [ O 'n n + 2 o q 6 s n d ~1 2 ( l - - v

( n - _ 1~2 , hi+ 2 13 rs.,.,,, n - + 2 c~ I ( Y 2 . , n d t~ + 2 [ ~ 1 (Ylm n -

f o r s = l , s ' = 2 , f or s = 2 , s ' = l , (7 )w h e r e

.O- PE h "h (d,~ + n 2) B K= - " ~ m , , - ~ , ~ - - - , 13 =r d ,Tz 2 r t r D 2 f o r D

kF ~31 K 1 tl Y,i~ / - 2 x r h ~ ~ 3 1 - 1 D " 6 I n = Z C O S 2 rj = l

k k , dm xJl k In 3) ~'~ 9 2 dm X j ,9 2,2 = E sin2 '31m Z cOs2 r = z~ s i n - - .I" r rj = l j t = l j l = lA n a n a l y s i s o f f o r m u l a s ( 7 ) s h o w s t h a t d e p e n d i n g o n t h e re l a t i o n s h i p s b e t w e e n k a n d n , k I a n d m , a n d , c o n s e q u e n t l y ,

b e t w e e n t h e w a v e l e n g t h o f t h e b u c k l i n g d e f l e c t i o n a n d t h e d is t a n c e b e t w e e n t h e r ib s , t h e re e x i s t q u a l i t a t iv e l y d i f f e r e n t p a t t e r n so f b u c k l i n g o f t h e r i b b e d s h e l l s d u e t o th e d is c r e t e a r r a n g e m e n t o f ri b s. I n A m i r o ' s t e rm s , t h e s e p a tt e r n s a r e c a l l e d d e f o r m a t i o nc a s e s . T h e r e a r e t h e f o l l o w i n g s i x d e f o r m a t i o n c a s e s f o r s h e ll s s t if f e n e d w i t h a r e g u la r n e t o f r ib s :

1 . T h e g e n e r a l d e f o r m a t i o n c a s e ( al l r i b s b e n d a n d t w i s t ) , m i s a l iq u a n t t o k ] + 1 a n d n i s a l i q u a n t t o k / 2 o r k ( f o re v e n o r o d d k , r e s p e c t i v e l y ) .

2 . T h e f i rs t s p e c i a l c a s e o f d e f o r m a t i o n ( l o n g i tu d i n a l r i b s b e n d a n d r i n g r ib s b e n d a n d t w i s t ) , m i s a l i q u a n t t ok 1 + 1 a nd n i s m u l t i p l e t o k / 2 o r k .

3 . T h e s e c o n d s p e c i a l c a s e o f d e f o r m a t i o n ( l o n g i tu d i n a l r i b s t w i s t a n d r in g r ib s b e n d a n d t w i s t) , m i s a l iq u a n t t ok I + 1 and 12 i s m ul t i p le to k / 2 o r k .

4 . T h e f o u r t h s p e c i a l c a s e o f d e f o r m a t i o n ( l o n g i t u d i n a l r i b s b e n d a n d t w i s t a n d r i n g r ib s t w i s t ) , m i s m u l t i p l e tok I + 1 a nd n i s a l i qu a n t t o k / 2 o r k .

5 . T h e f i f t h s p e c i a l c a s e o f d e f o r m a t i o n ( l o n g i t u d i n a l r i b s b e n d , t h e r i n g r i b s tw i s t ) , m i s m u l t i p l e t o k 1 + 1 a n d n i sm u l t i p l e t o k / 2 o r k .

5 6 4


7/27

6 . T h e e i g h th s p e c i a l c a s e o f d e f o r m a t i o n ( l o n g i t u d i n a l a n d r i n g r i b s t w i s t) , m i s mul t i p l e t o k 1 + 1 and n i s mu l t i p l eto k / 2 or k .

T h e t h ir d , s i x th , a n d s e v e n t h c a s e s o f d e f o r m a t i o n a r e r e a l i z e d u p o n b u c k l i n g o f r ib b e d s h e l l s t h a t a r e s t if f e n e d w i t ha r e g u l a r s y s t e m o f l o n g i t u d in a l r i b s a n d a s y s t e m o f r in g r i b s a r r a n g e d s o t h a t t h e d i s t a n c e s f r o m t h e f i r s t a n d t h e l a s t r i b s t ot h e e d g e s o f t h e s h e ll a r e e q u a l t o h a l f t h e d i s ta n c e b e t w e e n o t h e r r i n g r i b s .

N o t e t h a t t h e m e n t i o n e d c l a s s i f ic a t i o n i s o f f u n d a m e n t a l i m p o r t a n c e , s i n c e i f i t is n o t u s e d , i t i s i m p o s s i b l e t o c o r r e c t l yf o r m u l a t e p r o b l e m s o f o p t i m i z a t i o n o f t h e p a r a m e t e r s o f a r i b b e d s h e l l . A s i m i l a r c la s s i f i c a ti o n c a n a l s o b e c o n s t r u c t e d b a s e do n t h e e x a c t s o l u t io n s o f p r o b l e m s o f s t a b i l i ty o f c y l i n d r i c a l s h e l ls s t i f f e n e d w i t h l o n g i t u d i n a l o r r i n g r i b s i n o n e d i r e c t i o n .F o r e x a m p l e , t h e g e n e r a l c as e a n d t h e f i r s t an d s e c o n d s p e c i a l c a s e s o f d e f o r m a t i o n a r e r e a l i z e d in b u c k l i n g o f s h e l ls r e i n f o r c e dw i t h l o n g i t u d i n a l r i b s , an d t h e g e n e r a l c a s e a n d th e f o u r t h s p e c i a l c a s e o f d e f o r m a t i o n a r e r e a l i z e d i n b u c k l i n g o f s h e l l ss t i ff e n e d w i t h r i n g r i b s. I t s h o u l d b e n o t e d t h a t o v e r a w i d e r a n g e o f t h e r e a l p a r a m e t e r s o f r i b b e d s h e l l s , w a v e m o d e s a r er e a l i z e d f o r w h i c h r ib s o n l y t w i s t [ 2 5 , 8 1 ] (t h e s e c o n d s p e c i a l c a s e o f d e f o r m a t i o n f o r l o n g i t u d i n a l l y r e i n f o r c e d s h e l l s, t h ef o u r t h s p e c i a l c a s e f o r s h e ll s s t r e n g t h e n e d w i t h r in g r i b s , a n d t h e e i g h t h s p e c i a l c a s e f o r s h e l ls s t r e n g t h e n e d w i t h c r o s s r i b s ) .I n th i s c a s e , t h e a r r a n g e m e n t o f r i b s n o n s y m m e t r i c a b o u t t h e m e d i a n s u r f a c e o f t h e c a s in g d o e s n o t a f f e c t t h e v a l u e o f t h ec r i ti c a l s tr e s s e s , a n d t h e a s s u m p t i o n s a c c e p t e d i n d e r i v i n g f o r m u l a s ( 7 ) a r e q u i t e ju s t i f ie d .

W h e n t h e g e n e r a l c a s e o r s p e c i a l c a s e s o f d e f o r m a t i o n w h e r e t h e ri b s w o r k in b e n d i n g a r e r e a l i z e d , th ea b o v e - m e n t i o n e d f o r m u l a t i o n o f t h e p r o b l e m s h o u l d b e re f i n e d . I n t h is c o n n e c t i o n , I . Y a . A m i r o a n d h i s f o l l o w e r s h a v ed e v e l o p e d a n o t h e r te c h n i q u e f o r s t a b i l i ty a n a l y s i s o f r i b b e d s h e l l s t h a t i s b a s e d o n t h e s o l u t io n o f t h e i n d i c a t e d p r o b l e m i nd i s p l a c e m e n t s [ 2 5 , 29 , 3 0, 3 6 ]. L e t u s s t a te th i s t e c h n i q u e b y a n e x a m p l e o f c i r c u l a r c lo s e d c y l i n d r i c a l s h e l l s s u b j e c t to a x i a lc o m p r e s s i v e f o r c e s . T h e p o t e n t i a l s t ra i n e n e r g y o f t h e r i b b e d s h e l l u n d e r c o n s i d e r a t i o n i s w r i t te n a s

U2r~rf

f ~ ( a u + . . . . 2 ( l - v ) a u p__y_v_w2 ( l - v - ) 0 [ ( 3 x b y r 3 y r0

-a12~ r[/ 9 2Wd x d v + E h 3 a - w ~ 1 O vj -

" 1 2 ( 1 - V 2 ) 0 0 [ ( a x - a V 2 + r - f f T v J

[ ; ] }2 w ( a 2 w 1 a v ] ( a 2 w 1 a v. . . . . . _ { . _ - - - -2 ( 1 v ) 7 [ ~ + r 3 v J [ a x O y r 3 x d xd y2 :' P 2 w ,[ d u .+ ~ E I E F j + E l ) J ( d x 2 ) + E L j ( a x 2 ) + 2 ( l + v )j = l 0 Lax)

[ i , ,k, I I d -+ 4 2 1 ;j j = l 0 (d2w. w.I J l . 1 1- I- E ( v j I [ 7 ~ V 2 -} -+ E ~ j ,

and t he wor k o f ex t e r na l f o r ces i s

% Oi, + - - + dr,d y 2 r 2 (1 + v ) [ - ~ y 2 - -~ - y " ( 8 )

A = - ~ p f I d x d v - P Z F j ] + d x." _ d x d x0 0 j= l 0 (9 )

5 6 5


8/27

H e r e , u a n d v a re t he c o m p o n e n t s o f t h e v e c t o r o f d i s p l a c e m e n t s o f t h e m e d i a n s u r f a c e o f th e c a s i n g a l o n g t h e a x e s x a n d y .u j , v j , u j , , a n d V jl a r e t h e c o m p o n e n t s o f t h e d i s p l a c e m e n t s o f th e a x i s o f t h e j t h l o n g i tu d i n a l r i b a n d t h e j l t h r in g f i b , /v j , Ir j, a n dI k p a r e t h e m o m e n t s o f i n e rt ia o f t h e c r o s s s e c t i o n o f t h e j t h l o n g i t u d in a l r i b i n b e n d i n g i n t h e r a d i a l p l a n e , i n th e p l a n e p a r a l l e lt o t h e t a n g e n t t o t h e m e d i a n s u r f a c e o f t h e c a s i n g , a n d i n t o r s i o n , a n d / v j ~ , Iz j d a n d l k p j ~ a r e t h e c h a r a c t e r i s t i c s ( s i m i l a r t o( x j ' L j , and l k p j ) o f th e c r o s s se c t io n o f t h e j l t h r i n g r i b .

