A consistent first approximation in the general theory of thin eleastoc shells

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A c o n s i s t e n t f i r s t ajyproxima,tion i n t h e g e n e r a l t h e o r y o i t h i n e l a s t i c s h e l l s P a r t 1 , f o u n d a t i o n s and l i n e a r t h e o r y b y W.T, K o i t e r (Technolo^^'ical U n i v e r s i t y , D e l f t ) 3 1 . INTRODUCTION

2. FOUITOATIONS: LOVE'S APPROXIMATE STRAIN-ENERSY EXPRESSION

, 2.1 B a s i c a s s u m p t i o n s 2.2 D e r i v a t i o n o f s t r a i n - e n e r ^ ^ y e x p r e s s i o r i 2.3 J u s t i f i c a t i o n o f LOVE's e x p r e s s i o n a s a c o n s i s t e n t f i r s i ; a p p r o x i m a t i o n 2.4 I n c o n s i s t e n c y o f h i e i h e r a p p r o x i m a t i o n s " 2.5 D i s c u s s i o n o f r e s u l t s

LINEAR THEORY (IKFIITITESILiAL DEFLECTIONS) -." 3.1 E x t e n s i o n a l s t r a i n s and r o t a t i o n s 3.2 Changes o f c u r v a t u r e 3.3 S t r e s s - s t r a i n r e l a t i o n s 3.4 E q u a t i o n s o f e q u i l i b r i u m and b o u n d a r y c o n d i t i o n s . 3.5 D i s c u s s i o n and c o m p a r i s o n w i t l i e a r l i e r v / r i t e r s REFEitENCES

APPEIJDIX: TENSOR NOTATION AlTD GEOMETRICAL RELATIONS A l Geometry o f s h e l l s A2 C o v a r i a n t d i f f e r e n t i a t i o n A3 The s t r a i n t e n s o r A4 C o m p a t i b i l i t y c o n d i t i o n s A5 T r a n s l a t i o n i n t o n o n - t e n s o r i a l n o t a t i o n

'9/d lo

/o

L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L Au£u.ct 5 t h , 1959.

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P R E F A C E ,

T h i s r e p o r t was i n t e n d e d f o r p r e s e n t a t i o n a t t h e I.Ü.T.A.LÏ. Symposium on t h e T h e o r y o f T h i n E l a s t i c S h e l l s ( D e l f t , 2 4 t h t o 2 8 t h A u g u s t , 1 9 5 9 ) . However, when t h e r e p o r t was c o m p l e t e d i n i t s

p r e s e n t f o r m - o m i t t i n g , a n e x t e n s i v e d i s c u s s i o n o f n o n l i n e a r t h e o r y w h i c h was o r i g i n a l l y a l s o e n v i s a g e d t o be i n c l u d e d - i t t u r n e d out t o be t o o l o n g f o r p r e s e n t a t i o n a t t h e s y m p o s i u m , and more i n p a r t i c u l a r f a r t o o l o n g f o r i n c l u s i o n i n t h e P r o q e e d i n g s , The a u t h o r t h e r e f o r e d e c i d e d t o p r e p a r e , u n d e r t h e same t i t l e , a s e p a r a t e p a p e r f o r p r e s e n t a t i o n a t t h e s y m p o s i u m , c o n t a i n i n g o n l y t h e m o s t i m p o r t a n t r e s u l t s o f t h e a n a l y s i s and t h e b a s i c a s s u m p t i o n s and a r g u m e n t s i n t h e i r d e r i v a t i o n . S i n c e t h e symposium p a p e r f r e q u e n t l y r e f e r s t o t h e p r e s e n t r e p o r t f o r s u b s t a n t i a t i o n , t h i s r e p o r t i s a l s o i s s u e d t o p a r t i c i p a n t s i n t h e s y i i i p o s i u m . l t i s hoped t o p u b l i s h t h e p r e s e n t d e t a i l e d r e p o r t a t some f u t u r e d a t e , i f p o s s i b l e i n a more c o m p l e t e f o r m b y a d d i t i o n o f a f u l l d i s c u s s i o n o f n o n l i n e a r t h e o r y . D e l f t , 5 t h A u g u s t 1 9 5 9 , L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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2 -1. IIITROSUCTIO:i. The g e n e r a l e q u a t i o n s o f t h e t h e o r y o f t h i n e l a s t i c s h e l l s have o f t e n b e e n d i s c u s s e d i n r e c e n t y e a r s . The araount o f l i t e r a t u r e on t h i s s u b j e c t i s nov; n e a r l y o v e r w h e l i a i n g , and t h e l i s t o f r e f e r e n c e s a t t i e end o f t h i s ' r e p o r t q u o t e s o n l y some, i n o u r o p i n i o n , more i m p o r t a n t p a p e r s ; a c o m p r e h e n s i v e b i b l i o g r a p h y on s h e l l t h e o r y has been c o m p i l e d by MSH L33j« most a u t h o r s d e r i v e t h e i r e q u a t i o n s on t h e b a s i s o f t h e w e l l - k n o w n lOVE-KIRCHHOPP a s s u m p t i o n s :

( a ) p o i n t s w h i c h l i e on one and t h e same n o r m a l t o t h e u n d e f o r m e d m i d d l e s u r f a c e a l s o l i e on one and t h e same n o r m a l t o t h e d e f o r m e d m i d d l e s u r f a c e ; ( b ) t h e e f f e c t o f t h e n o r m a l s t r e s s on s u r f a c e s p a r a l l e l t o t h e m i d d l e s u r f a c e may be n e g l e c t e d i n t h e s t r e s s - s t r a i n r e l a t i o n s ; ( c ) t h e d i s p l a c e m e n t s 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 a r e a . j p r o x i m a t e l y e q u a l f o r a l l p o i n t s on t h e same n o r m a l . The w i d e v a r i e t y o f r e s u l t i n g e q u a t i o n s , t o be f o u n d i n t h e l i t e r a b i i r e , i s due t o v a r i a t i o n s i n r i g o r and t o d i f f e r e n t a p p r o x i m a t i o n s i n t h e s u b s e q u e n t a n a l y s i s . Some a u t h o r s c l a i m a h i g h e r a c c u r a c y f o r t h e i r e q u a t i o n s , as compared t o t h o s e o f o t h e r w r i t e r s , on t h e g r o u n d s o f a more r i g o r o u s d e r i v a t i o n f r o m t h e b a s i c a s s u m p t i o n s . I n v i e w o f t h e a p p r o x i m a t i v e c h a r a c t e r o f t h e LOVE-KIRCHHOFF a s s u m p t i o n s t h e s e c l a i m s a r e open t o some d o u b t . I t may w e l l happen, and t h i s i s even a c t u a l l y so i n most c a s e s , t h a t t h e r e f i n e m e n t s made a r e o f t h e same o r d e r o f m a g n i t u d e as t h e e r r o r s v/hich r e m a i n on a c c o u n t o f t h e b a s i c a s s u m p t i o n s , and such r e f i n e m e n t s a r e o f c o u r s e m e a n i n g l e s s i n a g e n e r a l t h e o r y . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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3 -I t i s t h e p r i m a r y p u r p o s e o f t h e p r e s e n t pcipei' t o eiive t h e f o u n d a t i o n s f o r s h e l l t h e o r y i n as s i m p l e a f o r m as i s c o n s i s t e n t w i t i . t h e b a s i c a s s u m p t i o n s ; a l t h o u g l i l a r g e d e f l e c t i o n s a r e p e r m i t t e d , i t w i l l be assumed t h a t t h e s t r a i n s a r e s m a l l e v e r y w h e r e . Our pua?pose i s a c h i e v e d b y a s t r a i n - e n e r g y a p p r o a c h ( s e c t i o n 2 ) . I t i s shown t h a t LOVE's s o - c a l l e d f i r s t a p p r o x i m a t i o n f o r t h e s t r a i n e n e r g y , as t h e sum o f s t r e t c h i n g o r e x t e n s i o n a l e n e r g y and bendxn^^i, o r f l e x u r a l e n e r g y , i s a c o n s i s t e n t f i r s t a p p r o x i m a t i o n , and t h a t no r e f i n e m e n t ot t h i s f i r s t a p p i - o x i m a t i o n i s j u s t i f i e d , i n g O i . e r a l , i f t h e b a s i c L0Y£-KIRCIiKOFI'' a s s u i u p t i o n s ( o r e q u i v a l e n t a s s u a p t i o n s ) a r e r e t a i n e d . T h i s f a c t i s p r o v e d by s h o w i n g t h a t t h e e r r o r s o f L0V2*s f i r s t a p p r o x i m a t i o n , as compared t o a r i g o r o u s e l a b o r a t i o n o f t h e LOVE-KHtCH-HO'FF a s s u m p t i o n s , a r e o f t h e same o r d e r o f m a g n i t u d e as t h e s t r a i n e n e r g y due t o t r s a i s v e r s e n o n n a l and s h e a r s t r e s s e s , w h i c h a r e i n l i e r e n t l y n e g l e c t e d i f t h e LOVE--KinCHHOF]? a s s u m p t i o n s a r e employed. M o r e o v e r , i t i s shown t h a t i t i s a l w a y s p e r m i s s i b l e , i n LOVE's f i r s t a p p r o x i m a t i o n , t o add t e r m s o f t y p e e/R (vvhere e i s any p h y s i c a l m i d d l e s u r f a c e s t r a i n component and R any r a d i u s o f c u r v a t u r e o r t o r s i o n o f t h e m i d d l e s u r f a c e ) t o e x p r e s s i o n s f o r t h e p h y s i c a l changes o f c m - v a t u r e . I n o t h e r w o r d s , e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e , w h i c h d i f f e r o n l y b y t e r m s o f t y p e e/R, a r e e q u i v a l e n t i n LOVE's c o n s i s t e n t f i r s t a p p r o x i m a t i o n . The f o u n d a t i o n s o f s e c t i o n 2 a r e a p p l i e d t o t h e l i n e a r t h e o r y o f s h e l l s i n s e c t i o n 3. The e x p r e s s i o n s f o r t h e e x t e n s i o n a l s t r a i n s and r o t a t i o n s a r e w e l l --known. Our e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e , i n t e n s o r f o r m i n i t i a l l y d e f i n e d as t h e c o v a r i a n t d e r i v a t i v e o f t h e s t r a i n t e n s o r i n s u r f a c e s p a r a l l e l t o t h e m i d d l e sui-face w i t h r e s p e c t t o t h e c o o r d i n a t 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 , a p p e a r t o be new and more g r a p h i c t h a n e x i s t i n g e x p r e s s i o n s . The s t r e s s - s t r a i n r e l a t i o n s have a p a r t i c u l a r l y s i m p l e f o r m , and t h e e q u a t i o n s o f e q u i l i b r i . u v u and b o u n d a r y c o n d i t i o n s . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A

