curvatura cascaras esfericas

7/28/2019 curvatura cascaras esfericas

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CURVATURE OF A SPHERICAL SHELL UNDER THEINFLUENCE OF CONSTANT ASYMMETRIC LOAD- USSR -[ F o l l o w i n g is a t r a n s l a t i o n o f a n a r t i c l e byl an gu ag e j o u r n a l I z v e s t i y a AN SSSR, Mekhanika(News of t h e Academy of S c i e n c e s USSR , Mech-a n i c s ) , M arch -A p ri l 1 96 5, No 2 , pages 154-159. ]

, .nip T. V, B ud ni ko va , L e n i n g r a d, i n t h e R u s s i a n -

T h e c u r v a t u r e of a s p h e r i c a l s h e l l u nd er t h e i n f l u e n c eof a s t a t i c load a r b i t r a r i l y c ha ng ing a l on g t h e m er id i an w i t ha s l i d i n g s e a l i n g o f t h e r i m (see F i g u r e ) i s examined . Byway of ex ampl e , t h e p r ob l em of t h e l o a d i n g of a s h e l l by mc o n c e n t r i c c i r cu m f er e n ce s of normal c o n s t a n t i n t e n s i t y l o a di s so lv ed . Under o r d i na ry methods of c a l c u l a t i o n i n t h i scase t h e prob lem i s r e d u c e d t o t h e s o l u t i o n o f a sy s t em o fa l g e b r a i c e q u a t i o n s of t h e 4 ( m + 1) - t h o r d e r , w h i c h r e p r e -s e n t t h e b o u n d a r y c o n d i t i o n s ana tne c o n d i t i o n s of csnjuga-t i o n of i n d i v i d u a l s e c t i o n s of t h e s h e l l . I n o r d e r t o a vo idt h i s , it i s proposed t h a t t h e M e is s ne r e q u a t i o n s ( 1 ) b e s o l v -ed i n t h e form of e x pa n si o ns i n e i g e n f u n c t i o n s s a t i s f y i n g t h eb o u n d a r y c o n d i t i o n s .

I'

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I n t h e p a r t i c u l a r c a s e , t h e d e f l e G t i o n of t h e s p h e r i -c a l s h e l l l o ad ed a t m c i r c u m fe r e n c e s, w i t h o u t a n o p e n i n gw i t h t h e e x t r e m a 1 cross s e c t i o n = *, i s c a l c u l a t e d .BoE q u a t i o n s of e q u i l i b r i u m of t h e s p h e r i c a l s e t down i n

M ei ss ne r v a r i a b l e s 8 and V a re of t h e form [ I ]L (V) +-pi- h H6 =RV ( f 3 ) (Y (P)=3-(I p) F v )

Here, h = t h i c k n e s s of s h e l l ; E = modulus of e l a s -t i c i t y ; p = c o e f f i c i e n t of t r a n s v e r s e c o m p r e s s i o n ; F,, Fn =components of t h e e x t e r n a l s u r f a c e load F(B) a l o n g t h e t en -s i o n t o t h e m e r i d i an a n d a l on g t h e e x t e r n a l n or mal t o t h es u r face.

T he i n t e r n a l c o n d i t i o n s a n d b e n d i ng moments a re d e t e r -mined from t h e f u n c t i o n s 6 and V a c c o r d in g t o f o r mu l as [ l ](pa ge 35), w he re t o the f o r c e s c a l c u l a t e d i n t h i s way, t h ef o r c e s of t h e non-momentum s t a t e must still be added .I n a s t a t e of s l i d i n g s e a l i n g , d e p i c t e d i n t h e f i g u r e ,t h e a n g l e of r o t a t i o n a nd t h e o v e r t u r n i n g f o r c e a t t h e r i mmust be a b s e n t . I n t h e f u n c t i o n s 8 a n d V t h i s amounts t ot h e r e q u i r e me n t t h a t

(9=0, v = o u h e r l p = p owe seek for s o l u t i o n s o f the system (1) under t h e cen-d i t i o n s (2 ) i n t h e form o f e x p a n s i o n s i n t h e s e r i e s

m M

Here, Lps ( p ) = e i g e n f u n c t i o n s of t h e e q u a t io n

w h e r e a2 = e i g e n v a l u e s , t h a t i s , t h o s e v a l u e s of t h e p a r a -meter unde r w h i c h t h e S t u r m - L i o u v i l l e p r o b l e m (4 ) has non-t r i v i a l s o l u t i o n s . We p l a c e i n e q u a t i on (4 )aa+i =v (v +i) i

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a nd o b t a i n t h e L e ge n d re e q u a t i o n w i t h t h e a s y e t unknownp a r a m e t e r J

The s o l u t i o n s of (5) bounded by ze r o w i l l be t h e as-s o c i a t e d f i r s t - o r d e r L eg en dr e f u n c t i o n s

T h e v a l u e s of t h e parameter V = Ys ( s = 1 , 2 , ...)a r e d e t e r m i n e d f r o m t h e t r a n s c e n d e n t a l e q u a t i o nI n s e a r c h i n g f o r t h e r o o t s of (7 ) w e c a n u s e t h e a s y -m p t o t i c f or mu la i n [2].T h e e i g e n f u n c t i o n


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I(13)= (v,' +V , - )' c p3f k k4 =12 (Ipa) R' 1 ha

Thus , t h e pr ob lem r educes t o e x p a n s i o n o f a knownf u n c t i o n of t h e load i n a s e r i e s u s i n g e i g e n forms.From t h e 4 and V found a l l t h e i n t e r n a l f o r ces andb e nd i ng moments c a n b e c a l c u l a t e d . Howe ver, i n s e v e r a l p r ac -t i c a l cases d e t e r m i n a t i o n of d e f l e c t i o n of t h e s h e l l i s ofg r e a t e s t i n t e r e s t . The d e f l e c t i o n c a n be r e p r e s e n t e d i n t h eform [ I ]

H e r e , 3 6 = d e f l e c t i o n o f t h e s h e l l c o r r e s p o n d i n g t ot h e non-momentum s t r e s s e d s t a t e .