I n a d d i ti o n t o c o n d i t i o n s ( 5 ) . t h e f o l l o w i n g c o n d i t i o n s o f a r i g i d c o n t a c t b e t w e e n t h e c a s i n g a n d t h e ri b s a re i m p o s e d :

u j ( x ) = [ u ( x . y ) + h j t P l ( x . Y ) ] [ y = ? ) ,v j ( x ) : [ v ( x , y ) + h j , l ,, ( x , y ) ] ly = y

u j, ( Y ) = [ u ( x ' y ) + h j , ~ 2 ( x ' Y ) ] l x = ~ . "

v j , ( Y ) = [ v ( x , y ) + h j , c P 2 ( x . Y ) ] [ x = x j , , ( 10)

w h e r e h j a n d h j~ a r e t h e d i s ta n c e s t o t h e m e d i a n s u r f a c e o f th e c a s i n g f r o m t h e a x e s o f t h e j t h l o n g i t u d i n a l r ib a n d t h e j l t h r i n gO wr i b . r e spec t i ve l y ( t he eccen t r i c i t ie s o f t he r i bs h j , h j ~ > 0 i f t he ri bs a r e l oca t ed on t he i n t e r io r su r f ace o f the cas ing) , t pl - ~ x '

[ ~ w vand ~ -, = - [ - 3 - f + r )M o r e o v e r , a s a b o v e , i t i s a c c e p t e d t h a t t h e e d g e s o f t h e s h e ll a r e h i n g e d , a n d . in a d d i t i o n t o ( 6 ), t h e f o l l o w i n g s e r i e s

w e r e u s e d f o r a p p r o x i m a t io n o f t h e c o m p o n e n t s u a n d v o f th e d i s p l a c e m e n t v e c t o r:ooE d r n X E ( V +n m n ~ )= COS m n COS n sin n ,

Fm =O n=O \oov y + n c o s n /s in d m r ~ l n ,z co s n r ~r '

m =l i

( l l )

w h e r e A , n n , B , n n ,


9/27

T h e s e t e c h n i q u e s w e r e c o n s t r u c t e d f o r s h e l l s s t r e n g t h e n e d w i t h r i b s a r r a n g e d a l o n g t h e c o o r d i n a t e l i n e s . T h et e c h n i q u e f o r s t a b i l it y a n a l y s i s o f c y l i n d r i c a l s h el l s r e i n f o r c e d w i t h s p ir a l r i b s a n d l o a d e d b y a x i a l c o m p r e s s i v e f o r c e s a n dt o r q u e s i s p r o p o s e d a n d e x p e r i m e n t a l l y ju s t i f i e d i n [ 8 1 ] o n t h e b a si s o f t h e m i x e d e n e r g y m e t h o d a n d t h e m o n o m i a la p p r o x i m a t i o n o f t h e d e f le c t i o n .

T o a p p r o x i m a t e t h e r e s o l v i n g f u n c t i o n s , t h e t e c h n i q u e s s t a t e d i n t h e a b o v e - m e n t i o n e d s t u d i e s , a s a r u l e , e m p l o yd o u b l e t r i g o n o m e t r i c a l s e r i e s . T h e r e f o r e , t h e r e s u l t s o b t a i n e d p e r t a i n e d t o t h e w i d e l y o c c u r r i n g - - o n e o f t h e p o s s i b l ec o n d i t i o n s o f f i x a t i o n o f t h e s h e l l e d g e s - - h i n g e d s u p p o r t .

T o e s t i m a t e t h e e f f e c t o f t h e g r i p c o n d i t i o n s o n t h e c r i t i ca l s t re s se s , t h e l i m i t e q u a t i o n s o f t h e t h e o r y o f s t r u c t u r a l l yo r t h o t r o p i c s h e l l s w e r e a n a l y z e d u s i n g t h e p r o c e d u r e s d e v e l o p e d i n [ 5 2 ] , a n d a p p r o x i m a t e d e s i g n f o r m u l a s w e r e d e r i v e d[ 25 , 62 ] .

T h e s e f o r m u l a s w e r e u s e d i n s o l v i n g a n u m b e r o f a p p l i e d p r o b l e m s , in p a r t i c u la r , d e te r m i n i n g t h e c r i t i c a l r i g id i t y o fr i n g r i b s [ 2 5 ] . T h e s u f f i c i e n t c o n d i t i o n s f o r a p p l i c a b i l i ty o f t h e t h e o r y o f s t r u c tu r a l l y o r t h o t r o p i c s h el l s f o r t h e s t a b i l i t y a n a l y s i so f r i b b e d c y l i n d r i c a l sh e l ls w e r e e s t a b l i s h e d i n [ 58 ]9

U s i n g t h e m i x e d e n e r g y m e t h o d , t e c h n i q u e s f o r s o l u ti o n o f s o m e n o n l i n e a r p r o b l e m s o f t h e t h e o r y o f r i b b e dcy l in dr ic a l she l l s and , in pa r t i cu la r , she l l s wi th in i t i a l de f l ec t io ns were dev e lop ed [6 , 7 , 31 , 38 , 39 , 75 , 76 , 83 , 90] . T he po ten t i a ls t r ai n e n e r g y o f t h e c a s i n g a n d r i b s i n t h i s c a s e w a s w r i t t e n i n t h e f o r m ( 2 ) , a n d t h e e q u a t i o n o f c o m p a t i b i l i t y o f s t ra i n s w a ss e l e c t e d i n t h e f o r m

2 24 { [ ~ 2 (W +W O )]S x -~ -;] ( ~ 2 w O )_O X 4 - O x - c ] y 2 O y 4

_ . _ 0 20 9 ( w + w 0 ) O ' ( W + W o ) O ' W o Wo 1 0 2 w9 "~ + ' - - 2 - ~ " ( 1 2 )O x - O y - ~ x 2 b y - ax 2w h e r e w 0 is the ini t ia l deflect ion .

I n [ 3 9 ] , t h e i n i t ia l d e f l e c t i o n w a s d e t e r m i n e d e x p e r i m e n t a l l y b y m e a s u r i n g s h e l l s on a s p e c ia l a p p l i a n c e . T h e r e s u l tso f t h e m e a s u r e m e n t s w e r e p r o c e s s e d o n t h e a s s u m p t i o n t h a t w0 ca n b e r e p r e s e n t e d i n t h e f o r m

7n fi ( 2d in x nOY 2dm o x n o v )w 0 = Z Z a m 0n 0c ~ r ~ c o s r +bmonoSin r sin--Z-'r ' (1 3)m o = O n o = O

m o r t rw h e r e d m o I and fit nd ~ are given numbers.T h e d e f l e c t i o n o f a s h e l l i n t h e s t ra i n s ta t e w a s d e t e r m i n e d a s fo l l o w s :

9 4 , x . 9 n v xW = C 1 s in d,n x s in n y + C, , sm am r sm - --7- + C 3 sin 4 d,n r "r r (14)T h e p a p e r [ 3 9 ] w a s p u b l i s h e d a l m o s t s i m u l t a n e o u s l y w i t h th e w i d e l y k n o w n s t u d y [ 1 0 6 ], th e d i s c r e te a r r a n g e m e n t

o f r i b s b e i n g t a k e n i n t o a c c o u n t i n [ 3 9 ] , i n c o n t ra s t t o [ 1 0 6 ] . T o s o l v e s im i l a r p r o b l e m s , t h e f i n i t e - d i ff e r e n c e m e t h o d ( r e v i e w so f r e l a t e d s tu d i e s a l e c o n t a i n e d i n [ 4 9 , 5 0 ] , s e e a ls o [ 5 5 ] ) a n d t h e f i n i t e - e l e m e n t m e t h o d [ 6 9 - 7 1 ] w e r e u s e d . P o g o r e l o v ' sg e o m e t r i c a l m e t h o d w a s u s e d i n [ 4 7 , 9 8 ] f o r d e s ig n o f r i b b e d s h e l ls w i t h i n i ti a l d e f l ec t i o n s .