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nFB 4 nFB o b t a i n e d by t h e v a r i a t i o n a l m e t h o d , a g r e e w i t h t h e v / e l l --known r i g o r o u s e q u a t i o n s . A c o m p a r i s o n w i t l i e x i s t i n g t h e o r i e s s l o w s t h a ^ most w r i t e r s have o b t a i n e d r e s u l t s w h i c h , a l t h o u g h dif±-erent i n a p p e a r a n c e , a r e e q u i v a l e n t f r o m t h e p o i n t o f v i e w o f t h e f i r s t a ^ j p r o x i m a t i o n i n s h e l l t h e o r y . On t h e o t h e r h a n d , some e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e g i v e n i n t h e l i t e r a t u r e a r e shov/a t o be i n a d e q u a t e f o r g e n e r a l a p p l i c a t i o n . The a d v a n t a g e s o f t e n s o r a n a l y s i s i n t h e t h e o r y of s h e l l s a r e now w i d e l y r e c o g n i z e d f e . g . 5,12,14,26,34, 36,42,4Q.;, and no a p o l o g y i s needed f o r i t s f r e e use i n

t h e p r e s e n t p a p e r . An e x p l a n a t i o n o f t h e n o t a t i o n s e m p l o y e d | m o s t l y t a k e n f r o m [12], i s g i v e n i n t h e a p p e n d i x , t o g e t h e r v / i t h some more i n v o l v e d g e o m e t r i c d e r i v a t i o n s . The m a i n r e s u l t s have a l s o been g i v e n i n t h e more u s u a l n o t a t i o n , i n o r d e r t o f a c i l i t a t e c o m p a r i s o n w i t h e x i s t i n g l i t e r a t u r e ; t h e t r a n s l a t i o n r u l e s f r o m g e n e r a l t e n s o r n o t a t i o n t o t h e more c o n v e n t i o n a l n o t a t i o n f o r g e n e r a l o r t h o g o n a l p a r a -m e t r i c c u r v e s on t h e -m i d d l e s u r f a c e a r e a l s o su-m:iiarized i n t h e a p p e n d i x . 2. F O U ' T S A T I Q H S : 1075*3 A P P R O X I L I A T ; :^ ' x t A I K - E H E u G Y EXPRBSoION. 2.1 B a s i c ass-umptions. A c o m p l e t e t h e o r y o f t h i n e l a s t i c s h e l l s i n a c o n -s i -s t e n t f i r -s t a p p r o x i m a t i o n , and v a l i d f o r d e f l e c t i o n -s o f ainy m a g n i t u d e , may be based on LOVE's a p p r o x i m a t e s t r a i n - e n e r g y e x p r e s s i o n . T h i s s t r a i n - e n e r g y e x p r e s s i o n v / i l l be d e r i v e d on t h e b a s i s o f t h r e e a s s u . i p t i o n s : ( a ) t h e s h e l l i s t h i n , i . e . h / R 4 c l , v/here h i s t h e s h e l l t h i c k n e s s , and R i s t h e s;.-allest p r i n c i p a l r a d i u s o f c u r v a t u r e o f t h e m i d d l e s u r f a c e ; ( b ) t h e s t r a i n s a r e s m a l l e v e r y w h e r e , a l t h o u g l i l a r g e d e f l e c t i o n s a r e a d m i t t e d , and t h e s t r a i n e n e r g y p e r u n i t volume o f t h e u n d e f o r m e d body i s r e p r e s e n t e d by t h e q u a d r a t i c f u n c t i o n o f t h e L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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s t r a i n components f o r an i s o t r o p i c s o l i d (HOGKE's l a w ) ; ( c ) t h e s t a t e o f s t r e s s i s a p p r o x i m a t e l y p l a n e , i . e . t h e e i f e c t o f t r a n s v e r s e s h e a r s t r e s s e s and o f t h e t r a n s v e r s e n o r m a l s t r e s s may be n e g l e c t e d . I t v / i l l be o b s e r v e d t h a t we have r e p l a c e d t h e u s u a l LOVE-KIRCHHOFE a s s u m p t i o n s ( ( a ) t o ( c ) i n s e c t i o n 1 ) by t h e s i n g l e a s s u m p t i o n o f p l a n e s t r e s s ( c ) above, w h i c h i s o f c o u r s e e f f e c t i v e l y e q u i v a l e n t t o t h e u s u a l assump-t i o n s . 2.2 D e r i v a t i o n o f s t r a i n - e n e r g y e x p r e s s i o n . On t h e b a s i s o f a s s u m p t i o n s (b) and ( c ) o f p a r . 2.1 t h e s t r a i n e n e r g y p e r u n i t volume o f t h e undeformed b o d y i s g i v e n b y ( l 2 , p.3841 where y^g c o v a r i a n t s t r a i n t e n s o r i n s u r f a c e s p a r a l l e l t o t h e m i d d l e s u r f a c e , E°•^^*^ i s t h e c o n t r a -v a r i a n t t e n s o r o f e l a s t i c m o d u l i , d e f i n e d b y E^^^'' = 2Grg^^^g^^ + ^ g^ V ^ J , g'^^ i s t h e c o n t r a v a r i a n t m e t r i c t e n s o r , G i s t h e s h e a r m o d u l u s , and v i s POISSON*s r a t i o , i n o u r a p p r o x i r a a t i o n o f p l a n e s t r e s s t h e t r a n s v e r s e s h e a r s t r a i n s a r e z e r o ( 2 . 2 ) Ya3= ° ' and -fche t r a n v e r s e n o r m a l s t r a i n i s g i v e n b y (2.3) '^"^The t e n s o r n o t a t i o n employed h e r e i s f u l l y e x p l a i n e d i n [ l 2 | and i n t h e a p p e n d i x . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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( 2 . 4 ;

M u l t i p l y i n g ( 2 . 1 ) b y ( c f . p a r . A l )

\/g/a = 1 - 2Hz + Kz^ , ( 2 . 5 1

where H and K a r e t l i e mean and GAUDSIAW c u r v a t u r e s o f t h e m i d d l e s u r f a c e , ,and z i s t h e d i s t a n c e t o t h e m i d d l e s u r f a c e , t h e s t r a i n e n e r g y p e r u n i t a r e a o f t h e undeformed m i d d l e s u r f a c e i s o b t a i n e d b y i n t e g r a t i n g w i t h r e s p e c t t o z V=

"C

[1 - 2Hz + Kz^ |dz .

4

( 2 . 6 l

The TAYLOR-expansion o f t h e e n e r g y d e n s i t y f w i t h r e s p e c t t o t h e c o o r d i n a t e x?=z may be w r i t t e n i n t h e f o r m ( c f . p a r . A2) f ( x ^ , z ) = $(x^,0)+ z f ^ 3 ( x ^ , 0 ) + | z ^ J ^ 3 3 ( x ^ , 0 ) + | ( x ^ , 0 ) + z$,|3(::^,0)+ ^ z ^ j , 3 3 ( x ^ , 0 ) - h .. ( 2 . 7 : The c o v a r i a n t d e r i v a t i v e s o f t h e t e n s o r o f e l a s t i c m o d u l i w i t l : r e s p e c t t o x"" a r e a l l z e r o , and ( 2. 7 ) i s r e d u c e d t o J ( x , z ) = ^ ^ + ZY^^„3 + 2^ Yai3||33 • (2.8)^ where s u b s c r i p t s and s u p e r s c r i p t s o i n d i c a t e v a l u e s a t t h e m i d d l e s u r f a c e . S u b s t i t u t i n g ( 2 . 8 ) i n t o ( 2 . 6 ) , p e r -f o r m i n g t h e i n t e g r a t i o n , and r e t a i n i n g o n l y tv/o t e r m s , v/e o b t a i n t h e e q u i v a l e n t i n t e n s o r n o t a t i o n o f LOVE's ^ ) _{I t v / i l l be shov/n i n p a r . 2.3 t h a t t h e s e tv/o t e r m s} i n d e e d p r o v i d e a f i r s t a p p r o x i m a t i o n . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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7 -a p p r o x i m -a t e s t r -a i n - e n e r g y e x p r e s s i o n (2.9) The f i r s t t e r m i n ( 2 . 9 ) r e p r e s e n t s t h e e x t e n s i o n a l s t r a i n ener,?;y , i . e . t h e e n e r g y due t o t h e m i d d l e s u r f a c e s t r a i n s o r e x t e n s i o n a l s t r a i n s y T h e second t e r m r e p r e s e n t s t h e f l e x u r a l s t r a i n e n e r g y , i . e . t h e e n e r g y due t o t h e f l e x u r a l o r h e n d i n ^ s t r a i n s , c o r r e s p o n d -i n g t o t h e changes o f c u r v a t u r e d e f -i n e d b y ( 2 . 1 0 ) LOVE'S a p p r o x i m a t e r e s u l t ...ay t h e r e f o r e be d e s c r i b e d b y t h e s t a t e m e n t t h a t t h e e n e r g y p e r u n i t a r e a o f t h e m i d d l e s u r f a c e i s t h e sura o f e x t e n s i o n a l and f l e x u r a l s t r a i n e n e r g i e s V= V^ + V, ; V^- .^hL YagYjiv • (2.11) (2.12) Our r e s u l t i n t e n s o r n o t a t i o n i s e a s i l y t r a n s l a t e d i n t o t h e more u s u a l n o t a t i o n f o r o r t h o g o n a l p a r a m e t r i c c u r v e s on t h e m i d d l e s u r f a c e b y means o f t h e t r a n s l a t i o n scheme g i v e n i n pai'. A 5 o f t h e a p p e n d i x . L e t e-j^, Eg and

