Cylinder to Cylinder

8/8/2019 Cylinder to Cylinder

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S T R E S S E S F R O M P R E S S U R E , R A D I A L , A N D M O M E N T L O A D S I NC Y L I N D E R - T 0 - C Y L I N 9 E R V E S S E L B Y A F I N I T E P L A T E M E T H O D

N U C L E A R E Q U I P M E N T D I V I S I O NB A B C O C K & W I L C O X C O M P A N Y*-"" A K R O N , O H I O

A u t h o r s : S . J . B r o w nM . E . F o x

M I I tut te U M awnU W M W I * . *>jitet\* * U M M Sam M tta (MM S U M 1 < Ia u w ^ i t i ^ M M M M I I M H H , m m if t i

MMiifMyMnKi,tifcn,imM M iWB 4, IIIMmrH MW Mt'WWM MX


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t ; i n u u c t i o i iA struct ural problem that has received continued intrn-ist and drvclo piwut w r itic

late several decades Is the determination of stresses In two normally intersectip cyli n-drical shells) subjected to Internal pressur e and extern al lendi ng. Thi s type of stru ctur ein tnanifent In mnnwny s, handholes, view ports , nozcle-plping attachments, etc . Over theyen rs moot an alyti cal deve lopm ents ha ve been orlnt>1 (3, 41 , Van Dyke (5 ), anil tokkc rke rkr r J6J. utils.-itip.


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, ." ' T h a a d v a n t a g e o f b o t h M e t h o d s i a t h a r e p r e s e n t a t i o n o f v a r i a b l e t h i c k n e s s , b r a n c h;. _ : . p o i n t s , s n d e a a a i n h a n d l i n g l a r g a c o m p l e x c o m p o n e n t s . T h a s h e l l f o r m u l a t i o n s g e n e r a l l y

i < ' a s s u m e c l a s s i c a l t y p e s h a l l s i m p l i f i c a t i o n s w h e r e a s t h e f i n i t e e l e m e n t c h a r a c t e r i s e s K M>\ f i n i t e r e g i o n b y a d i s p l a c e m e n t p o l y n o m i a l . T h e s w a t v e r s a t i l e n u m e r i c a l m e t h o d o f p e r -

, ' f o r m i n g a s t r u c t u r a l a n a l y s i s o f a c y l l n d e r - t o - c y l l n d e r c o m p o n e n t s i s b y t h e 3 - D f i n i t ee l e m e n t m e t h o d , s i n c e t h a a y a a w t r l c g e o m e t r i c a n d s t r e s s n a t u r e c a n f e e m o s t a c c u r a t e l yi d e a l i s e d . I t s l s a d v a n t a g a i s t h a t I t ' s s o l u t i o n s r e p r e s e n t t h e m a n i p u l a t i o n o f a v a r yl a r g a s y s t e m o f . a l g e b r a i c e q u a t i o n s , h e n c e l a g e n e r a l l y m o r e e x p e n s i v e t h a n t h a o t h e r n u -

, m e r l c e l s h e l l o r f i n i t e e l e m e n t m e t h o d s . T h e a x i s / M e t r i c s e r - ^ t r i e a e d d i s p l a c e m e n ta s s u m p t i o n f o r t h e n u m e r i c a l s h e l l s n d f i n i t e e l e m e n t m e t h o d s w a s t h a e a r l i e s t a p p r o a c h u s e dt o e v a l u a t e , c y l l n d e r - t o - c y l l n d e r s t r u c t u r e s a n d h a s b e e n t h e m o a t p o p u l a r . I t h a a i b ee nf o u n d : o b e a p o o r a p p r o x i m a t i o n o f t h a s t r e s s a t a d i s c r e t e l o c a t i o n i n t h a j u n c t u r e f r o *i n t e r n a l p r e s s u r e l o a d i n g , h o w e v e r , i t l a a d e q u a t e I n r e p r e s e n t i n g s o m e c i r c - i n f c r e n m l l ya v e r a g e d s t r e s s v a l u e . T h e r e a r e v a r i o u s w a y a i n w h i c h t h e h o s t c y l i n d e r i s u s u a l l y r e p r e -s e n t e d . T h e r e a r e a h e m i s p h e r e w i t h t w i c e t h e r a d i u s o f t h e c y l i n d e r a n d a h e m i s p h e r e w i t hi t s r a d i u s e q u a l t o t h e c y l i n d e r w i t h a d o u b l e d i n t e r n a l p r e s s u r e . A s t u d y b y R o d a b a u g h1 2 0 ' d i s c u s s e s t h e l i m i t a t i o n s o f u t h a p p r o x i m a t i o n M e t h o d s , a n d i n e l u a e s s s a t u M n t