T h e o r e t i c a l i n v e s t i g a t i o n s , a s a r u l e , w e r e a c c o m p a n i e d b y e x p e r i m e n t s 9 T h e r e s u l t s o f t h e e x p e r i m e n t s w i l l b ed e t a i l e d b e l o w o n l y t o t h e e x t e n t s u f f i c i e n t t o s u b s t a n t ia t e t h e r e l i a b il i ty o f t h e d a t a o b t a i n e d t h e o r e t ic a l l y .

W e w i l l f u r th e r d e s c r i b e s t u d i e s o f t h e f e a t u r e s o f t h e d e f o r m a t i o n o f f i b b e d s h e l ls t h a t a r e d u e t o t h e d i s c r e t ea r r a n g e m e n t o f t h e f i b s ,

1 .2 . C y l i n d r 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 . T h e n e ce s si ty o f a l lo w i n g f o r th e d i sc r e te a r r a n g e m e n t o f r ib sin the s t ab i l i ty ana lys i s o f r ibbed she l l s fo l low s , in pa r t i cu la r , f ro m the re su l t s p resen ted in Tab les 1 and 2 [31 ] . Tab le 1 re la te sto a she l l s t i f fened wi th lo ng i tu d ina l r ibs ( h = 0 . 004 , o~ = 9 . 06 , ]3 = 5 . 23 , and ~ ' = 0 . 0278) and T able 2 to a she l l s t r eng then ed

5 67


10/27

TABLE 1

I / r

1 . 5 7

3.14

6.28

General

Deformat ion C aseSpec i a l

1 2

q0 m n q I m n q 2 rn n

16 1.445 1 6 0.564 1 8 4.56 1 824 1.647 1 6 1.152 1 12 13.8 3 1232 1.866 1 6 1.852 2 16 32.0 8 1616 0.995 1 5 0.564 2 8 4.56 2 824 1.107 1 5 1.064 3 12 13.8 6 1232 1.194 1 5 1.728 3 16 32.0 16 1616 0.780 1 4 0.564 4 8 4.56 4 824 0.827 1 4 1.023 5 12 13.8 13 1232 0.863 1 4 1.728 6 16 32.0 32 16

TABLE 2

General

110 /T/ II

2.28 1 32.47 3 4

Deformation Case

4

114 m n

1.15 2 51.44 3 51 . 8 9 4 6

Special

5

115 m n

1.29 6 121.53 6 121.85 4 12

with longitudinal and ring r ibs (- /= 6.26, ~ = 9.06, 13j = 5.23, and k = 24 ); 7= / , h, o~, 13, and "t' are the same as for the shel lrconsidered above. Note that the data presented indicate that in the examples considered, the critical-stress parameter isminimum for special cases of deformation, which confirms the necessity of allowing fo r the discrete arrangement of ribs. Thedata in Tables 1 and 2 were obtained using formula (7). In using the monomial approximatio n of the resolving functions forshells with rigid ribs, heavily overestimated values of the indicated parameter may be obtained. Table 3 contains the dataobtained in [44] for both mo nomia l and polynomial approximation of the deflection. A shell stiffened with longitudinal ribs(p =pr/Eh,"y= 0.0625, k = 8, h = 0.25-1 0-2 ,7= 2, n * is the maximum value o f n retained in the series app roximat ing thedeflection) is being considered. An analysis of the table shows that even for relatively small rigidities of ribs and in the

568


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T A B L E 3

D e f o r m a t i o nC a s e

G e n e r a l

F i r s t

S e c o n d

8 .9 3 8 9 .3 8 9 .3 8 9 .3 8 9 .3A p p r o x i m a t i o n

1 .37 1 .37 1 .37 1 .37 1 .370 .4 6 7 0 .5 8 8 0 .9 5 8 3 .1 5 1 1 .8 0 0

M o n o m i a l m 1 1 1 1 1n * 7 7 7 5 7

0 . 3 5 1 0 . 5 3 6 0 . 5 4 0 0 . 5 4 0 0 . 6 0 5P o l y n o m i a l m 1 1 1 1 2 3

n * 25 25 35 35 350 . 3 3 7 0 . 5 1 9 0 . 5 1 9 0 . 5 1 9 2 . 3 3 5

M o n o m i a l m 1 1 1 1 1

n * 8 8 8 8 80 . 3 2 7 0 .5 1 7 0 . 5 1 7 0 . 5 1 7 0 . 6 0 7

P o l y n o m i a l m 1 1 1 1 2 1n * 6 0 3 6 3 6 3 6 6 0

0 . 6 4 6 0 . 6 4 6 1 . 00 8 4 . 6 3 2 0 . 6 4 6M o n o m i a l m 2 3 2 3 2 3 2 3 2 3

n * 4 4 4 4 40 . 6 0 6 0 . 6 0 6 0 . 6 0 6 0 . 6 0 6 0 . 6 0 6

P o l y n o m i a l m 2 3 2 3 2 3 2 3 23n * 3 6 3 6 3 6 3 6 3 6

m o n o m i a l a p p r o x i m a t i o n o f t h e d e f l e c t i o n , t h e c r i t ic a l s t re s s e s o f th e g e n e r a l c a s e o f d e f o r m a t i o n a r e d e t e r m i n e d w i t h a l o wa c c u r a c y . A t t he s a m e t i m e , in t h e m o n o m i a l a p p r o x i m a t i o n o f t h e d e f l e c t i o n , t h e m i n i m u m c r i t i c a l s t r e s se s a r e d e t e r m i n e da c c u r a t e l y e n o u g h i n th e r a n g e o f p a r a m e t e r s b e i n g c o n s i d e r e d .T h e v a l u e s o f p a r e g i v e n i n T a b l e 4 f o r a s h e l l s t i f fe n e d w i t h c r o s s r i b s . T h e s e v a l u e s w e r e a l s o o b t a i n e d i n [ 4 4 ] i nt h e m o n o m i a l a n d p o l y n o m i a l a p p r o x i m a t i o n o f t h e d e f l e c t io n ( m * i s th e m a x i m u m v a l u e o f m r e t a i n e d i n th e s e r i e sa p p r o x i m a t i n g t h e d e f l e c t i o n , k I = 4 , k = 8 , c~ = 8 9 .3 , [3 = 1 .3 7 , a I = 9.1,13 1 = 1 .4 , ~ ,= 0 .0 6 2 5 , h = 0 . 2 5 .1 0 - 2 7= 2). H e r e ,P m in c o r r e s p o n d s t o t h e s e c o n d s p e c i a l c a s e o f d e f o r m a t i o n a n d c o r r e c ts t h e v a l u e o b t a i n e d i n th e m o n o m i a l a p p r o x i m a t i o no f t h e d e f l e c t i o n b y 8 % . N o t e t h a t f o r t h e c o n s i d e r e d s h e l l w i t h r a t h e r ri g i d r i b s , th e c r i t i c a l s t r e s se s f o r t h e g e n e r a l a n d s p e c i a lc a s e s o f d e f o r m a t i o n d i f f e r b y n o m o r e t h a n 5 % , w h i c h i n d i c a t e s t h at th e s h e l l b u c k l e s a s a p a n e l r i g i d l y f a s t e n e d a l o n g t h ee d g e s ( a s i m i l a r c o n c l u s i o n f o l l o w s f r o m [ 2 2, 2 3 , 25 , 3 3 ] ) .

5 6 9


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T A B L E 4Monom ial Approximation Polynom ial Approx imationDeformation Case

Gen er a l 2 .96 2 6 0 .650 18 21Fi rs t 2 .27 1 4 0.62 0 23 32

S eco nd 0 .67 23 4 0 .618 23 32F our t h 4 .30 5 11 0 .647 40 22F i f t h 5 .65 5 12 0 .646 70 32

E i gh t h 0 .71 25 4 0 .644 60 32

4

3 /

2 / / 0

1 " ~ - - . . _ . ._ ' . . _ _ . _ _ . ,. ~ . L - . - . ~

4 8 hl/hFig. 2

T h e e f f e c t o f t h e r i b e c c e n t r i c i ty o n t h e c r i t i c a l a x i a l - c o m p r e s s i o n s t r e s s e s i n r i b b e d c y l i n d r i c a l s h e l l s is s t u d i e d i n[ 2 5 , 3 0 ] . T h e m o s t t y p i c a l r e s u l t is sh o w n i n F i g . 2 , p r e s e n t e d i n [ 25 , 3 0 ]. A s h e l l s t r e n g t h e n e d w i t h l o n g i t u d i n a l e d g e s( h = 0 .25 - 10 - 2 , 7= 2 .1 , k = 32 , F = 4 .75 mm 2 , I v = 17 .4 m m 4, I c r = 0 . 4 m m 4 ) i s c o n s i d e r e d . T h e c a l c u l a t i o n s w e r e c a r r i e do u t w i t h m o n o m i a l a p p r o x i m a t i o n o f th e d is p l a c e m e n t c o m p o n e n t s . I n th e f i g u re , p = P , w h e r e P 0 is th e c r it ic a l s t r e s se s f o rP 0t h e u n s t i f f e n e d s h e ll , 0 , 1 , a n d 2 a r e c u r v e s p l o t te d f o r t h e g e n e r a l , f i rs t , a n d s e c o n d s p e c i a l c a s e s o f d e f o r m a t i o n , r e s p e c t i v e l y ,a n d t h e c o n t i n u o u s a n d d a s h - d o t t e d l in e s p e r t a i n t o s h e ll s w i t h i n t e m a l a n d e x t e r n a l r i b s , r e s p e c t i v e l y . A n a n a l y s i s o f th ec u r v e s i n F i g . 2 a l l o w s u s t o n o t e t h a t w h e n th e r i b s a r e a t t a c h e d t o th e o u t s i d e o f t h e c a s i n g , t h e c r i t i c a l s t r e s se s f o r t h e g e n e r a la n d f i r s t s p e c i a l c a s e s o f d e f o r m a t i o n a r e a l w a y s h i g h e r t h a n f o r t h e in t e r n a l ri b s . H o w e v e r , t h e s e c o n d s p e c i a l c a s e o f

I h l ld e f o r m a t i o n i s r e a l iz e d f o r ~ > 4 , a n d t h e c o r r e s p o n d i n g c r i t ic a l s t r e ss e s d o n o t d e p e n d o n t h e e c c e n t ri c i t y o f t h e ri b s .S i m i l a r r e s u l t s w e r e a l s o o b t a i n e d f o r s h e ll s s t re n g t h e n e d w i t h c r o s s r ib s [ 2 5 ] .