Jt* d e n o t e t h e e x t e n s i o n s a l o n g t h e p a r a m e t r i c c u r v e s and t h e s h e a r s t r a i n b e t w e e n t h e s e c u r v e s , and l e t K ^ , K g and X d e n o t e t h e p h y s i c a l components o f t h e changes o f c u r v a -t u r e and -t o r s i o n ; we have Y22= S ^2 ' •^22 ^12= ^21= ^^^^ ' A 2 = -^21= ABT . (2.13 (2.14

* ) _{The minus s i g n i s i n t r o d u c e d f o r t h e sake o f c o n v e n i e n c e J}

L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A

DER

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Expressions ( 2 . 1 2 ) may now be v / r i t t e n i n JJOVE'S form \= | c [ ( e i + E 2 ) ^ - 2 ( l - ^ ^ ^ ^ 1 ^ 2 - T'^'^^J» ^2-^5) V^= |D [ ( K 3 _ + K 2 ) ^ - 2 ( l - v ) ( K ^ X 2 _ t 2 ) , ( 2 . 1 6 ) where

C = - l i ^ ,

D=

,

( 2. 1 7 )

' 12(1-7)

and E=2G(l+v) i s YOUNG'S modulus»

2.3 J u s t i f i c a t i o n o f LOVE's e x p r e s s i o n as a c o n s i s t e n t f i r s t a p p r o x i m a t i o n .

I n order t o j u s t i f y LOVE's a p p r o x i m a t i o n ( 2 , 9 ) or ( 2 . 1 1 ) i t w i l l be shown t h a t t h e o r d e r o f maé;iiitude of the terms which have been o m i t t e d a f t e r intet2,rating ( 2 . 6 ) i s indeed n e ^ ^ l i g i b l e compared w i t h ( 2 . 9 ) o r ( 2 . 1 1 ) , A l l n e g l e c t e d terms, which might c o n c e i v a b l y be c r i t i c a l i n t h i s r e s p e c t , a r e l i s t e d below

' LOVE'S d e r i v a t i o n o f h i s approximate expression ( 2 . 1 1 ) was r e s t r i c t e d t o i n f i n i t e s i m a l d e f l e c t i o n s , and h i s

j u s t i f i c a t i o n was based on more o r l e s s q u a l i t a t i v e , a l t h o u g h e n t i r e l y sound, p h y s i c a l a.rgimients. The

f o l l o w i n g argument i s v a l i d f o r d e f l e c t i o n s o f any magnitude, p r o v i d e d t h a t the s t r a i n s remain s m a l l , Reference should a l s o be made t o [ 2 0 ] , where a j u s t i -f i c a t i o n i s g i v e n , s t a r t i n g -from the usual

LOVS--KIRCHHOPP ass\miptions. L A B O R A T O R I U M V O O R

T O E G E P A S T E M E C H A N I C A DER

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(2,20)

(2.21)

ïïü^ % YaPj|3Yiiv||33 ' (2,22)

RÜ^^ 1 YaB||33Y^v||33 • (2.23)

I t i s almost obvious t h a t t h e f i r s t two n e g l e c t e d terms are indeed n e g l i g i b l e i n a t h i n s h e l l . I n f a c t , i f

R denotes t h e s m a l l e s t p r i n c i p a l r a d i u s o f c u r v a t t i r e o f t h e m i d d l e s u r f a c e , we have t h e estimates |(2.18)| £ , 1 ( 2 . 1 9 ) | < - ^ V , ^ . (2.24) By means o f SCIIV/ARZ's i n e q u a l i t y , h o l d i n g f o r t h e i e s s e n t i a l l y p o s i t i v e s t r a i n - e n e r g y d e n s i t y ( 2 . 1 ) , we o b t a i n an e s t i m a t e f o r t h e t h i r d n e g l e c t e d term 1 • . ov. 1/2 (2.2C) <-|i2^(V_V^) ' ' ~ V3 R (2.25)

I n order t o assess t h e order o f magnitude of t h e l a s t t h r e e neglected terms (2.21) t o ( 2 . 2 3 ) , i t i s

convenient t o determine f i r s t t h e order of magnitude of t h e second c o v a r i a n t d e r i v a t i v e s o f the s t r a i n t e n s o r v^ith r e s p e c t t o the c o o r d i n a t e nomaal t o the m i d d l e s u r f a c e Yaf,||33* ^^^^ o b j e c t i s achieved by

a p p r o p r i a t e use o f t h e c o m p a t i b i l i t y c o n d i t i o n s ; t h e

somewhat l a b o r i o u s a n a l y s i s i s g i v e n i n par. A4 of t h e appendix. Let e denote t h e ( i n a b s o l u t e v a l u e ) l a r g e s t e x t e n s i o n i n the m i d d l e s u r f a c e , K t h e l a r g e s t change of c u r v a t u r e , R t h e s m a l l e s t r a d i u s o f c u r v a t i i r e o f t h e undeformed middle s u r f a c e , and h t h e s m a l l e s t "w ave l e n g t h " of t h e d e f o r m a t i o n p a t t e r n on t h e m i d d l e s u r f a c e , d e f i n e d by L A B O R A T O R I U M V O O R 1 T O E G E P A S T E M E C H A N I C A 1 OER TECHNISCHE H O G E S C H O O L 1

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d E p d ^ (3s i » _{d s} f _{d s} dK-j^ dT d s t _{d s} f ( 2 . 2 6 ) ( 2 . 2 7 ) where ds i s any a r c e l e m e n t a l o n g t h e m i d d l e s u r f a c e , -1/2 o The o r d e r o f m a g n i t u d e o f a "^aBII33 exceed t h e o r d e r o f m a g n i t u d e o f nov.' does n o t ^ o r R ( 2 . 2 8 ) w h i c h e v e r o f t h e s e may be c r i t i c a l ( c f . p a r . A 4 ) . I t i s now e a s i l y seen t h a t t h e r e l a t i v e e r r o r i n t h e n e g l e c t i o n o f ( 2 . 2 1 ) t o ( 2 . 2 3 ) compared w i t h ( 2 . 9 ) o r ( 2 . 1 1 ) does n o t exceed t h e o r d e r s o f m a g n i t u d e i i o r h K R ( 2 . 2 9 ) w h i c h e v e r o f t h e s e may be c r i t i c a l ; o u r assixcnptioi: o f s m a l l s t r a i n s i m p l i e s o f c o u r s e t h a t t h e l a s t o r d e r g i v e n i n ( 2 . 2 9 ) i s alv/ays n e g l i g i b l e . Combining ( 2 . 2 4 ) , ( 2 . 2 5 ) and t h e o r d e r s o f m a g n i -t u d e ( 2 . 2 9 ) f o r t h e n o t^ l e c t e d t e r m s ( 2 . 2 1 ) t o ( 2 . 2 3 ) , v/e o b t a i n ouir f i n a l r e s u l t t h a t t h e r e l a t i v e e r r o r i n n e g l e c t i n g a l l t e r m s ( 2 . 1 8 ) t o ( 2 . 2 3 ) does n o t exceed t h e o r d e r s o f m a g n i t u d e h / L o r h/R, w h i c h e v e r o f t h e s e may be c r i t i c a l . Hence LOVE's s t r a i n - e n e r g y e x p r e s s i o n ( 2 . 1 1 ) i s i n d e e d . j u s t i f i e d as a c o n s i s t e n t f i r s t a p p r o x i m a t i o n on t h e b a s i c a s s u m p t i o n o f p l a n e s t r e s s , and t h e r e l a t i v e o i - r o r i n t h i s a p p r o x i m a t i o n does n o t exceed h /L o r h/R, v/hichever o f t h e s e may be c r i t i c a l .