I' o f t n a u a a o f t h e t v x i s y m m e t r i c g e o m e t r i c a n d a s y m m e t r i c l o a d i n g a p p r o x i m a t i o n u c l n g K a l n l n s'' p r o g r a m . I n a p a p e r t>y B r o w n , ( 2 1 ] t h i s l a t t e r a p p r o x i m a t i o n , t h e r e p r e s e n t a t i o n o f a


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' s w n t . T h e d u r a b i l i t y o f B i j l a a r d ' e w o r k U a t t e s t e d t o b y t h e f a c t t h a t I t I s s t i l l w i d e l y1 w e d t o a s s e s s r e c t a n g u l a r a n d c i r c u i t s a t t a c h m e n t s t o c y l i n d r i c a l s h e l l s . I m p r o v e m e n t * ,

s w d l f l c a t i o n a , t a b u l a t i o n f o l l o w e d i n p a p e r a b y X e a p n o r ( 2 7 ] , a i j l a a r d 1 2 8 ] , W l c h m a n e t a l1 3 9 ) , D o d g e ( 3 0 ] , a n d S e l l a r a ( 3 1 ) . I n 1 9 6 3 H a n a b e r r y a n d J o n e a ( 3 2 ] g a v e a t h e o r e t i c a lt r e a t m e n t o f t h e i n p l a n e M o m e n t l o a d i n g o f t w o i n t e r s e c t i n g c y l i n d e r s , w h e r e t h e m o m e n t 1 *a p p l l o d t o t h e n o t t l e t e r m i n u s . I n t h e I n s t a n c e o f r e i n f o r c e m e n t o f t h e I n t e r a e e t l o n , w eh a v e t h e a c m e l i m i t a t i o n a a t h e t h e o r e t i c a l p r a a a u r a a o l u t l o n a ; n a m e l y , I n a b i l i t y t o d e a -i! n i b s s c c u r a t e l y t h e s t r e a a e a w i t h i n t h e j u n c t u r e .

I n t h e c a a e o f t h e n u m e r i c a l a e t h o d e c i t e d p r e v i o u s l y , t h e 3 - D f i n i t e e l e m e n t m e t h o di s p a r t i c u l a r l y s u i t a b l e t o d e a e r l b a t h e a s y m m e t r i c n a t u r e o f t h e p r o b l e m w i t h r e i n f o r c e -m e n t ; h o t u v e r , C h e c o m p u t e r c o a t l a u s u a l l y s i g n i f i c a n t l y I n c r e a s e d f o r t h e a o t c l e t e r m i n u s

' s u b j e c t e d t o f o r c e a n d m o m e n t l o a d i n g v e r s u s a s i m p l e i n t e r n a l p r e s s u r e l o a d . T h e c o s ti n c r e a s e l a a r e s u l t o f t h e i n c r e a s e i n t h e m e s h s i t e n e e d e d t o i d e a l i z e t h e n o n s y a m e t r i cl o a d s a n d r s s u l t i n g b o u n d a r y c o n d i t i o n s .

T h e a x l e y a m a t r l c g e o m e t r i c a n d d i s p l a c e m e n t f o r m u l a t i o n s c a n n o t r e a s o n a b l y a p p r o x i m a t et h e e x t e r n a l M o m e n t a n d t r a n s v e r s e l o a d c a s e s ; h o w e v e r , b y u a l n g a F o u r i e r a e r i e s e x p a n s i o n( o r d i s p l a c e m e n t , a n a s y a m e t r i c l o a d i n g v l t h a n a x l s y a m c t r l c g e o m e t r i c a s s u m p t i o n o f f a r e ap o s s i b l e a p p r o x i m a t i o n m e t h o d .


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j A simple t wo erm expansion' s chosen which results In a uniform biaxial stre ss litthe plot* and la everywhere equivalent toth e tresa In tha ylinder aa a approachea sero .The w knowr. hoop membrane stress6gt at r a of t ht nnular pAate ta rlten aa a function

Tr CYLINDER THICKNESS'*P~INTERWL PRESSURETtom Van tyke'a solution of atreasea In a hole in a cylinder (see figure 2)g f fa W H E R E j f rv were fcble to obtain with aquation (o)

/ AND ^These functions at* sraphlcally ahown In fltura 3.With P . K A J 3 tha outer radius of the late and he f*brne loads a y be etermined. '

' I n a manner similar to the nplane load paraaeterc , It as shown for he bending hoopatress In tha late at /*"a that 9 t ' - r { p . G . 9 . p ). F o r endini, the equilibrium eqn. takes tha orm of (2), where 0 la transvarse dla -placement.