T h e p o s s i b i l i ty o f u s i n g t h e t h e o ry o f s t r u c t u r a l ly o r t h o t r o p i c s h e l ls f o r d e t e r m i n a t i o n o f t h e c r i ti c a l s t r e s s e s in a s h e l ls t i f f e n e d w i t h l o n g i t u d i n a l r i b s m a y b e j u d g e d f r o m T a b l e s 5 - 7 , w h i c h p r e s e n t t h e v a l u e s o f th e c r i t ic a l s t r e s s es f o r t h e g e n e r a l ,

5 7 0


13/27

T A B L E 5Exact Solution Formulas for Structurally Ortho~'opic Shells

km??/ PI

I n t e r n a l R i b s

12182 43 64 8

0 . 9 40 .9 10 . 8 60 . 8 60 . 8 20 . 7 9

101010

0 . 9 60 . 9 20 . 8 90 . 8 70 . 8 30 . 7 9

1 01 0101 0101 0

E x t e r n a l R i b s

12182 43 64 86 0

7 28 49 6

1 .001.00

1 .0 01 .001.011 .161.55

1.85

1.87

1.88

10101110

1 0

I117232 9

12"1212

1 .20

1.35

1 .46

1.56

1.69

1.78

1 .84

1.87

1.89

1 .90

1 0111111111112

1 21 21 2

f ir s t, a n d s e c o n d s p e c i a l c a se s o f d e f o r m a t i o n , r e s p e c t i v e l y , w h i c h w e r e o b t a i n e d u s i n g t h e e x a c t s o l u t i o n o f th e p r o b l e m . T h ed a ta p r e s e n t e d a r e ta k e n f r o m [ 2 1 , 2 2 , 2 5 ] . T h e c a l c u l a t i o n s w e r e c a r r i e d o u t f o r s h e l l s w i t h t h e f o l l o w i n g p a r a m e te r s:

h = 0 . 2 5 . 1 0 - " 7 = 1 , F 0 . 1 5 9 - 1 0 - l I c r_ , ~ _ 0 . 8 2 8 9 . 1 0 - 6 ,2 r r r h 2 r r r 3 hz _ z . I h l t- - = 0 .1 3 2 6 - 1 0 - 6 , - - - 0 . 5 3 0 5 - 1 0 - 6 , = 0 . 1 3 7 5 -1 0 - -2 r o . 3 h 2rcr3h r

h I i s t h e e c c e n t r i c i t y o f a r i b .F r o m T a b l e s 5 - 7 , i t is s ee n t h a t t h e t h e o r y o f s t r u c t u r a ll y o r t h o t r o p i c s h e l l s i s q u it e a p p l i c a b l e t o th e d e t e r m i n a t i o n

o f t h e c r i t i c a l s t r e s s e s i n s h e l l s s t i f f e n e d w i t h i n t e r n a l r i b s . I n th i s c a s e , t h e c r i t i c a l s t r e s s e s f o r th e f i r s t a n d s e c o n d s p e c i a l

571


14/27

TABLE 6Exact Solution

k ~ m m M ~bsi n n 1

6 1.00 11 312 1.01 11 018 1.04 11 024 1.12 11 036 1.54 12 048 2.29 14 0

ExternM ~bsm l l 1

1.00 11 31.01 11 01.04 11 01.12 11 01.53 12 02.27 14 0

App~'oximateFormula

m t l l

1 .0 0 1 1 3

1 .0 1 1 1 6

1 .0 3 1 1 9

1.09 11 121.37 11 181.90 13 24

TABLE 7Exact solution Approximate Formulak

p m nl p m n l6 1.00 11 3 1.00 11 312 1.01 11 0 1.01 11 61 8 1 .0 2 1 1 9 1 .0 3 1 1 9

24 1.05 10 12 1.09 11 1236 1.27 10 18 1.37 1 ! 1848 1 . 7 9 12 24 1.90 13 24

cases of deformat ion are higher than for the general case. As for shells strengthened with external ribs, they can be designedonly in terms of a theory that takes into account the discr ete arrangement o f ribs.

Tables 5 -7 conf irm the conclusion followi ng fro m Table 4 that the critical stresses in shells stiffened with rigid ribsare close to the critical stresses in a panel rigidly fastened along the edges (the latter were calculated by the approximateformula derived using the monomial approximation o f the displacement components and are presented in Tables 6 and 7).

Thus, we can prop ose a simplified technique for determination of the critical stresses of axial compression - - usingdesign formulas obtained by monomial ap proximati on o f the resolving functions supplemented by formulas obtained on theassumption that the edges of the shell are rigidly fastened [33].

Note that works [22, 25] were the first to show that a longitudinally reinforced shell under axial compression canbuckle with formation o f more than one half-wave along the generatrix (Fig. 3).

The stability of ribbed cylindrical shells subject to eccentric compres sion and bending was considered in [4, 31,89] .The basic result of these studies is that if the approximation of the deflection is quite accurate, then the critical stresses areclose to those under axial compression.

572


15/27

7

2Z~ . . . 60. ~ - 4 89 13 m

F i g . 3

T A B L E 8

Shell Ma terial r, nma l , m m h , m i l l ~ ' E ~ ' r A ~ o0 . 3 4 0 . 3 5 3

S m o o t h

S m o o t h

R i b b e d( k 1 = 4 ,k = 32)

S t e e l

A M G - 6 M

A M G - 6 M

2 0 0

2 0 0

2 0 0

3 9 0

4 5 0

4 5 0

0 .5

0 .8

0 .5

0 . 3 5 0 . 3 5 00 .4 4 0 . 3 8 - 1 60 . 3 0 0 . 3 4 1 20 .3 1 0 .3 4 9

0 . 4 2.4 9 - 1 70 .4 5 0 .5 3 1 50 . 4 7 0 . 5 6 1 60 .4 9 0 .5 5 1 10 . 5 2 0 . 5 8 1 00 .5 3 0 . 5 5

T h e e f f e c t o f t h e er r o r i n tr o d u c e d b y t h e m o n o m i a l a p p r o x i m a t i o n o f t he d i s p l a c e m e n t c o m p o n e n t s o n t h e cr i t ic a ls t r e ss e s in s h e l l s s u b j e c t t o e x t e r n a l p r e s s u r e w a s s t u d i e d i n [ 4 4 ]. T h e c o n c l u s i o n s d r a w n a r e c l o s e t o t h o s e p r e s e n t e d a b o v ef o r s h el l 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 .

T h e r e s u l t s o f t h e s t u d i e s o f t h e e f f e c t o f i n i t ia l d e f l e c t i o n s o n t h e c r it i c a l l o a d s a r e g i v e n i n [ 3 1 , 3 9 ] a n d i n T a b l e 8 .E l e v e n s h e l l s w e r e m e a s u r e d ( f o r r i b b e d s h e l ls , cr = 0 . 3 8 7 , [3 = 0 . 8 3 4 - 1 0 - 2 7 = 0 . 5 9 7 - 1 0 - 2 (z I = 8 .6 , [31 - - - 0 . 4 1 7 -1 0 1

X ,., = P c lr P e r a n d p T a r e c r i ti c a l l o a d s d e t e r m i n e d w i t h o u t a n d w i t h r e g a r d f o r t h e i n i t i a l d e f l e c t i o n s , P cE a r e t h e~ 'E = G ' ' l Pc r "

5 7 3


16/27


17/27

Fig. 4

TABLE 9

k q. kg/cm pT T P~cr- T P~cr/PT0 7.70 4.3 0.56

0.4 7.75 5.8 0.75

24

32

0.8 7.991.2 7.751.6 6.75

6.2 0.786.5 0.845.8 0.86

2.0 5.75 5.9 1.020 8.30 5.3 0.64

0.4 8.770.8 9.041.2 8.201.6 7.20

7.0 0.806.9 0.767.2 0.887.3 1.02

2.0 6.20 6.3 1.020 10.0 7.2 0.72

0.4 11.1 8.4 0.750.8 10.1 8.3 0.82481.2 9.10 8.6 0.941.6 8.10 7.9 0.972.0 7.10 8.3 1.17

575


18/27

T A B L E 1 0

4

M m 910- 6H " cm

M c r . 1 0 6 ,H " c m

0 . 1 7 00 . 1 7 0

0 . 1 8 60 . 1 9 0 0 . 2 3 20 . 1 5 0 0 . 2 6 5

0 . 2 7 50 , 2 8 9

0 . 2 3 5 0 . 2 9 40 . 2 9 4

0 . 1 4 0

0 . 2 0 6

0 . 2 3 4

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T h e e x p e r i m e n t a l i n v e s t i g a t i o n s a n d c a l c u l a t i o n s w e r e c a r r ie d o u t fo r s h e ll s m a d e f r o m O T 4 - 1 t i t a n i u m a l l o y( h = 0 .3 m m , r = 2 0 0 m m , a n d 1 = 3 9 0 m m l , R i n g r i b s h a v i n g a z - s h a p e d p r o f i le , a w a l l h e i g h t a n d w i d t h f l a n g e o f 5 r a m , a n da t h i c k n e s s o f 0 .5 m m w e r e p l a c e d o n t h e o u t s i d e s u r f a c e o f th e c a s i n g .