DER

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2.4 I n c o n s i s t e n c y o f h i g h e r a p p r o x i r a a t i ons. Many w r i t e r s f e . g . 13,23,24,25,36j r e t a i n i n t h e i r energy e x p r e s s i o n s , d e r i v e d on t h e b a s i s o f t h e s i n g l e a s s u m p t i o n o f p l a n e s t r e s s o r i t s c o m b i -n a t i o -n w i t h t h e KIRCKHOPP h y p o t h e s i s , a d d i t i o n a l t e r m s , e q u i v a l e n t t o some o r a l l o f t h e t e r m s ( 2. 1 8 ) t o ( 2 . 2 3 ) n e g l e c t e d i n LOVf*s e x p r e s s i o n . Such a s u p p o s e d l y h i g h e r a p p r o x i m a t i o n i s a l s o i m p l i e d i n t h e a n a l y s i s o f o t h e r w r i t e r s [ e . g . 4,6,10,11,19,48]» ^'^^'1° d e v e l o p t h e t h e o r y w i t h o u t d i r e c t r e f e r e n c e t o t h e s t r a i n e n e r g y . I t has been p o i n t e d out r e p e a t e d l y b y s e v e r a l w r i t e r s ( e . g . 12, 20 21,22,30,40j t h a t such a r e f i n e m e n t may be m e a n i n g l e s s i f t h e a s s u m p t i o n o f p l a n e s t r e s s a n d / o r t h e KEIiCHHOFF h y p o t h e s i s ( w h i c h a s s u m p t i o n s a r e o f c o u r s e o n l y a p p r o -x i m a t e l y v a l i d ) a r e r e t a i n e d . I n f a c t , t h e t r a n s v e r s e sheai" s t r e s s e s , o b t a 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 , a r e i n g e n e r a l o f o r d e r h/L t i m e s t h e b e n d i n g s t r e s s e s , and n e g l e c t i o n o f t h e c o r r e s p o n d i n g s t r a i n e n e r g y t h e r e f o r e a l r e a d y i m p l i e s a r e l a t i v e e r r c r o f o r d e r h / L , M o r e o v e r , t h e t r a n s v e r s e 2 2 n o r m a l s u r e s s i s , i n g e n e r a l , o f o r d e r h / L o r h/R t i m e s t h e b e n d i n g o r d i r e c t s t r e s s e s , and i t s n e g l e c t i o n i n t h e s t r a i n - e n e r g y d e n s i t y ( 2 . 1 ) i n v o l v e s r e l a t i v e e r r o r s o f t h e same o r d e r s . Hence a r e f i n e m e n t o f LOVE's a p p r o -x i m a t i o n ( 2 . 1 1 ) i s i n d e e d m e a n i n g l e s s , i n ' t ^ e n e r a l ,

i m l e s s t h e e f f e c t s o f t r a n s v e r s e s h e a r and noi-mal s t r e s s e s

a r e t a k e n i n t o a c c o u n t ai: t h e same t i m e ,

T h i s g e n e r a l c o n c l u s i o n i s c o n f i r m e d by a d e t a i l e d a n a l y s i s b y JOHNSON and REISoNER [lj] o f a s e m i - i n f i n i t e c y l i n d r i c a l s h e l l u n d e r a x i s y m m e t r i c r a d i a l l o a d s and

b e n d i n g moments a t i t s end c r o s s - s e c t i o n . The t h r e e

-d i m e n s i o n a l s t r e s s d i s t r i b u t i o n i s expanded w i t h r e s p e c t

t o t h e s m a l l p a r a m e t e r X=h/2a, where a i s t h e mean r a d i u s . •jf) ' S e v e r a l w r i t e r s have d e v e l o p e d t h e o r i e s i n w h i c h t h e e f f e c t s o f t r a n s v e r s e s t r e s s e s a r e taicen i n t o a c c o u n t e,g, 1,15,31,32,40,41 ; a d i s c u s s i o n o f t h e s e t h e o r i e s f a l l s o u t s i d e t h e scope o f t h e p r e s e n t p a p e r , L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER

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The t e r m s o f o r d e r \°represent t h e c l a s s i c a l s h e l l s o l u -t i o n , and i n o r d e r t o o b t a i n a l l t e r m s o f o r d e r i t i s n e c e s s a r y t o talce i n t o a c c o u n t b o t h t h e t r a n s v e r s e

s l i e a r s t r e s s e s and t h e t r a n s v e r s e n o r m a l s t r e s s . Our g e n e r a l argument on t h e o r d e r o f mae,nicude o f t h e e r r o r s , i n t r o d u c e d b y t h e a s s u m p t i o n o f p l a n e

s t r e s s , g i v e s o f c o u r s e an e s t i m a t e f o r t h e g e n e r a l c a s e . I n some p r o b l e m s i t may happen t h a t t h e e r r o r s o f t h e

p l a n e s t r e s s a s s u m p t i o n a r e o f s m a l l e r o r d e r t h a n h/R, I n s u c h p r o b l e m s r e t e n t i o n o f some o r a l l o f t h e a d d i -t i o n a l -t e r m s ( 2 . 1 8 ) -t h r o u g h ( 2 . 2 3 ) may c o n c e i v a b l y

r e s u l t i n a b e t t e r a c c u r a c y . Hov/ever, such an i m p r o v e m e n t i s b y no means c e r t a i n , s i n c e i n such cases t h e n e g l e c t e d t e r m s i n LOVE's e x p r e s s i o n may a l s o be o f a s m a l l e r o r d e r o f m a g n i t u d e . E.g., i f t h e e x t e n s i o n a l s t r a i n e n e r g y A i s o f o r d e r h /R t i m e s t h e f l e x u r a l s t r a i n e n e r g y A^^ ( o r v i c e v e r s a ) , and i f L i s o f t h e same o r d e r as R, t h e e r r o r i n LOVE's a p p r o x i m a t i o n t h r o u g h n e g l e c t i o n o f ( 2 . 1 8 ) t h r o u g h ( 2. 2 3 ) i s o f o r d e r h /R i n s t e a d o f o r d e r h/R. An example o f sue:, h i g h e r a c c u r a c y o f LOVE's e x p r e s s i o n

o c c u r s i n t h e case o f th.. h e l i c o i d a l s h e l l under n o n a a l l o a d i n g [S,i2f vvhere r e t e n t i o n o f a d d i t i o n a l t e r m s i n t h e e n e r g y e x p r e s s i o n i s a g a i n m e a n i n g l e s s u n l e s s t h e e f f e c t o f t r a n s v e r s e s h e a r s t r e s s e s i s a l s o t a k e n i n t o a c c o u n t a t t h e same t i m e . 2.5 D i s c u s s i o n o f r e s u l t s .

The f a c t t h a t LOVE's e x p r e s s i o n ( 2 . 1 1 ) has i n h e r e n t e r r o r s o f t h e t y p e o f t h e n e g l e c t e d t e r m s , has i m p o r t a n t consequences f o r p r a c t i c a l a p p l i c a t i o n s o f t h e t h e o r y . These i n l i e r e n t e r r o r s i m p l y t h a t t h e a c c u r a c y o f LOVE's e x p r e s s i o n i s n o t a f f e c t e d b y t h e a d d i t i o n o r s u b t r a c -tlon o f c e r t a i n t e r m s o f t h e same t y p e . I n p a r t i c u l a r , i t i s t h e r e f o r e p e z - m i s s i b l e ( c f . ( 2 . 2 0 ) ) t o add t o t h e e x p r e s s i o n s f o r t h e p h y s i c a l components o f t h e changes o f c u r v a t u r e and t o r s i o n (/C-j^, i<2 and x ) t e r m s o f t y p e

e/R (v/here e i s any o f t h e p h y s i c a l m i d d l e s u r f a c e s t r a i n s

OER

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13 -e^, ^2 °^ ^» ^ "^^ ^•^'^ r a d i u s o f c u r v a t u r e o r t o r s i o n o f t h e m i d d l e sui^iace R-j_, R2 o r T ) , m u l t i p l i e d b y a n u m e r i c a l f a c t o r , p r o v i d e d t h i s f a c t o r i s n o t l a r g e compared t o u n i t y . T h i s f r e e d o m i s o f c o n s i d e r a b l e i m p o r t a n c e f o r two r e a s o n s . ï'irstly, i t a l l o w s t o d i s t i n g u i s h b e t w e e n s i g n i f i c a n t and n o n - e s s e n t i a l d i f f e r e n c e s i n t h e w i d e v a r i e t y o f e x p r e s s i o n s f o r t h e changes o f c u i v a t u r e i n l i n e a r s h e l l t h e o r y , o b t a i n e d b y v a r i o u s w r i t e r s . D i f f e -r e n c e s o f t y p e e/R may be r e g a r d e d as u n i m p o r t a n t , v;hereas d i f f e r e n c e s w h i c h a r e n o t r e d u c i b l e t o t h e f o r m e/R s h o u l d be r e g a r d e d as e c s e n t i a l d i f f e r e n c e s w h i c h may i m p l y a s i g n i f i c a n t l o s s i n a c c u r a c y f o r a t l e a s t one o f t h e c o r r e s p o n d i n g s t r a i n- e n e r g y e x p r e s s i o n s ; a d e t a i l e d c o m p a r i s o n o f v a r i o u s e x p r e s s i o n s i s d i s c u s s e d i n p a r . 3.5 and siimmarized i n t a b l e 1 . On t h e o t h e r h a n d , t h e e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e can o f t e n be s i m p l i f i e d b y a d d i t i o n o r s u b t r a c t i o n o f s u i t a b l e t e r m s o f t y p e e/R; t h i s f r e e d o m i s o f p a r t i c u l a r v a l u e i n s e l e c t i n g the' a p p r o p r i a t e s i m p l e s t e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e f o r a p p l i c a t i o n s t o s p e c i f i c p r o b l e m s c f . 20,21_ .