From Van Dyke'a solution of th a banding atresa at the utout Inthe ylinder


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i Equation s (20) and (21).are Illustrated la figure .

aaentaon a Wossle Att ac he dIn ref. [ 2 2 ] , . BlJlaard praaenta the aerlea solution o f longitudinal a n d ircuafcrontiaiaoasnta ( M ^ n d M ^ ) applied o n a cylinder. The aoasnts are effectuated b y a pressuredistribution over the fttJfCfXZCgon the cylinder (figure 5 ) . I n the manner o f t h e

; axial solution, w e ill u s e Bljlaard'e aolutlon to i teralne the aaxlaua internal aoaenta: a n d orcea (Af* n d/ V O t o e lkposed upo n t h e circular plate. This l a done In order t o:' calculate the external loads ori the circular plate. D p . until thla point w e have been deal-; ing with aclf equilibrating loada applied t o t h e plate. For a n external bending aoaent'. applied to the plate , vet will have equilibrium froa equal and pposite aoaenta via a prcs-

aure distribution overTftf^nd a tranaverae ahear distributi on along77'>(flgure 9 ) .. ' F o r his type of loading, o u r eries solutions will consist of odd sraonica. T o b e;' consistent with the philosophy o f a n Inexpensive approximation aetho d, w e will consider only

the caae otC&iiffuhenM*/. The ma xi ma Internal aoaenta and forcea In the cylinder occur: a t C i o r C 2 1 depending upon the application o f M ^ rA ( r . As an exaaple, a longitudinali aoaacjc applied o n * C f would reault In sxlnua awaenta and orcea Mfjjand/ |/ J) *tbGi.O,O)1 T h e tessec at X * b , Y* t are found t o e usually negligible o r a n order o f magnitude lower. A circumferential external aoaent at th e * C j erea results In the ma xi ma internnl aoaent


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, {hence

Equating Ato to /tf< aniNp we awy solveO R Ar* -yL an

(2S)

ExPla of Axial Force and ttowint toadlnt. Theanalyt ical and experlacntal evaluation of aodel #3 (seefigure 11) teated at OakMdge

la compared to anFPM an al ys is usin g B*W fi n it e elo aen t prograa MAS (363 The curva-ia 0.11O

\*0.4, b*/. / 9 3 / M *F*SOOlb..Me *800iN-ib.. X c Z4.5*j3i*0. //. The forces at r-b foreach load la calculated as fol lows:

a) for TB (radial force) - 629- 42.712X30 lb/


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' - I

' t o - e y l i n d t r j u n c t u r e * . S i n c e t h e a p p r a a e h v a c s a h a l l o w a h e l l f o w u l a t l o n s ( b y V a n C j - k c a n dM j l u a r d ) a n d a t h r o e t e w s e r i e s e x p a n s i o n p l a t e C o w u l a t l o n , t h e j a n j c o f a p p l i c a b i l i t yi a r o u a h l y H a l t e d * * ( i t f T ) H Q , & / f t p . S a n d c u r v a t u r e 1 ., I t i f a i t t h a t t h a v a l u a o f t h a M t h o d l a I t a a c c u r a c y * c c o n o a y , atyi e a s e i n a n d c l i n ta a t r u c t u r a w h i c h f a l l s w i t h i n t h a r a n t t o f a p p l i c a b i l i t y . A n o c h a r a p p e a l i n g f e a t u r e o f t h e a t b o d i s t h a t i t a a i a p l i a t i e a p p r o a c h o l w p a r p o a i t t o n o f r e s u l t s p e w i t s a n a a s y r t t a n -a l o n t o - l n e l u ^ a a d d i t i o n a l l o a d s n e t t r a a t a d . F o r t h o s a a c b a n l c a l l c a d l n g a n o t d o v a l o p c d ,i t i a f a i t t h t t t h e i r e f f e c t c a n e i t h e r b a a c c o u n t e d f o r b y t h e a w c h a n i s M d i s c u s s e d o r b y l a p l a e e l c u l a t l o n s . C e n e r a U y , t h a s t r e s s e s r e s u ^ c i o c i t e m t o r s l o n a l o r t r a n s v e r s e s h e a ra r a a a a l l c o a p a r a d t o t h a l o a d s d i s c u s s e d , h o w e v e r , t h e s e s h e a r a f f e c t s a a y b e l a c J v . d e d .F i n a l l y , i n t h a i n s t a n c e o f t h e n t a l a t r a a a w i t h i n t h a c y l l n d e r - t o - c y l i a d e r s t r u c t u r e , i th a s b a a n s h o w n I n . a n u n p u b l i s h e d a t u d y b y B r o w n t h a t t h a f F K y i e l d s v e r y f o o d r e s u l t s f o rt h a x a n s e o f c u r v a t u r e s d i s c u s s e d .