F r o m T a b l e 1 0 , i t i s s e e n t h a t t h e d i f f e r e n c e b e t w e e n t h e 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 v a l u e s o f c r i t ic a l t a n g e n t i a ls t r e s s e s is m u c h l e s s th a n u n d e r a x i a l c o m p r e s s i o n . T h i s , i n o u r o p in i o n , i n d i c a t e s t h a t t h e e f f e c t o f p r o c e s s i n g f a c t o r s o n t h es t a b i l i ty o f s h e l ls i s c o n s i d e r a b l y l e s s u n d e r t o r s i o n t h a n u n d e r a x ia l c o m p r e s s i o n . T h i s c o n c l u s i o n i s s u p p o r t e d b y a s i m i l a rc o m p a r i s o n f o r s h e ll s s t r e n g t h e n e d w i t h l o n g i t u d i n a l r i b s o r c r o s s r i b s.

1 . 4. C y l i n d r i c a l S h e l l s w i t h O p e n i n g s . E x t e n s i v e s t u d ie s o f t h e s t a b il i ty o f r i b b e d c y l i n d r i c a l s h e l ls w i t h l a r g er e c t a n g u l a r o p e n i n g s w e r e c a r r i e d o u t [ 3 4 , 8 6 , 8 7 ] . T h e d e s i g n p r o c e d u r e w a s c o n s t r u c t e d o n t h e a s s u m p t i o n t h a t t h e t o t a le n e r g y o f a s y s t e m c a n b e d e t e r m i n e d a s th e d i f f e r e n c e b e t w e e n t he e n e r g i e s o f a c lo s e d s h e ll a n d a p a n e l t h a t h a s b e e n c u to u t f r o m i t an d h a s d i m e n s i o n s e q u a l t o th o s e o f t h e o p e n i n g . T h e m a i n d i f fi c u l ty i n fo r m u l a t i n g t he p r o b l e m w a s a s s o c i a t e dw i t h t h e n e c e s s i t y o f d e s c r i b i n g t h e s u b c r i ti c a l s t a t e a s c o r r e c t ly a s p o s s i b l e . A n a n a l y s i s o f t h e e x p e r i m e n t a l d a t a s h o w s t h a tt h e f i e ld o f s u b c r it ic a l s t r e ss e s i s h o m o g e n e o u s a l m o s t e v e r y w h e r e e x c e p t f o r t h e c o m e r s o f t h e o p e n i n g s , T o d e s c r i b e t h ee x p e r i m e n t a l l y r e v e a l e d h e t e r o g e n e i t y o f th e s u b c r i t i c a l s t a te n e a r th e c o m e r s o f t h e o p e n in g s , i t w a s a s s u m e d t h a t p o i n t f o r c e sa r e a p p l i e d t o t h e s e lo c a t i o n s a n d t h e i r v a l u e s c a n b e d e t e r m i n e d f r o m e q u i l i b r i u m c o n d i t i o n s . T o e v a l u a t e t h e c r i t i c a l s t re s s e s ,t h e m i x e d e n e r g y m e t h o d w a s u s e d w i t h m o n o m i a l a p p r o x i m a t i o n o f th e d e f l ec t io n . T h e o b j e c t i v e o f t h e s t u d ie s w a s t o s e l ec ta n e f f i c ie n t r e in f o r c e m e n t o f t h e e d g e s o f t he o p e n i n g s . S e v e r al v a r i an t s o f re i n f o rc e m e n t w e r e c o n s id e r e d . T h e r e i n f o r c e m e n to f t h e o p e n i n g s b y b o r d e r i n g l o n g i t u d i n a l a n d r i n g r i b s o f h i g h r ig i d i t y t u r n e d o u t to b e th e m o s t e f f ic i e n t , S h e l l s s u b j e c t t oa x i a l c o m p r e s s i v e fo r c e s [3 4 , 8 6 ] a n d t o t h e c o m b i n e d a c t i o n o f a x i a l c o m p r e s s i v e f o r c e s a n d t o r q u e s [ 8 7 ] w e r e c o n s i d e r e d .

T a b l e 1 1 , t a k e n f r o m [ 3 4 ] , p r e s e n t s t h e r e s u l t s o f c o m p a r i s o n o f th e t h e o r e ti c a l a n d e x p e r i m e n t a l c r i ti c a l l o a d s f o re i g h t s e r i e s o f s h e l ls w e a k e n e d b y f o u r r e c t a n g u l a r o p e n i n g s , d i s t i n g u i s h i n g b y th e r ig i d i ty o f r i b s b o r d e r i n g t h e o p e n i n g a n db y t h e p r e s e n c e o r a b s e n c e o f l o n g i t u d i n a l o r r i n g r i b s l o a d e d w i t h a x i a l c o m p r e s s i v e f o r c e s . T a b l e 1 1 t e s t i f ie s t h a t t h ea p p r o x i m a t e d e s i g n p r o c e d u r e d e s c r i b e d a b o v e a l l o w s u s t o o b t a i n v a l u e s o f t h e c r i t i c a l l o a d s c l o s e t o e x p e r i m e n t a l . I t w a sr e v e a l e d d u r i n g e x p e r i m e n t s a n d c a l c u la t i o n s t h a t t h e g e n e r a l b u c k l i n g o f a s h e ll m a y b e p r e c e d e d b y l o c a l b u c k l i n g o f th ec a s i n g b e t w e e n r i b s .

1 . 5. C o n i c S h e l l s . T h e s t a b i l i t y o f r i b b e d c o n i c s h e l ls h i n g e d a l o n g t h e e d g e s a n d 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 so r e x t e r n al p r e s s u r e a n d t h e c o m b i n e d a c ti o n o f t h e s e l o a ds w a s s tu d i ed . R a t h e r s im p l e w o r k i n g f o r m u l a s w e r e d e r i v e d i n[ 2 0 , 9 3 , 9 4 ] w i t h m o n o m i a l a p p r o x i m a t i o n o f d i s p l a c e m e n t c o m p o n e n t s . P r a c t i c a ll y e x a c t v a l u e s o f t h e c r i ti c a l s t r e s s e s w e r e

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T A B L E 1 1Series P t h e o r , k g Pexp, kg P t h e o r / P e x p

B 960 990 1 .03C 2350 1480 0 .67D 1740 1910 1.10E 4 0 0 0 2 9 7 0 0 . 7 4

B 1 960 750 0 .78Cl 1560 1000 0 .64D1 1420 1790 1.25E l 1930 2450 1.27

o b t a i n e d u s i n g th e i r p o l y n o m i a l a p p r o x i m a t i o n [ 9 1 , 9 2 ]. S o m e r e s u l t s f r o m t h e s e s t u d i e s a re g i v e n b e l o w . F i g u r e 5 p r e s e n t st h e d e p e n d e n c e o f t h e d i m e n s i o n l e s s c r i ti c a l p r e s s u r e i n a c o n i c s h e l l s u b j e c t e d t o e x te r n a l p r e s s u r e a n d r e i n f o r c e d w i t hm e r i d i o n a l a n d r i n g r i b s o n t h e n u m b e r o f th e m e r i d i o n a l r i b s ( q l i s th e c r i t ic a l p r e s s u r e i n a s h e ll s t r e n g t h e n e d w i t h ri n g r i b s) .T h e i n d i c a te d d e p e n d e n c e i s o f a n o n m o n o t o n i c c h a r a c t e r , w h i c h i s d u e t o th e f e a s i b i l i ty o f s p e c i a l c a s e s o f d e f o r m a t i o n ,w h i c h a r e in d i c a t i v e o f t h e d e t e r m i n i n g i n f l u e n c e o f t h e d i s c r e t e a r r a n g e m e n t o f m e r i d i o n a l r i b s o n t h e c r i ti c a l p r e s s u r e . F ig u r e6 d e m o n s t r a t e s h o w t h e p o s i t i o n o f o n e f r a m e o f p e r m a n e n t w e i g h t a f f e c t s t h e c r i t ic a l p r e s s u r e (q 0 .5 is t h e c r i ti c a l p r e s s u r ew h e n t h e f r a m e i s l o c a t e d i n t h e m i d s e c t i o n o f th e s h e l l ) . T h e e f f i c i e n t d i s p o s i t i o n o f th e f r a m e m a y i n c r e a s e t h e c r i ti c a lp r e ss u r e b y a p p r o x i m a t e l y 4 0 % .