3. LINEAR THEORY (ITTFIITITEoILIAL DEPLSCTIONS).

3,1 E x t e n s i o n a l s t r a i n s and r o t a t i o n s . I n l i n e a r t h e o r y t h e g e n e r a l e x p r e s s i o n f o r t h e s t r a i n t e n s o r i n s u r f a c e s p a r a l l e l t o t h e m i d d l e s u r f a c e ( c f . p a r . A3) i s s i m p l i f i e d i n t o 2Yal3= ^allfi + ^ B l l a ( 3 . 1 ) I n t h e m i d d l e s u r f a c e i t s e l f t h e s p a t i a l c o v a r i a n t d e r i v a t i v e s may be e x p r e s s e d i n s u r f a c e d e r i v a t i v e s ( c f . p a r . A 2 ) . The r e s u l t i n g l i n e a r e x p r e s s i o n f o r t h e m i d d l e s u r f a c e s t r a i n t e n s o r i s L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A OER TECHNISCHE H O G E S C H O O L

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- 14 ~

(3.2)

The -fcranslation i n t o t h e u s u a l n o n — t e n s o r i a l n o t a t i o n i s obtained by means of (2.13) and par. A5; the r e s u l t i n g well-known expressions f o r t h e exten-s i o n exten-s e^. Eg a l o n g , and t h e exten-shear exten-s t r a i n ^ iiet^-veen the o r t h o g o n a l p a r a m e t r i c curves ( a , 13) are

1 l U . V ^ w ^1=

X

f a

3S

f f - ïï^ ' 1 3V . u ?B w UJ 1 . 1 ^v u 9 A V 3_B

T5

-^a 2w ( 3 . 3 ) ( 3 . 4 ) ( 3 . 5 )

where u, v and w a r e t h e p h y s i c a l displacement

compo-nents i n t h e d i r e c t i o n s o f t h e p a r a m e t r i c curves a,I3 and o f t h e n o m a l .

The r o t a t i o n i n the middle s u r f a c e around t h e

normal i s described by t h e antisyümietrie surface t e n s o r

( c f . p a r . A3)

2ü3„„= u^

The r o t a t i o n o f -the nonaal a t t h e middle s u r f a c e i s d e s c r i b e d by t h e s u r f a c e v e c t o r (f^ , g i v e n by

where t h e l a s t step i s a. consequence o f ( 2 . 3 )

- u„ o. ( 3 . 6 )

(3.7)

2Ya3= ^alU ^3lla = 0 . (3.8)

Expressing t h e s p a t i a l d e r i v a t i v e i n a s u r f a c e d e r i v a t i v e by means o f p a r . A2 "we have

(3.9)

DER

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1 5 -T r a n s l a t i n g i n t o t h e u s u a l n o n - t e n s o r i a l n o t a t i o n t h e p h y s i c a l r o t a t i o n a r o u n d t h e normal_j j O _ ^ i s g i v e n b y , 0 1 J V 1 3 u u 0 A . V a B

J Ta ' ^ W ~ TS JE ^ JS da *

( 3. 1 0 ) and t h e p h y s i c a l r o t a t i o n components o f t h e n o r m a l ^2 a r e g i v e n h y ^ _ l 3 W . U . v J s _ l 3 W . V . U ï^2- 3ÏÏ + IT: •*• T • ( 3 . 1 1 ) ( 3 . 1 2 ) 3.2 C h a n g e s o f c u r v a t u r e . I n o r d e r t o o b t a i n t h e t e n s o r o f c h a n g e s of c u r v a -t u r e we d i f f e r e n -t i a -t e ( 3 . 1 ) c o v a r i a n t l y w i t h r e s p e c t t o x-^, and i n t e r c h a n g e t h e o r d e r o f c o v a r i a n t d i f f e r e n t i a t i o n , w h i c h i s a d m i s G i h l e i n EUCLIDEAN s p a c e ( 3 . 1 3 ) A p p l y i n g t h e h a s i c f o r m u l a e o f c o v a r i a n t d i f f e r e n t i a t i o n , and u s i n g ( 3. 8 ) , we o h t a i n 2Yaf3|l3= - ^ ^ 3| l a \ l B - ^ ^ 3 l l l B \ a + ^^aQ''3\\\ ^ - 2 r^ B^ 3 l l 3 - r V a J l X - r 3 a- I ^ i X • ( 3 . 1 4 ) Prom ( 2 , 4 ) we h a v e ( 3 . 1 5 ) and s i n c e a t t h e m i d d l e s u r f a c e ( c f . p a r . A 2 ) ( 3. 1 6 ) we may n e g l e c t t h e t e r m i n v o l v i n g u ^ j j ^ "the e v a l u a t i o n o f ( 3 . 1 4 ) a t t h e m i d d l e s u r f a c e on a c c o u n t o f OTJT d i s c u s s i o L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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16 -In p a r . 2.5. S i r a i l a r l y , we may v / r i t e (3.17) where t h e f i r s t t e r m may a g a i n he n e g l e c t e d a t t h e m i d d l e s u r f a c e on a c c o u n t o f p a r . 2.5. R e w r i t i n g (3.14) a t t h e m i d d l e s u r f a c e i n t e r m s o f s u r f a c e d e r i v a t i v e s , i n t r o d u c i n g ( 3 . 7 ) , and r e m e m b e r i n g t h a t a t t h e m i d d l e s u r f a c e ( c f . p a r . A2) = - h ^ ' 313 °B » ( 3 . 1 8 ) we o b t a i n f i n a l l y (3.19) T h i s r e s u l t i s a g a i n e a s i l y t r a n s f o r m e d i n t o t h e u s u a l n o n - t e n s o r i a l n o t a t i o n f o r o r t h o g o n a l p a r a m e t r i c c u r v e s , hy means o f ( 2 . 1 4 ) and t h e t r a n s l a t i o n r u l e s o f p a r . A5; we o b t a i n i 5 * > i

*2^^^_a

^ TEJa ' TT •

1 F " 3B 1 T = _{aa "^ïï -jB} aA _^2

_3S

(3.20) (3.21) The r e s u l t i n t h i s f o r m ( a s i n t h e t e n s o r f o r m ( 3 . 1 9 ) ) i s more g r a p h i c t h a n t h e u s u a l e x p r e s s i o n s , s i n c e i t c l e a r l y dOKiOnstrates t h e dependence o f t h e changes o f c u r v a t u r e on t h e r o t a t i o n s <|)j^, (fig, / L and t h e i r d e r i v a -t i v e s . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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3.3 S t r e s s - s t r a i n r e l g . t i o n s .

S t r e s s r e s u l t a n t s and s t r e s s c o u p l e s may be i n t r o -duced i n v a r i o u s manners. I n o u r energy a p p r o a c h t h e most

o b v i o u s manner i n a d e f i n i t i o n o f s t r e s s r e s u l t a n t s and s t r e s s c o u p l e s as p a r t i a l d e r i v a t i v e s o f t h e s t r a i n e n e r g y p e r u n i t a r e a o f t h e m i d d l e s u r f a c e w i t h r e s p e c t t o t h e m i d d l e s u r f a c e s t r a i n s and t h e changes o f c u r v a t u r e . I n t h i s v/ay v/e o b t a i n t h e s y i a m e t r i c c o n t r a v a r i a n t t e n s o r s o f s t r e s s r e s u l t a n t s and s t r e s s c o u p l e s and t h e c o r r e s p o n d i n g s t r e s s- s t r a i n r e l a t i o n s n«^= 2 ^ = h f , ( 3 . 2 3 ) Por t h e p u r p o s e o f d e r i v i n g t h e e q u a t i o n s o f e q u i -l i b r i u m b y v a r i a t i o n a -l m e t h o d s , i t i s more c o n v e n i e n t t o i n t r o d u c e a m o d i f i e d a s y m m e t r i c t e n s o r o f s t r e s s r e s u l t a n t s , d e f i n e d b y * ^ ^ B l a ^ ^ e i a ^^B|a

By means o f ( 3 . 2 ) , ( 3 . 6 ) and ( 3 . 1 9 ) vve have nov/

aQ aS> . lt,fi„aX 1, a_B\ /, r.(-\

= n + ^ j ^ n i - •J'^x^ ° ^3.26; A s i m i l a r l y m o d i f i e d d e f i n i t i o n o f t h e t e n s o r o f s t r e s s c o u p l e s = = ^ — = m . ( 3 . 2 7 ) does n o t r e s u l t i n a m o d i f i c a t i o n o f t h i s t e n s o r . L A B O R A T O R U J f " . V O O R T O E G E P A S T E F R S E C H A N I C A DEft T E C H N I S C H E HOGF.SCHOOL

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18 -The s j i T m i e t r i c a l p h y s i c a l s t r e s s r e s u l t a n t s N

1 '

N g , S ' and s t r e s s c o u p l e s M-j^, Mg, W f o r o r t h o g o n a l p a r a -m e t r i c c u r v e s , c o r r e s p o n d i n g t o ( 3 . 2 3 ) and ( 3 . 2 4 ) , and t h e m o d i f i e d a s y n u a e t r i c a l p h y s i c a l s t r e s s r e s u l t a n t s S?-,, c o r r e s p o n d i n g t o ( 3 . 2 6 ) , may he o b t a i n e d

^2» ^12» "21

f r o m t h e t r a n s l a t i o n scheme i n p a r , A5. A l t e r n a t i v e l y , t h e s e q u a n t i t i e s may be f o u n d d i r e c t l y b y p a r t i a l d i f f e -r e n t i a t i o n o f t h e s t -r a i n - e n e -r g y e x p -r e s s i o n s ( 2, 1 1 ) ,

(2.15)

and ( 2, 1 6 ) . The r e s u l t s a r e

'21

N„= 2 - = ^ ( e g- H v e i ) , s =|I = | { i - v ) c y / ,

l - f ^ = ^ ( V

^ ^ 2 ^

M

_{2 - | ^}

_{= ^ ( V ^ ^ l ^}

2W= I I = 2 ( 1 - V ) D T ; N^ = N::= 9V

TTTiA = ^^1

9V

^12-av _ c ^ ^^i"^^2 i,„/ 1 i \ (A 3 ^ } ^ 1 2/

^V _ c ^'^l

''^''^2 ^ 1 « / 1 1 \

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

(20)

19

-3.4 EquationG o f e q u i l i b r i u m and boundary c o n d i t i o n s .