A c f c n o w l e d t j a a n t a

I l l p p r a c l a t l o n i a a l a o e x p r e a s e d t o J . L . R e c h a e r o f t h e K u d e s r E q u l p a e n t D i v i s i o n o f


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I R e f e r e n c e * . .( 1 ) L u r i s , *.. I . , " S t a t i c * o f T h i n - W a l l e d E l a s t i c S h e l l s " ( S t a t e P u b l i s h i n g H o u s e o fT e c h n i c a l a n d T h e o r e t i c a l L i t e r a t u r e , M o s c o w , 1 9 4 7 ) , T r a n s l .O f W i t h u m , D . , " O n t h e S h e a r S t r e s s i n a C y l i n d r i c a l S h a l l w i t h * C i r c u l a r O p e n i n g , "I n g e n i e u r - A r c h l n 2 6 , 4 3 5 ( 1 9 5 8 )( 3 ) K l i n e , L . V . , D i x o n , R . C , J o r d a n , M . F . , a n d E r i j e n , A . C , " S t r c s s e i n P r e s s u r l x e dC y l i n d r i c a l S h e l l w i t h C i r c u l a r C u t o u t , " G e n e r a l T e c h n o l o g y C o r p o r a t i o n T e c h n i c a lR e p o r t M o . 3 - 1 ( A u g . 1 9 6 2 ) . ( a c e a l s o R e p o r t 3 - 2 , 3 - 3 , 3 - 4 , 3 - 5 ) .( 4 ) E r l n g e n , A . C , N e g t i d l , A . X . , a n d T h i e l , C . C , " S t a t e o f S t r e s s i n a C i r c u l a r C y l i n -

d r i c a l S h e l l w i t h a C i r c u l a r B o l e , " W e l d i n g R e s e a r c h C o u n c i l B u l l e t i n M o . 1 0 2( J a n . 1 9 6 5 ) .( 5 ) V a n D y k e , P . , " S t r e s s e s A b o u t a C i r c u l a r H o l e I n a C y l i n d r i c a l S h e l l , " A I A A J o u r n a l ,V o l . 3 , H o . 9 , 1 9 6 5 , p p 1 7 3 3 - 1 7 4 2 .( 6 ) L e k k e r k e r k e r , J . G . , " S t r e s s C o n c e n t r a t i o n A r o u n d C i r c u l a r H o l e s I n C y l i n d r i c a l S h e l l s "P r o c e e d i n g s ' o f t h e E l e v e n t h I n t e r n a t i o n a l C o n g r e s s o f A p p l i e d M e c h a n i c s , S p r i n g e r -V e r l a g , B e r l i n , 1 9 6 4 , p p 2 8 3 - 2 8 B .( 7 ) R e l d e l b a c h , W . , " T h e S t a t e o f S t r e s s a t t h e P e r p e n d i c u l a r I n t e r s e c t i o n o f T w o R i g h t

C i r c u l a r T u b e s , " I n g r . - A r c h . , V o l . 3 0 , 1 9 6 1 , p p 2 9 3 - 3 1 6 .( 8 ) E r l n g e n , A . ' C . a n d S u h u b l , E . S . , " S t r e s s D i s t r i b u t i o n a t T w o N o r m a l l y I n t e r s e c t i n gC y l i n d r i c a l S h e l l s , " N u c l e a r S t r u c . E n g i n e e r i n g , 2 : 2 5 3 - 2 7 0 , 1 9 6 5 .( 9 ) G r e e n , A . E . a n d Z r c , W . , " T h e o r e t i c a l E l a s t i c i t y , " O x f o r d U n i v . P r e s s , 1 9 5 4 .( 1 0 ) 8 o f f , N . J . , " B o u n d a r y - V a l u e P r o b l e m s o f t h e T h i n - W a l l e d C i r c u l a r C y l i n d e r , "J o u r n a l o f A p p l i e d M e c h a n i c s , D e c . 1 9 5 4 , p p 3 4 3 - 3 5 0 .( l l ) ' E r l n g a n , A . C , N a g h d l , A . X . , M a h a o o d , S . S . , T h l e l , C . C . , a n d A r l a a n , T . , " S t r e s sC o n c e n t r a t i o n s I n T w o N o r m a l l y I n t e r s e c t i n g C y l i n d r i c a l S h e l l s S u b j e c t e d t o I n t e r n a lP r e s s u r e , " W e l d i n g R e s e a r c h C o u n c i l B u l l e t i n H o . 1 3 9 , A p r i l 1 9 6 9 .