1 . 6. S p h e r i c a l S h e l l s . T h e s t a b i l i ty o f r i b b e d s p h e r i c a l s h e l l s u n d e r e x t e r n a l p r e s s u r e w a s s t u d i e d . T h e t e c h n i q u e f o rd e t e rm i n a t io n o f t he c r i ti c a l s t r e s se s i s b a s e d o n t h e e n e r g y m e t h o d w i t h a p p r o x i m a t i o n o f d i s p la c e m e n t c o m p o n e n t s b y d o u b l et r i g o n o m e t r i c s e r i e s . I n t h i s c a s e , t h e b o u n d a r y c o n d i t i o n s v = w = 0 a r e s a t i s f i e d e x a c t l y a n d u = 0 ( T 1 = 0 ) a n d(p = 0 ( M 1 = 0 ) a r e s a t i s f i e d i n t e g r a l l y . T h e c a l c u l a t i o n s w e r e r e s t r i c te d t o m o n o m i a l a p p r o x i m a t i o n o f t h e c o m p o n e n t s o ft h e d i s p l a c e m e n t v e c t o r . F i g u r e 7 s h o w s t h e d e p e n d e n c e o f t h e d i m e n s i o n l e s s p a r a m e t e r o f t h e c r i ti c a l s tr e s s e s o n th e f l e x u r a ls t if f n e ss o f th e m e r i d i o n a l r i b s ( t h e s o l i d l in e c o r r e s p o n d s t o t h e g e n e r a l c a s e o f d e f o r m a t i o n a n d t h e d a s h e d l in e t o th e s e c o n ds p e c i a l c a s e o f d e f o r m a t i o n ) . T h e d a t a o b t a i n e d i n d i c a t e t h a t t h e g e n e r a l c a s e o f d e f o r m a t i o n i s r e a l iz e d o n l y f o r v e r y s m a l lf l e x u ra l s t if f n e s s e s o f r i b s . A s i m i l a r r e s u l t w a s o b t a i n e d i n v a r y i n g t h e e c c e n t r i c i t y o f t he m e r i d i o n a l r i b s ( F i g . 8 , t h e s o l i dl in e s c o r r e s p o n d t o t h e g e n e r a l c a s e o f d e f o r m a t i o n o f s h e l l s r e i n f o r c e d w i t h e x t e r n a l r ib s a n d t h e d a s h - d o t t e d l in e s t or e i n f o r c e m e n t w i t h i n t e r n a l r i b s ) . S h a l l o w s p h e r i c a l s h e l ls w i t h i n i t ia l d e f l e c t i o n s w e r e c o n s i d e r e d i n [ 1 4 , 1 7 , 1 8] .

1 . 7. S h e l l s o f O t h e r S h a p e s . T h e t e c h n i q u e , s t a t e d i n S e c t i o n 1 .1 , w a s g e n e r a l i z e d i n [4 5 ] t o s h e l l s o f r e v o l u t i o n o fa p o s i t i v e G a u s s i a n c u r v a t u r e . T h e s o l u t i o n o f t h e p r o b l e m i n d i s p l a c e m e n t s w a s u s e d o n t h e a s s u m p t i o n t h a t t he ri g i d i ti e so f ri b s i n b e n d i n g i n t h e p l a n e t a n g e n t t o t h e m e d i a n s u r f a c e o f t h e c a s i n g a r e e q u a l t o z e r o . A p a r a b o l i c s h e l l r e i n f o r c e d w i t hr i n g r i b s w a s c o n s i d e r e d . I t w a s a s s u m e d t h a t t h e e d g e s o f t h e s h e l l w e r e h i n g e d a n d t h e d i s p l a c e m e n t c o m p o n e n t s c o u l d b ed e t e r m i n e d i n th e f o r m o f d o u b l e t r i g o n o m e t r i c s e ri e s . T h e r e s u l t s o f s t u d y i n g t h e i n f l u e n c e o f t h e n u m b e r a n d a r r a n g e m e n to f th e ri b s a r e t a b u l a te d i n T a b l e 1 2 ( q i s t h e d i m e n s i o n l e s s c r i t i c a l p r e s s u r e a n d ~ % i s t h e d i m e n s i o n l e s s d i s t a n c e f r o m t h e j l t hr ib t o th e u p p e r e n d o f t h e s h e l l ). A n a n a l y s i s o f T a b l e 1 2 s h o w s t h a t , a s e x p e c t e d , t h e v a l u e o f th e c r it i c a l p r e s s u r e s u b s t a n t ia l l yd e p e n d s o n t h e p o s i t i o n o f a r i b ( ~ ,j t = 0 . 7 2 5 i s th e c o o r d i n a t e o f t h e m a x i m u m d e f l e c t i o n o f a n u n s t i f f e n e d s h e l l i n b u c k l i n g ) .

1 .8 . D e t e r m i n a t i o n o f the Ef f i c i ent P a r a m e t e r s o f S h e l l s . T h e p r o b l e m w a s a s fo l lo w s : it is n e c e s s a ry t o fi n d t hem a x i m u m r e l a t i v e - e f f i c i e n c y c o e f f i c i e n t e q u a l t o t h e r a t i o o f t h e c r i t i c a l l o a d s f o r r i b b e d a n d u n s t i f f e n e d s h e l ls o f e q u a l

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q / q l

4 l I 18 25 k

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0.5 0 0.2 0.4 0.6 0.8 ~4 ~Fig. 5 F ig. 6

11

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1.00.5 1.0 1.5 i 0 2 4 6 I h l I /h

Fig. 7 F ig. 8

w e i g h t , I t w a s s h o w n t h a t t h e m i n i m u m w e i g h t o f t h e s h e l l c o r re s p o n d s t o th e m a x i m u m o f th i s co e f f ic i e n t. T h e m i n i m u ml o a d o b t a i n e d b y c o m p a r i n g t h e r e s u l t s o f c a l c u l a t i o n s f o r a ll th e c a s e s o f d e f o r m a t i o n d e s c r i b e d i n S e c ti o n 1 .1 w a s t a k e n a st h e c r i t ic a l l o a d . T h e r e l a t i v e - e f f i c i e n c y c o e f f i c i e n t w a s c a l c u l a t e d b a s e d o n o n e o f t h e w e l l - k n o w n s t r e n g t h c o n d i t i o n s a n dw a s c o m p a r e d w i th t h e c o e f f i c i e n t m e n t i o n e d a b o v e . T h e p r o c e d ur e f o r d e te r m i n i n g t h e r e l a ti v e - e ff i c ie n c y c o e f f i c i e n t iss t a t e d i n [ 8 , 3 1 ] . T h e n e c e s s i t y o f a l l o w i n g f o r th e d i s c r e t e a r r a n g e m e n t o f r i b s in d e t e r m i n i n g t h e e f f i c i e n t p a r a m e t e r s o fr i b b e d s h e l l s is c l e a r f r o m F i g . 9 . F i g u r e 9 s h o w s t h e d e p e n d e n c e s o f th e r e l a t i v e - e f f i c i e n c y c o e f f i c ie n t s ( la ) o n t h e r a t i o o ft h e w e i g h t o f a l l r i b s to t h e w e i g h t o f t h e s h e l l i t s e l f ( to ) f o r t h e g e n e ra l a n d s p e c i a l c a s e s o f d e f o r m a t i o n a n d f o r t h a t d e r i v e df r o m t h e s t r e n g t h c o n d i t i o n o n t h e a s s u m p t i o n t h a t t h e s h e l l a n d t h e ri b s a r e m a d e f r o m a n i d e a l l y p l a s ti c m a t e r i a l ( la T ) . F o ro3 < 1 . 5, t h e q u a n t i t y ~ tm i i s d e t e r m i n e d b y s p e c i a l c a s e s o f d e f o r m a t i o n , w h i c h i n d i c a t e s t h a t th e e f f i c i e n t p a r a m e t e r s o fr e i n f o r c e m e n t c a n b e f o u n d o n l y o n t h e b a s i s o f a t h e o r y t h a t t a k e s in t o a c c o u n t t h e d i sc r e te a r r a n g e m e n t o f r i b s . T h i s t e c h n i q u ew a s d e v e l o p e d i n [ 3 2 , 3 6 , 3 7 , 6 4 , 7 7 - 8 0 ] .

2 . T h e S t a b i li ty o f O r t h o t r o p i c S h e ll s u n d e r S t a ti c L o a d i n g . T h e d e s ig n p r o ce d u re s p r o p os e d b y I. Y a . A m i r ow e r e e x t e n d e d t o o r t h o t r o p i c l a m i n a t e d s h e l l s i n [ 4 0 , 4 3 , 5 6 ] ,

2 . 1 . C y l i n d r i c a l S h e l l s . F o r a n o r t h o t r o p i c s h e l l w i t h t h e e n d s s u p p o r t e d b y e l a st i c r i n g s , t h e e x a c t s o l u t i o n o f t h es t a b i l i ty e q u a t i o n s h a s b e e n d e r i v e d i n th e f o r m o f u n a r y t r ig o n o m e t r i c s e r ie s [ 4 1 ] . I t i s s h o w n t h a t i f t h e r ig i d i t ie s o f t h e r i b sa r e c l o s e t o t h e a c t u a l v a l u e s , t h e c r i t ic a l s t r e s s e s a r e c l o s e t o th o s e f o r a s h e l l h i n g e d a l o n g t h e e d g e s , w h i c h c o i n c i d e s w i t ht h e c o n c l u s i o n d r a w n i n [ 7 2 ] . T h e i n f l u e n c e o f t h e r i g i d i t y o f r ib s i n b e n d i n g i n t h e i r p l a n e o n t h e c r i ti c a l s tr e s s e s w a s s t u d i e d .T h e r e su l t s o f t h i s s t udy a r e d ep i c t ed i n F i g . 10 ( p i s t he r a t i o o f t he c r i t i ca l s t r e s ses t o "~--E l E 2 , E j and E 2 a r e t he e l a s t i cm o d u l i o f t h e s h e l l m a t e r i a l i n t h e a x i a l a n d t r a n s v e r s a l d i r e c ti o n s , r e s p e c t i v e l y , a n d ~, i s th e r a t i o o f th e f l e x u r a l s t i f f n e s s o f

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TABLE 12k j ~ h q . 1 0 4 m * n

0 0.113 10 211 0 . 5 0 . 1 2 5 1 4 241 0.725 0.176 14 232 0.33, 0.66 0.150 14 273 0.25, 0.50, 0.75 0.183 14 30

1.5

1.0

0.5

." 4 1 1

0 2Fig. 9

[aT

(0

the fibs to some given one). It follows from Fig. 10 that a rib can have so me optimal flexural stiffness such that the criticalstresses are maximum.