The equations o f e q u i l i b r i u m and boundary c o n d i t i o n s are o b t a i n e d by t h e v/ell-Icnov/n v a r i a t i o n a l method. The f i r s t v a r i a t i o n o f the t o t a l e l a s t i c energy U, i . e . t h e i n t e g r a l o f V (2,11) over the e n t i r e middle s u r f a c e o f the s h e l l , i s g i v e n by

6,

fi

a dx-j^dXg. ( 3 . 3 Ö )

On account o f t h e syimnetry of b^^ we may r e p l a c e n aB aB

i n the second t e n n by t h e asymmetric t e n s o r n^ . A p p l y i n g GREEN'S theorem ( c f . p a r . A2), and d e f i n i n g the c o n t r a

-B v a r i a n t v e c t o r o f t r a n s v e r s e shear f o r c e s q by

t h e f i r s t v a r i a t i o n ( 3 . 3 8 ) i s reduced t o

SU=

[K^SUQ

+ m'^^^^Je^^dx^ +

u

- i K I a

^^B - ^""^"PB *

^r

^aB^^]

^ ^ ^ ^ ^ ' ( 3 . 3 9 ) ( 3 . 4 0 ) where t h e f i r s t i n t e g r a l i s t a k e n a l o n g t h e boundary of t h e s h e l l ' s middle s u r f a c e . Remembering ( 3 . 9 ) , t h e d e r i v a t i v e s 6Wjg=fiw ^ i n t h e s u r f a c e i n t e g r a l are

removed 'bj a second a p p l i c a t i o n o f GREEK'S theorem. I n order t o remove the same d e r i v a t i v e s , as f a r as p o s s i b l e , from the l i n e i n t e g r a l , i t i s convenient t o choose a s p e c i a l c o o r d i n a t e system i n which the boundary o f t h e middle s u r i a c e i s a c o o r d i n a t e l i n e , say an x - l i n e

( i . e . a l i n e x'^=constant), Reaiembering. t h e d e f i n i t i o n of t h e antisyiuiiietrie e-tensor ( c f , p a r , A l ) , v;e have

(21)

by i n t e g r a t i o n by p a r t s m'^^w pe„„dx^= fm^-^v^Sw . - ( m ^ ^ v ^ ) ^ S w l d x ^ . (3^.41) Our f i n a l r e s u l t f o r t h e f i r s t v a r i a t i o n o f t h e e l a s t i c energy i s or ~

- J J t

< f a

- ^ X

) K ^ ( ^ r ^ f i

+ ^ ^ ) ^ ^ { j ^

d^^d^^. (3.42) Let- p*^, p-^ d e n o t e t h e c o n t r a v a r i a n t v e c t o r o f e x t e r n a l s u r f a c e l o a d s p e r u n i t a r e a o f t h e m i d d l e

s u r f a c e , and l e t n ^ , q , and m"^ denote t h e c o n t r a v a r i a n t v e c t o r o f edge f o r c e s , and t h e edge b e n d i n g moment, b o t h p e r u n i t l e n g t h o f t h e edge. The work 5TJ, p e r f o r m e d b y t h e s u r f a c e l o a d s and edge l o a d s i n any v a r i a t i o n o f t h e d i s p l a c e m e n t s i s t h e n g i v e n b y

SU=J n^Sug+q^w+m-^8(p-j^\/^ dx^+J p^Sug+p%^^v/a d x ^ d x 2 . ( 3 . 4 The c o n d i t i o n s o f e q u i l i b r i u m a r e now o b t a i n e d

f r o m t h e p r i n c i p l e o f v i r t u a l w o r k , i . e . SU=&ÏÏ f o r a l l k i n e m a t i c a l l y a d m i s s i b l e v a r i a t i o n s o f t h e d i s p l a c e m e n t s . The f u n d a m e n t a l leüima o f t h e c a l c u l u s o f v a r i a t i o n s now r e s u l t s i n t h e e q u a t i o n s o f e q u i l i b r i u m

' I t has been assumed h e r e t h a t t h e e n t i r e b o u n d a r y 2 i s a smooth x - l i n e . I f t h e b o u n d a r y has c o r n e r s , s t o c k t e r m s a r i s e i n t h e s e c o r n e r s as a r e s u l t . o f t h e i n t e g r a t i o n b y p a r t s ; t h e c o e f f i c i e n t s o f 8w i n t h e s e s t o c k t e r m s r e p r e s e n t c o n c e n t r a t e d f o r c e s n o r m a l t o t h e s h e l l m i d d l e s u r f a c e , w h i c h a r e w e l l

known t o occi.ur i n c o r n e r s o f p l a t e s and s h e l l s , b u t we s h a l l n o t p u r s u e t h i s c o m p l i c a t i o n h e r e .

DER

(22)

21 -fi + + p- ( 3 . 4 5 ) h o l d i n g i n t h e i n t e r i o r o f t h e m i d d l e s u r f a c e . M o r e o v e r , u n l e s s t h e b o u n d a r y d i s p l a c e m e n t s u ^ , w and t h e r o t a t i o n cp-j^ a r e s p e c i f i e d ( i n w h i c h case t h e i r v a r i a t i o n s a r e z e r o a l o n g t h e b o u n d a r y ) , t h e b o u n d a r y c o n d i t i o n s (ni^+m^^^b^)v^ = ( n ^ + n H ^ ) / ^ , ( 3 . 4 6 ) c^fa -(m^^\/I)^2= " ^ ^ 2 ' (3.47)

ïiP-fa. = m^v/a^ (3.48)

must h o l d a l o n g t h e edge o f t h e m i d d l e s i i r f a c e .

I t s h o u l d be n o t e d t h a t o u r e q u a t i o n s o f e q u i l i -b r i u m ( 3. 4 4 ) , ( 3 . 4 5 ) , and t h e d e f i n i t i o n o f t h e s u r f a c e v e c t o r o f t r a n s v e r s e s h e a r f o r c e s a g r e e c o m p l e t e l y w i t h t h e r i g o r o u s e q u a t i o n s o f e q u i l i b r i u m , as d e r i v e d b y

GREEN and ZERNA [±21 f r o m t h e g e n e r a l t h r e e d i m e n s i o n a l e q u a t i o n s ^ . L i k e w i s e , GREEN's and ZERNA's e q u a t i o n f o r e q u i l i b r i u m o f moments a r o u n d t h e n o r m a l i s e q u i v a l e n t t o o u r e q u a t i o n ( 3 . 2 6 ) . Hence, i n s p i t e o f t h e a p p r o x i m a -t i v e c h a r a c -t e r o f o u r -t h e o r y , b a s e d on'LOVE's a p p r o x i m a -t e s t r a i n - e n e r g y e x p r e s s i o n , our e q u a t i o n s o f e q u i l i b r i u m a r e c o m p l e t e l y r i g o r o u s . The t r a n s l a t i o n o f o i i r r e s u l t s i n t o t h e u s u a l n o n - t e n s o r i a l n o t a t i o n f o r o r t h o g o n a l p a r a m e t r i c c u r v e s i s a ^ a i n e a s i l y o b t a i n e d by means o f p a r . A5 o f t h e a p p e n d i x . The n o n - t e n s o r i a l f o r m u l a t i o n may o f c o u r s e a l s o be a c h i e v e d by a d i r e c t a p p l i c a t i o n o f t h e c a l c u l u s o f v a r i a t i o n s t o t h e e n e r g y e x p r e s s i o n i n n o n - t e n s o r i a l f o r m . L e t Q-^ onci d e n o t e th.; t r a n s v e r s e s h e a r f o r c e s ^ ) _Due_{a l l o w a n c e}_{s h o u l d o f}_{c o u r s e be made f o r t h e} d i f f e r e n c e i n s i g n b e t w e e n our d e f i n i t i o n o f m and t h e d e f i n i t i o n i n | l 2 ] . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

(23)