(15)

8.

Here F, = a x i a l p r o j e c t i o n of t h e e x t e r n a l s u r f a c el o a d F ( B ) .have U s in g t h e e x p an s io n of (3), and a l s o ( l l ) , w e

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I n t h e g e n e r a l c a s e , t h e load c an va ry i n s t e p - l i k ef a s h i o n a l o n g t h e m e r i d i a n , and t h e n i t i s n e c e s s a r y t o i m -prove t h e convergence of t h e expans ion (17) by s u b t r a c t i n ga nd s umming t h e p oor l y c onve rg i ng se r ies 131. By way o f a nexample , w e examine t h e fo rma t i on o f a s p h e r i c a l s h e l l w i t he x t r e m a 1 cross s e c t i o n /lo = gz unde r t h e i n f l u e n c e of t h enormal load qj e ve nl y d i s t r i b u t e d a l o n g m c o n c e n t r i c c i r c u m -f e r e n c e s rj ( J = 1 , 2, ..., m ; rj = R s i n B j ) .

The e i g e n f u n c t i o n s i n t h i s problem a r e t h e a s s o c i a t e df i r s t - o r d e r L eg en dr e f u n c t i o n s w i t h a n i n t e g r a l , e ve n i n de x[41 \

The e i g e n f u n c t i o n s Pan1 ( c o s 8 ) a r e o r t h o g o n a l w i t h i nt h e i n t e r v a l (04 os! G 1) and have & normal N2 e q u a l t o

W it h t h e examined load w e have

L e t u s d e t er m i n e t h e c o e f f i c i e n t of e x pa n si o n 8 2 n ,c o r r es p o n d in g t o t h e l o a d qj

From t h e fo rm ul a s (11)w e o b t a i n

(21)' q . 4 , +Im D A I 2n, 6 = -J - sinB.P ( cosB j ). ( 4 n -1I)4n' +2n - $- p)

A s i n BjP,, (cos P j ) ( 21 ),, =- q jA =(4n*+2n - )2 - a+A* ' ( 1 3 )

Based on (14) and (17) w e f i n d t h e d e f l e c t i o nm I

w = x w j ( 2 4 )j=i ~

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wheres in fl os PSI+siIi* PS, +Eh

=0 (B > j). ( 2 7 )

W employ the f a m i l i a r e x p a n s i o n

W e x p r e s s t h e d i v e r g in g s e r i e s S2 by t h e d e l t a - f u n c-t i o n(30)6 B - j )SI= sin pi - I

To improve convergence of S3 w e u s e t h e w el l- kn ow nr a t i o of t h e t h e o ry of s p h e r i c a l f u n c t i o n [ 2 ]

From (31) i t fo l lows that

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L e t t i n g r = 2 n, w e have

The i n t e g r a n d e x p r e s s i o n (33) c a n b e r e p r e s e n t e d i nt h e f o r m

To c a l c u l a t e (34) we u s e t h e i d e n t i f y

(s a i s o b t a i n e d f r o m s b y r e p l a c i n g x by -x).I n t e g r a t i n g member by member t h e s e r i e s (35) w i t hr e s p e c t t o x w i t h i n t h e l i m i t s ( O . l ) , w e o b t a i n i n a c l o s e df o r m e x p r e s s i o n s for t h e members of t h e r i gh t -h a n d s i d e of(34) . As a r e s u l t w e h av e

We p l a c e t h e e x p r e s s i o n o b t a i n e d (36) i n (33) a n dc a r r y o u t t h e c a l c u l a t i o n of i n t e g r a l s of t h e form

F i n a l l y , w e o b t a i n

In summing up t h e ser ies S1 w e c a n b eg i n i n t h e sameorder, e x c e p t t h a t w e u s e i n p l a c e of (321, t h e a u x i l i a r yr a t i o- 7 -


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9 '

wher e

The const ant component of t he def l ect i on i s* ' .The r emai ni ng ser i es i n f or mul a (46) conver ges as2n)-3.

Recei ved 23 J une 1964

BIBLIOGRAPHY1. Lur ' ye, A . I . , St at i ka t onkost ennykh upr ugi kh obol ochek( St at i cs of Thi n- Wal l ed El ast i c Shel l s ) , Gost ekhi zdat ,

1947.2 . Hobson, E. Y., Teor i ya s f er i cheski kh i el l i ps oi dal ' nykhf unkt si v (Theor y of Spher i cal and El l psoi dal Func-.t i ons ) , Yor ei gn- L i t er at ur e Publ i shi ng House , 1952.3. Lur ' ye, A . I . , Pr ost r anst vennyye zadachi t eor i i upr ugost i( Thr ee Di mensi onal Pr obl ems of t he Theor y of El ast i -ci t y) , Gost ekhi zdat , 1955.4 . Casacci , S. , " Et ude de l a f l exi on des coques de r evol u-t i on char ges axi symet r i quent , " ( St udy of t he bendi ng

of r evol vi ng shel l s l oaded axi symmet r i cal l y) , Tr avaux,1959, Vol 43, No 301, 302.

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