The stability of laminated orthotropic longitudinally reinforced shells subject to axial compr essive forces was studiedusing the working formulas derived in [42] on the basis of the mixed energy method. The effect of the number of ribs andthe thicknesses of layers on the critical stresses in a three-layer shell is investigated in [43] with monomial approximation ofthe deflection. The calculations have revealed that, as expected, the critical stresses are minimum in special cases ofdeformation for a comparatively small number of ribs and in the general case of deformation for a high nu mber of fibs. It wasshown that the critical stresses for ribbed shells cons idered are higher than for unsti ffened shells of the same weight.

The exact solutions of the problem in displacements for a laminated shell reinforced with either longitudinal or ringribs have been derived in [63]. In contrast to Section 1.1, a refined theory of shells based on the Timoshenko model was used.Longi tudina lly reinforced shells under axial compress ive forces and shells strengthened with ring fibs under external pressurewere considered. It was shown that as the lateral-shear modulus changes, the shape of the buckling wave changes too, whichis indicative of the significant effect of the discrete arrangement of the ribs.

A typical dependence of the critical stresses on the lateral-shear modulus is shown in Fig. 11 (the data presentedpertain to a laminated shell strengthened with ring ribs and subjected to external pressure. El I is the elastic modulus o f thematerial of the first layer in an axial direction, E r and G r are the elastic modulu s and the lateral-shear modu lus of the materialof the ring fibs). The horizontal section of the curve corresponds to the fourth special case of deformatio n and the droopingsection to the general case of deformation.

2.2. Shells of Revolution with a Positive Gaussian Cur vat ure , The difference of the design procedure used fromthat accepted in Section 1.7 lies in the fact that the refined theory based on the Timoshenko model was used in the formulationof the problem. The results of studies are reported in [56]. The influence of lateral-shear strains and the thicknesses of the

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p . 1 03I

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~3cr 9 03

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2 4 6 8 10 n 26 80 160 240 E r / G rF i g . 10 F i g .11

l a y e r s o n t h e c r i t ic a l st r e s s e s a n d b u c k l i n g m o d e s o f s h e ll s h i n g e d a l o n g e d g e s a n d s u b j e c t e d t o e x t e r n a l p r e s s u r e w a s s t u d ie d .A s e x p e c t e d , t h e b e h a v i o r o f th e c r it i c a l s tr e s s e s d e p e n d i n g o n t h e l a t e r a l - s h e a r m o d u l u s t u r n e d o u t t o b e s i m i l a r t o t h a t i nFig. 11.

3 . T h e S t a b il i ty o f S h e ll s u n d e r D y n a m i c L o a d i n g . I n [ 1 0 - 1 5 ], I . Y a . A m i r o p r o p o s e d a n um b e r o f o r ig i n ala n a l y t i c a l c ri t e ri a o f s t a b i li t y o f s h e l ls u n d e r d y n a m i c l o a d i n g . T h e s e c r i t e r i a a r e b a s e d o n a n a p p r o x i m a t e d e s t i m a t e o f t h el o a d c a u s i n g t h e d e f l e c ti o n s o f a s h e l l t o g r o w r a p i d l y . S i m p l e w o r k i n g f o r m u l a s w e r e d e r i v e d f o r a ll o f t h e m o s t t y p i c a lv a r i a n ts o f t i m e - v a r i a t i o n o f l o a d ( l i n e a rl y i n c r e a s i n g , l i n e a r l y d e c r e a s i n g , s t e p , a n d p a r a b o l i c ) .

T h e s t a b il it y o f c y l i n d r i c a l ( i n c l u d i n g o r t h o t r o p i c l a m i n a t e d [ 4 3 , 5 9 , 6 4 ] ) s h e l ls , s h a l l o w s p h e r i c a l s h e ll s [ 1 3 ] , a n ds h e ll s o f r e v o l u t i o n [ 4 6 ] w a s t h o r o u g h l y s t u d i e d . S h a l l o w s p h e r i c a l s h e l l s h a v i n g i n i ti a l d e f l e c t i o n s w e r e a n a l y z e d f o r s t a b il i ty[14, 15, 19].

S i n c e t h e b u c k l in g w a v e - f o r m a t i o n d i f f e rs u n d e r s ta t ic a n d d y n a m i c l o a d i n g , t h e d i s c re t e a r r a n g e m e n t o f r i b s a f f e c tst h e c ri t ic a l s t r e s s e s d i f fe r e n t l y a l s o. F i g u r e 1 2 p r e s e n t s t h e r e s u l t s o f c a l c u l a t i o n o f p f o r a n o r t h o t r o p i c c y l i n d r i c a l s h e l ls t r e n g tf i e n e d w i t h c r o s s r ib s ( t h e s o l i d li n e s c o r r e s p o n d t o a d y n a m i c l o a d l i n e a r l y i n c r e a s i n g i n t i m e a n d t h e d a s h e d l i n e t oa st a ti c l o a d ) . I n th e e x a m p l e c o n s i d e r e d , t h e c r i t i c a l s tr e s s e s f o r th e s e c o n d a n d e i g h t h s p e c i a l c a s e s o f d e f o r m a t i o n a r em i n i m u m . N o t e t h a t u n d e r s t a ti c a n d d y n a m i c l o a d i n g , t h e c h a r a c te r o f w a v e - f o r m a t i o n c h a n g e s f o r d i f fe r e n t v al u e s o f o ).

C o n c l u s i o n s . I . Y a . A m i r o a n d h i s d i s c i p l e s a n d c o l l e a g u e s d e v e l o p e d t e c h n i q u e s f o r t h e d e s i g n o f r ib b e d s h e l l s w i t ht h e i r b u c k l i n g f e a tu r e s d u e t o t h e d i s c r e t e a r r a n g e m e n t o f r i b s ta k e n i n t o a c c o u n t .

I t w a s s h o w n t h a t o n l y d o e s a l l o w a n c e f o r t h e i n d i c a te d f e a t u r e s e n a b l e u s t o o b t a i n r e l i a b l e v a l u e s o f t h e c r it i c a ls t re s s e s , a n d i t i s o n l y t h is c a s e w h e r e a c o r r e c t f o r m u l a t i o n o f t h e p r o b l e m o n t h e o p t i m a l p a r a m e t e r s o f r i b b e d s h e l l s i sp o s s i b l e .

T h e d e s i g n p r o c e d u r e s o r i g i n a l l y d e v e l o p e d f o r r i b b e d c y l i n d r i c a l s h e l l s w e r e e x t e n d e d t o ri b b e d c o n i c a n d s p h e r i c a ls h e ll s a n d r i b b e d s h e l ls o f r e v o l u t i o n w i t h a p o s i t i v e G a u s s i a n c u r v a t u r e a n d a n a r b i t r a r y f o r m o f t h e m e r i d ia n .

R i b b e d s h e l l s m a d e f r o m s i n g l e - l a y e r i s o t r o p i c o r l a m i n a t e d o r t h o t r o p i c m a t e r i a l s w e r e a n a l y z e d f o r s t a b i l i t y . B o t ht he c l a s si c a l t h eo r y o f r i b b e d s h e ll s a n d t h e r e f i n e d t h e o r y b a s e d o n t h e T i m o s h e n k o m o d e l w e r e u s e d .

C u r r e n t l y , i n t h e a u t h o r ' s o p i n i o n , t h e t e c h n i q u e s f o r d e s i g n i n g r i b b e d s h e l l s o f r e v o l u t i o n o n t h e a s s u m p t i o n t h a tt he s u b c r i ti c a l s ta te i s h o m o g e n e o u s m a y b e c o n s i d e r e d a l m o s t c o m p l e t e l y d e v e l o p e d . I n th e f u t u r e , i t m a y o n l y b e n e c e s s a r yt o a d j u s t c a l c u l a t e d r e s u lt s u s i n g m o r e p e r f e c t a p p r o a c h e s , i n c l u d i n g n u m e r i c a l t e c h n i q u e s .

I t w a s s h o w n t h a t c r it i ca l l o a d s c l o s e t o e x p e r i m e n t a l v a l u e s c a n b e o b t a i n e d i f t h e d e s i g n p r o c e d u r e a l l o w s f o r th ei n it ia l d e f l e c t i o n s m e a s u r e d o n a s h e ll , u s e d f u r t h e r f o r e v a l u a t i o n o f t h e c r i ti c a l l o a d s . C r i t i c a l l o a d s c l o s e r t o e x p e r i m e n t a lv a l u e s w e r e o b t a i n e d w h e n d e f l e c t i o n s m e a s u r e d u n d e r l o a d s c l o s e t o c r i t i c a l , e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s w e r e u s e da s t h e i n i ti a l d e f l e c ti o n s . T h e e f f e c t o f l o c a l b u c k l i n g o n t h e c r i t ic a l g e n e r a l - b u c k l i n g l o a d s w a s a s s e s s e d .