- 2 2 p e r u n i t l e n g t h ; t h e y a r e e x p r e s s e d i n t h e b e n d i n g and t o r s i o n a l .uoraents b y t h e t r a n s l a t i o n o f ( 3 . 3 9 ) L e t P-J-, P J J and P j j j = P"^ d e n o t e t h e p h y s i c a l e x t e r n a l l o a d s p e r u n i t ai^ea o f t h e m i d d l e s u r f a c e . The e q u a t i o n s o f e q u i l i b r i u m ( 3 . 4 4 ) a r e now t r a n s l a t e d i n t o X ~ ÏÏ IS 3^^^12^^21^'^ - | i - ^ + P j = 0 , ( 3 . 5 1 ) Q and e q u a t i o n ( 3 . 4 5 ) now r e a d s - é - ' T ^ P l l = ^ » ^ ^ * 5 2 )

Nj^

1 ^2% ^lV

^2

"l 1

^ ^ 1

1 ^^2 ^ 1

..B,

1 9A,

E q u a t i o n s ( 3 . 4 9 ) t o ( 3 . 5 3 ) and ( 3 . 2 6 ) a g r e e c o m p l e t e l y ^ w i t h t h e s i x e q u a t i o n s o f e q u i l i b r i u m i n t h e i r w e l l k n o w n -f o r m -f o i " a r b i t r a r y o r t h o g o n a l p a r a m e t r i c c u r v e s j3«g» 6 , 7 , 1 1 , 1 8 , 1 9 ] . * ^

'Here a g a i n , a l l o w a n c e must o f c o u r s e be made f o r d i f f e r e n c e s i n s i g n i n t h e d e f i n i t i o n s employed

• h e r e and t h o s e i n t h e c i t e d r e f e r e n c e s ; m o r e o v e r ,

no d i s t i n c t i o n i s made i n our a n a l y s i s b e t w e e n

t h e t o r s i o n a l moments Wjg and W^j, v/hich a r e b o t h

e q u a l t o \'l i n o u r a p p r o x i m a t i o n .

DER

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The g e o m e t r i c b o u n d a r y c o n d i t i o n s a l o n g a b o u n d a r y c o i n c i d i n g w i t h a B - l i n e ( a = c o n s t a n t ) a r e i n t h e u s u a l n o t a t i o n s i m p l y t h a t p r e s c r i b e d v a l u e s o f u , v , w and | ^ a r e s p e c i f i e d . The dyna-dc b o u n d a r y c o n d i t i o n s a r e o b t a i n e d by a t r a n s l a t i o n o f ( 3 . 4 6 ) t o ( 3 - 4 0 ) . L e t N^, Ng, Q and M d e n o t e t h e s p e c i f i e d f o r c e s and b e n d i n g moment p e r u n i t l e n g t h o f t h e b o u n d a r y ; we have

Nj=An''", No=Bn^ Q=q, lT=Am-'-, ( 3 . 5 4 )

and t h e dynamic b o u n d a r y c o n d i t i o n s r e a d , i f ( 3 . 4 8 ) i s used i n o r d e r t o s i m p l i f y ( 3 . 4 6 ) N

J + I =

N J (3.55) ^12

1^

- ^2 » •^1 - ÏÏ ^ = ^ » Mj = M . ( 3 . 5 6 ) ( 3 . 5 7 ) ( 3 . 5 8 )

Dynamic b o u n d a r y c o n d i t i o n s have been f o r m u l a t e d e x p l i -c i t l y by o n l y a fev/ w r i t e r r ^ , , 7^ 2 2 , 2 ^ and t h e y a g r e e w i t h t h e p r e s e n t c o n d i t i o n s , i f a g a i n due allov;ance i s made f o r d i f f e r e n c e s i n s i g n i n t h e d e f i n i t i o n s . 3.5 D i s c u s s i o n and c o m p a r i s o n w i t h e a r l i e r w r i t e r s . A l l a u t h o r s a r e i n agreement on t h e f o r m u l a e f o r t h e m i d d l e s u r f a c e s t r a i n s , ( 3. 2 ) i n t e n s o r n o t a t i o n , and ( 3 . 3 ) t o ( 3 . 5 ) i n t h e more c o n v e n t i o n a l n o t a t i o n . L i k e w i s e , f u l l agreement e x i s t s on f i v e e q u a t i o n s o f e q u i l i b r i u m , (3.39) and ( 3. 4 4 ) , (3.45) i n t e n s o r n o t a t i o n , and (3.49) t o (3.53) i n t h e u s u a l n o t a t i o n . C o n s i d e r a b l e d i f f e r e n c e s o c c u r , however, i n t h e e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e , i n t h e s t r e s s - s t r a i n r e l a t i o n s , and i n t h e i m p o r t a n c e a t t a c h e d t o t h e e q u a t i o n o f e q u i -l i b r i u m o f moments a r o i m d t h e n o r m a l , e x p r e s s e d b y L A B O R A T O R I U M V O O R TOEGEPASTE MECHANICA DER TECHNISCHE HOGESCHOOL

(25)

2 4

-(3,26) i n t e n s o r n o t a t i o n , and by ( 3 . 3 6 ) , (3.37) i n the u s u a l n o t a t i o n ,

3,5.1 Chanè4'e3 o f c u r v a t u r e . The expressions g i v e n i n 13 papers and books are coiapared w i t h our r e s u l t s i n t a b l e 1 ; the d i f f e r e n c e s have been reduced t o t h e i r s i m p l e s t form by a p p r o p r i a t e use o f the JiAINARDI-CODAZZl equations ( c f . p a r , A5). I t appears t h a t not l e s s than 10 d i f f e r e n t

expressions have been proposed, a l t h o u g h a l l o f them

are based on the same b a s i c L C V E - I C I A C H H O P F assumptions,

or, as i n our t h e o r y , t h e e q u i v a l e n t assumption of plane s t r e s s . F o r t u n a t e l y , a l l d i f f e r e n c e s o f type e/R are

i r m a a t e r i a l from t h e p o i n t o f view o f a f i r s t a p p r o x i m a t i o n ( c f , p a r , 2,5), Some a u t h o r s present t h e i r r e s u l t s as

"second" o r " b e t t e r " a p p r o x i m a t i o n s ; i n t h e absence o f a complete a n a l y s i s i n t h e i r papers o f the e f f e c t o f t r a n s v e r s e normal and shear s t r e s s e s , these c l a i m s l a c k adequate s u b s t a n t i o n . On t h e o t h e r hand, d i f f e r e n c e s l i s t e d i n t a b l e 1 which are not o f t y p e e/R are e s s e n t i a l d i s -crepancies-r Since our expressions have been proved t o p r o v i d e a c o n s i s t e n t f i r s t a p p r o x i m a t i o n i f used i n

con-j u n c t i o n w i t h L O V E ' S s t r a i n - e n e r g y e x p r e s s i o n ( 2 . 1 1 ) , i t

f o i i o w s t h a t the expressions i n t h e c i t e d r e f e r e n c e may be s e r i o u s l y i n e r r o r . These d o u b t f u l expressions are marked by an a s t e r i s k i n t a b l e 1 .

and i n a subsequent check GOKEN discovered an a r i t h m e t i c a l e r r o r i n h i s o r i g i n a l a n a l y s i s ; the term (3.59) should be d e l e t e d from h i s e q u a t i o n ( 1 . 6 , 2 1 ) ,

The q u e s t i o n a b l e term i n COHEN's change of t w i s t [6, eq. (1,6.21)3 i s o b v i o u s l y t (3.59)

^ )

L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A OER TECHNISCHE H O G E S C H O O L

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-I n p a p e r s "by RE-ISSNEii [38ƒ and b y KROV/-IES and REISSïiER f i ö j t h e q u e s t i o n a b l e t e r m s a r e t h o s e i n w h i c h t h e ( p h y s i c a l ) r o t a t i o n a r o u n d t h e n o r m a l ^ o c c u r s , d i v i d e d b y a r a d i u s o f c u r v a t u r e o r t o r s i o n . These d i s c r e p a n c i e s a r e a g a i n u n i m p o r t a , n t, i f t h e r o t a t i o n i s s m a l l , i . e . o f t h e same o r d e r o f m a g n i t u d e as t h e m i d d l e sui^iace s t r a i n s . T h i s i s a c t u a l l y so i n t h e m a j o r i t y o f i n t e r e s t i n g s p e c i f i c p r o b l e m s , and t h e e x p r e s s i o n s o f ^ l 8 ] and a r e t h e n e s s e n t i a l l y e q u i v a l e n t t o o u r e x p r e s s i o n s . The e x c e p t i o n a r i s e s e.£-> i f t h e s h e l l g e o m e t r y and b o u n d a r y c o n u i t i o n s j e r m i t i n e x t e n s i o n a l d e f 0 i\ . i. a ti o n s o f t h e mi.,ule s u r -f a c e . I n such cases / L may be l a r g e , even i n f i n i t e , compared w i t h t h e m i d d l e s u r f a c e s t r a i n s , and t h e e x p r e s s i o n s i n f i ö j and f ^ o j , used i n c o n j u n c t i o n w i t l i LOVE's s t r a i n - e n e r g y e x p r e s s i o n ( 2. 1 i ) , a r e i n a d e q u a t e as a f i r s t a p p r o x i m a t i o n .