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loaabaua h, E. C., "Applicability of Axlsyaaetrlc Geometry Analytic Methods to N ozzl esI K Cylindrical Shells with Internal Pressure Loading," Battciic Report 117-8,. M M .subcontract Ho. 3131, Columbus, Ohio. July 197 3.

, 1

Br ow n> . J., "A Finite Plate Method to Sclve Cyllndcr-to-Cyllmder Structures Sub -jected t o S Mr n a l Pressure," ASMS paper Mo. 76-PVP-C, Journal of Pressure Vessel andPiping.(22) Bljlsard, i. P., "Str es Wa From Local Loadlnga in Cylindrical Pressure Vessels," ASKEpaper Mo. 54-PIT-7, Sept. I H i ,(23) U jl aa rt , P. P., "Stresses From Radlsl Loads in Cylindrical Pressure Vessels," Veld-Ing Journal Research)Supplement 33, 1954.(24) Bijlaard, P. P., "Stresses From Radial Loads and External Moments in Cylindricalv Pressure Vessels," Welding Journal, 34, H 5 S .(25) Bljlaa rd, P.. ., "Additional Data on Stresses In Cylindrical Shells Voder Local Loa d-Ing," Ibid, Mo . SO. May 1959: . ' - .(26) Bijlaard,' P. P., "On the Effect of Tangentia l Loads on Cylindrical and SphericalShells," PVRC, Welding Research Council.(27) Kenpn er, J., et al "Tables and Curves for Deformations, nd Stresses in Circular- Cylindri cal Sh eila trader Localised L oad ing s} " 3. Ae r o . S e . M . 2 4 , 1 9 5 7 .(28) Bijlaa rd, P. Pi and Cr anch, E. T., "Interpretive Commentary o r the. Application ofTheory to Experimental.Results for Streeaes and Deflections Due to Local Loads onCylindrical Shells," WRC Bulletin He . 60 , 1960.(29) Vichm an, K. R., Hopper, A. C;, and Merahon, J. L., "Local Stresses In Spherical andCylindrical Shells d ue to External Loadings, " WRC Bulletin 107 , 1965.(30) Dod ge, W. C. , "Secondard Stress Indices for Integral Structural Attachments toStraight P ipe, " WRC Bulletin 19 8, 1974.(31) Sellara, F., "A Not e on the Correlation of Photoelastic and Still Model Data for


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O P t l o m1.- Uniform biaxia l loading idealisation.2 . Circular cylinder1 with circular cutout. -3. U n i t * plat* paraaatera for dl Mi wi on (X), biaxial loading ( 4 ) , vertical

l o a d i n j * (P/p and o/pi varaua tba curvatur* paraawtar (0) .4. K'S notcle5. Circular cylindar unit n l y loaded ever, rectangular area (2Ci x 2 C 2 ) .6. Circular plata uniformly loaded over a circular area ( U t c 1 ) .7 . Circular plate inpiane load paraatra for axial force (r ).8. Circular plate transverae load paraaetera for axial force (F ).9. Circular plate tranaverse load paraaeters for aoaents (Me or h.).

1 0 . Circular plata inpiane load parameters for atmenta CHc or H U 1 1 . Oak Ridge teat aodtl #3 .


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J


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V


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u


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DIMENSIONS-ABCDEFGH1JKLMN

INCHES27.64

4.533.542.605.200.200.390.798.661.373.941.970.4920.168

mm742115906613251020220501005012.54.3


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J

'.. I


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u...


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J

8" 1 -


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fc. J

00I


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00COS

2|HIIfc

. Z* w21^-/ IV o


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H - thickness

J . Thickness

Model .Ma^rDirnerisionsNo- ^T~T3 ' 10.0 1.29 0.2 0.168

Cylinder to Cylinder

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