I n t h e a u t h o r ' s o p i n io n , f u r t h e r d e v e l o p m e n t o f t e c h n i q u e s f o r s t a b i l i t y a n a l y s i s o f r i b b e d s h e l l s s h o u l d f o l l o w t h ep a t h s i d e n t i f ie d b e l o w .

S t u d y o f t h e m u t u a l e f f e c t o f t h e h e t e r o g e n e i t y o f th e s u b c r i t i c a l s t a t e a n d t h e d i s c r e t e a r r a n g e m e n t o f r ib s , w h i c h ,a p p a r e n t l y , i n s o m e c a s e s m i g h t b e c o n s i d e r a b l e . S u c h s t u d i e s w o u l d y i e l d d a t a o n t h e n u m b e r s a n d r i g i d it i es o f ri b s s u c h

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I0. 4 0.2 0.8 1.4 co

Fig. 12

t h a t t h e c r i ti c a l l o a d s a r e m a x i m u m . T h i s o b j e c t i v e c o u l d b e a c h i e v e d o n l y i f a n a c c u r a t e t e c h n iq u e f o r d e t e r m i n a t i o n o f th ec h a r a c t e r i s t i c s o f th e s u b c r i t ic a l s t a t e w o u l d b e f o u n d . P r o b a b l y , w e s h o u l d r e t u r n t o th e d ir e c t so l u t io n o f t h e c o r r e s p o n d i n gn o n l i n e a r p r o b l e m [ 39 ] a n d t o t h e u s e o f w e l l - k n o w n m e t h o d s f o r so l v in g n o n l i n e a r p r o b l e m s o f th e t h e o ry o f s h e ll s .

S t u d y o f t h e m u t u a l i n f l u e n c e o f t h e s h e l l - g ri p c o n d i t i o n s a n d t h e d i sc r e t e a r r a n g e m e n t o f ri b s o n t h e c ri t i c a l s tr e s s e s .A s f o l l o w s f r o m [ 2 5 ], w h e n t h e d i s c r e t e a r r a n g e m e n t o f r ib s i s a l l o w e d f o r, th e i n f l u e n c e o f t he g r i p c o n d i t i o n s f o r a s h e l l o ni ts s t r e s s - s t r a i n s t a te i n c r e a s e s , a n d i n c r e a s e d i n f l u e n c e o f t h e i n d i c a t e d c o n d i t i o n s o n t h e c r it ic a l s tr e s s e s m a y b e e x p e c t e d .

I t w o u l d b e d e s i r a b l e t o c o n t i n u e s t u d y i n g t h e i n f l u e n c e o f t h e r i b h e i g h t o n t h e c ri t ic a l s tr e s s e s o f l o c a l a n d g e n e r a lb u c k l i n g . T o s o l v e th i s p r o b l e m , i t i s n e c e s s a r y t o d i s c a r d t h e a s s u m p t i o n t h a t r i b s a r e o n e - d i m e n s i o n a l e l a s ti c e l e m e n t s a n dt o c o n s i d e r t h e m a s p l a t e s r i g i d ly a t t a c h e d t o t h e c a s in g a l o n g t h e c o n t a c t l in e . A f t e r it s s o lu t io n , i t i s p o s s i b l e t o d i s c a r d t h ec o n s t r a i n t s o n t h e r i b h e i g h t , w h i c h a r e u s e d i n m a n y s t u d i e s d e v o t e d t o th e d e t e r m i n a t i o n o f t h e o p t i m a l p a r a m e t e r s o f s h e l l s .

T h e m e n t i o n e d p r o b l e m s a r e t o p i c a l b o t h f o r s h e l ls m a d e f r o m s i n g l e -l a y e r i so t r op i c m a t e ri a ls a n d f o r l a m i n a t e ds h e l l s m a d e f r o m o r t h o t r o p i c m a t e r i a l s . I n t h e l a tt e r c a s e , t h e e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e s tr e n g t h c h a r a c t e r i s t i c s o fs t r u c t u r a l e l e m e n t s i s a l s o i m p o r t a n t . I t i s o b v i o u s t h a t d a t a o b t a i n e d o n s t a n d a r d s p e c i m e n s m a y n o t a l w a y s b e u s e d f o re v a l u a t i n g t h e l o a d - c a r r y i n g c a p a c i t y o f s t ru c t u r e s . T h e i n d i c a t e d c h a r a c t e r is t i c s s h o u l d b e k n o w n f o r f i n d i n g t h e c a u s e s o fs t r u c tu r a l c o l l a p s e ( b u c k l i n g o f s t r u c t u r a l e l e m e n t s , l o s s o f s t r e n g t h o f th e m a t e r i a l , e t c. ) . S u ch s t u d i e s a r e o f s p e c i a li m p o r t a n c e f o r c o m p l e x s t ru c t u r a l e l e m e n t s w h i c h a r e r i b b e d s h e l l s.O f c e rt a i n in t e r e s t is f u r t h e r d e v e l o p m e n t o f m e t h o d s o f d e s i g n i n g r i b b e d p l a t e s a n d s h a ll o w f i b b e d s h e l l s w i t ha l l o w a n c e f o r th e d i s c r e t e a r r a n g e m e n t o f fi b s . P r e l i m i n a r y s t u d i e s s h o w e d t h a t i n th i s c a s e e f f e c t s a r e p o s s i b l e t h a t d i f f e rf r o m t h o s e r e v e a l e d e a r l i e r i n s t a b i l it y a n a l y s i s o f r i b b e d s h e l l s o f re v o l u t i o n .

REFERENCES

1 . I . Ya . Am i r o , "S t ab i l i ty ana l ys i s o f a r i bbed cy l i ndr i ca l she l l unde r l ong i t ud i na l comp r es s i on , " Prikl. Mekh., 6, N o . 3 ,2 7 2 - 2 8 1 ( 1 9 6 0 ) .

2 . I . Y a . A m i r o , " 'T h e s t a b i l i t y o f a r i b b e d c y l i n d r i c a l s h e l l u n d e r l o n g i t u d i n a l c o m p r e s s i o n , " Dop. Akad. Nauk Ukrainy,N o . 1 0 , 1 3 4 4 - 1 3 4 8 ( 1 9 6 0 ) .

3 . I . Ya . A mi r o , " 'Ana l ys i s o f t he u l t i ma t e l oad f o r r i bbed c y l i ndr i ca l she l l s und e r t he concur r en t ac t i on o f ax i a l f o r ces an di n t e r na l p r e s sur e , " PrikL Mekh., 7, N o . 5 , 4 9 6 - 5 0 2 ( 1 9 6 1 ).

4 . I . Ya . Am i r o , "S t ab i l i t y ana l y s i s o f a r i bbed cy l i ndr i ca l she l l unde r eccen t r i c com pr es s i o n , " Prikl. Mekh., 8, N o . 4 , 3 5 9 - 3 6 7( 1962) .

5 . I . Ya . Ami r o , " 'T he s t ab i l i t y o f t h i n cy l indr i ca l she l l s , " Prikl. Mekh., 9 , N o . 3 , 2 6 4 - 2 6 8 ( 1 9 63 ) .6 . I . Ya . Am i r o , "An a l ys i s o f t he s t ab i l i t y and supe r c r i t i ca l s tr a i ns o f ri bbed cy l i ndr i ca l she ll s unde r ax i a l com pr e s s i o n , "

Prikl. Mekh., 1 , No . 10 , 30 - 37 ( 1965) .

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7 . I . Ya . Am ir o , "The e f f ec t o f the in it ia l de f lec t ions on the s tab i l ity o f r ibbed cy l indr ic a l she ll s under ax ia l com pr es s ion , "Prikl. Mekh., 2 , No. 1 , 53 - 5 8 ( 1966) .

8 . L Y a . A m ir o , "S tudy ing the in f luence o f the r e in f or cem en t pa r am ete r s o f r ibbed cy l ind r ica l she l l s on the c r it i ca l ax ia ll o a d , " Prikl. Mekh., 5 , No. 2 , 71- 78 ( 1969) .

9 . I . Ya . A m ir o , "S tab i l i ty ana lys i s o f r ibbed cy l indr ica l she l l s ," Prikl. Mekh., 8 , No. 12 , 15- 24 ( 1972) .10 . I . Ya . Am ir o , "De te r m ina t ion of c r it i cal com pr es s ive f o r ces r ap id ly inc r eas ing in t im e , " Prikl. Mekh., 1 5 , N o . 5 , 5 4 - 6 0

( 1 9 7 9 ) .11. I . Ya . A m ir o , "De te r m in a t ion of the c r i ti ca l dyn am ic load (a s tep impulse),"Prikl. Mekh., 16 , No. 8 , 75- 83 ( 1980) .12 . I . Ya . A m ir o , "D e te r m ina t ion of the c r i ti ca l pa r am ete r s o f a load r ap id ly dec r eas in g in t im e ( a t r i angula r im pu lse ) , " Prikl.Mekh., 16 , No. 1 , 70 - 76 ( 1980) .13 . I . Ya . Am ir o , "De te r m i

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