V.henever t h e r o t a t i o n i s s m a l l , o f t h e same o r d e r o f m a g n i t u d e as E-j^, e^, *//, v/e may s i i r i p l i f y ( 3 . 1 9 ) i n t o

( 3. 6 0 )

R-membering ( 3 . 9 )f we may now w r i t e

( 3. 6 1 )

where t h e MAIITARDI-CODAZZI e q u a t i o n s ( p a r . A2) have been u s e d . From ( 3 . 2 ) and ( 3 . 6 ) we have i n t h e p r e s e n t appir o x i m a t i on

( 3 . 6 2 )

and hence

( 3 . 6 3 )

GREEN and ZERNA {^12]] i n t r o d u c e a f u r t h e r a p p r o x i m a t i o n b y r e t e n t i o n o f o n l y t h e f i r s t t e r m i n ( 3 . 6 3 ) . T h i s L A B O R A T O R I U M V O O R

T O E G E P A S T E M E C H A N I C A DER

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26

-a p p r o x i m -a t i o n i s e q u i v a l e n t t o DONRELL's j^9j i n t h e case o f c y l i n d r i c a l s h e l l s . I t has been n o t e d p i 6 , 2 9 ] t h a t DONNELL'S a p p r o x i m a t i o n i s sometimes i n a c c u r a t e , even t h o u g h t h e r o t a t i o n SL may r e m a i n s m a l l o f t h e same o r d e r as t h e s t r a i n s , and i n s u c h cases t h e second t e r m i n ( 3. 6 3 ) s h o u l d e v i d e n t l y n o t be o m i t t e d ; t h e l a s t t e r m i s o i c o u r s e alv/ays z e r o i n c y l i n d r i c a l o r s p h e r i c a l s h e l l s . LOVE's e x p r e s s i o n s f o r t h e changes o f c u r v a t u r e have o f t e n b e e n c r i t i c i z e d i n t h e l i t e r a t u r e j^e.g. 23, 35,473 f o r t h e l a c k o f syrmaetry i n h i c change o f t v / i s t and f o r supposed o t h e r d e f e c t s . Kence i t may be w o r t h -v / h i l e t o r e c o n f i r i i i . by means o f t h e c o m p a r i s o n i n t a b l e 1 t h a t LOVE's r e s u l t a c t u a l l y i s c o n s i s t e n t and a d e q u a t e w i t h i n t h e f r a m e w o r k o f t h e f i r s t a p p r o x i m a t i o n i n s h e l l t h e o r y £cf. a l s o 2 0, 2 l J ; i t i s e n t i r e l y e q u i v a l e n t t o o t h e r c o n s i s t e n t r e s u l t s . 3.5.2 S t r e s s - s t r a i n r e l a t i o n s and e q u i l i b r i u m o f moments a r o u n d t h e n o r m a l . C o n s i s t e n t s t r e s s - s t r a i n r e l a t i o n s f o r s t r e s s r e s u l t a n t s and s t r e s s c o u p l e s t o be used i n t h e e q u a t i o n s o f e q u i l i b r i u m , a r e g i v e n b y ( 3 . 2 6 ) and (3.27) i n t e n s o r f o r m , and b y ( 3 . 3 1 ) t o ( 3 . 3 7 ) i n t h e more c o n v e n t i o n a l n o t a t i o n . I n many i n v e s t i g a t i o n s t h e a s y m i i i e t r i c s h e a r s t r e s s r e s u l t a n t s a r e r e p l a c e d b y t h e i r s y m m e t r i c p a r t ( 3 . 3 0 ) The c o n d i t i o n o f e q u i l i b r i u m o f moments a r o u n d t h e n o r m a l i s o f c o u r s e v i o l a t e d i n t h i s s i m p l i f i c a t i o n . I n o u r e n e r g y a p p r o a c h , n e g l e c t i o n o f t h e a s y m i T i e t r i c t e r m s i n t h e s h e a r s t r e s s r e s u l t a n t s ^ The c o r r e s p o n d i n g s t r e s s - s t r a i n r e l a t i o n s ( 3 . 2 8 ) t o ( 3. 3 3 ) a r e o f t e n r e f e r r e d t o i n l i t e r a t u r e as"LOVE*s f i r s t a p p r o x i m a t i o n " . V/e s h a l l n o t use t h i s t e r m i n o -l o g y i n o r d e r t o a v o i d c o n f u s i o n w i t h o u r d e f i n i t i o n o f LOVE*s f i r s t a p p r o x i m a t i o n , i . e . t h e s t r a i n e n e r g y e x p r e s s i o n ( 2 . 1 1 ) v / i t h any a p p r o p r i a t e c o n -s i -s t e n t e x p r e s s i o n s f o r t h e e x t e n s i o n a l s t r a i n s and changes o f c u r v a t u r e . L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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(3.33) and

(3.34)

i s o b v i o u s l y e q u i v a l e n t t o o m i t t i n g t h e t e r m s i n v o l v i n g t l i e r o t a t i o n ^ f L i n oor e x p r e s s i o n s

f o r t l i e chax.ges o f cux-vature ( 3 . 2 0 ) t o ( 3 . i i 2 ) . The s i m p l i f i c a t i o n i s t h e r e f o r e j u s t i f i e d , i f t h e r o t a t i o n _/L i s s m a l l , o f t h e saiue o r d e r o f m a g n i t u d e as t h e m i d d l e s u r f a c e s t r a i n s e^^, Eg» y • On t h e o t h e r h a n d , c o n s i d e r a b l e e r r o r s may be' i n t r o d u c e d i f a s y m m e t r i c s h e a r s t r e s s r e s u l t a n t S i s assuiaed i n p r o b l e m s where t h e r o t a t i o n SL ..lay be l a r g e compared w i t h t h e e x t e n -s i o n a l -s t r a i n -s . T h i s c o n c l u s i o n i s b o r n e o u t b y COHEN*s d e t a i l e d a n a l y s i s o f a h e l i c o i d a l s h e l l t 6, 7 j . Nuiuerous more c o m p l i c a t e d s t r e s s - s t r a i n r e l a t i o n s have been d e r i v e d i n t h e l i t e r a t u r e , and i t i s o f t e n

b e l i e v e d thaA t h e a d d i t i o n a l t e r m s w o u l d i m p l y a h i g h e r a p p r o x i m a t i o n . Hov/ever, t h i s o b j e c t can a c t u a l l y n o t be a c h i e v e d w i t h o u t due a c c o u n t o f t h e e f f e c t o f t r a n s v e r s e n o r m a l and s h e a r s t r e s s e s ( c f . p a r . 2. 4 ) , and i t i s m e a n i n g l e s s t o use more " r e f i n e d " s t r e s s - s t r a i n r e l a t i o n s t h a n ( 3 . 3 1 ) t o ( 3 - 3 7 ) i f t h e LOVE-iaRCHHOPP a s s u m p t i o n s a r e r e t a i n e d i n t h e i r d e r i v a t i o n . T h i s v i e w i s a g a i n c o n f i r m e d b y COHEN's a n a l y s i s i n t h e case o f t h e h e l i -c o i d a l s h e l l

ZlJ*

On t h e b a s i s o f t h e s t r e s s - s t r a i n r e l a t i o n s ( 3 . 3 l ) t o ( 3 . 3 7 ) t h e same r e s u l t i s o b t a i n e d as f r o u i t h e more e l a b o r a t e s t r e s c - s t r a i n r e l a t i o n s i n

r e i .

L A B O R A T O R I U M V O O R T O E G E P A S T E M E C H A N I C A DER TECHNISCHE H O G E S C H O O L

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T a b l e 1 C o m p a r i s . o f v a r i o u s e x p r e s s i o n s f o r t h e ) ] L . y r ; i c a l chan.'^es o f c u r v a t u r e . NoBo T h e e n t r a n c e s i n h i s t a b l e i n d i c a t e t h e c o r r e c t i o n s AK^, ^ . « 2 , Li v/ h i c h m u s t b e a d d e d t o o u r e x p r e s s i o n s r K-^, i n . o r d e r t o o b t a i n t h e e x p r e s s i o n s i n t h e c i t e d r e f e r e n c e s * Where n e c e s s a r y ^ j u s t m e n t s f o r s i ^ ' n a n d / o r a n u , . > j r i c a l f a c t o r 2 h a v e b e e n made t o a c h i e v e c o n f o r m i t y v / i t h r n o t a t i o n o E s s e n t i a l d i f f e r c e s i n t h e s e n s e o f p a r a s . 2 . 5 ' i n d 3.5 a r e m a r k e d b y a n a s t e r i s k . R e f e r e n c e s emplo, i g t h e l i n e s o f c u r v a t u r e a s p a r a m e t r d c c u r v e s a r e m a r k e d b y a s m a l l c i r c l e . AUTHORS a n d REPER^hCES _A Kg LOVE, 1888 [27]

—

L A I f f i , 1891 ^ 2 ] REI3SNER, 1942 [39] ^ WLASSOY/, ,1949 {47I L A I f f i , 1891 ^ 2 ] REI3SNER, 1942 [39] ^ WLASSOY/, ,1949 {47I OSGOOD a n d JOSEPH, 1^50 HAT.VOOD a n d WILSON^ 1.958 REISSNER, 1941 [38)

—

KOITER, 1945 (20] "2 GOLDENÏÏEISER, 1953 [ l l ] , .54

—

COHEN, 1955 Cd • R^^"5? " 2 7 COHEN, 1959 \j\ ^1 ^2 1 1 1 KNOWLES a n d REISSNER, 195 [18]

^ r

KNOvVLES a n d REISSNER, 19^ _[19] Rg ^ 2T ^ ^ l " - ^2 ^ 1^,

l

^ J , .

r 2 Ï 4^^R-L R i ^ ^ l " - ^2 ^ 1^,

l

^ J , .

r 2 Ï 4^^R-L R

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DER

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- 3ü ~

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e x p r e s s i o n f o r t h i n e l a s t i c s h e l l s . J . A p p l . Mech,

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