Stress InStress Intensity Factors for Threaded Connectionstensity

7/27/2019 Stress InStress Intensity Factors for Threaded Connectionstensity

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PergamonEngineering Fracture Mechanics Vol. 50, No. 4. pp. 545-567, 1995Copyright 1995 Elsevier Science Ltd00 13 -7 94 4( 94 )0 01 71 -5 Pr in t ed in Grea t Br it ain . Al l r igh ts r eserved0013-7944/95 $9.50 + 0 . 0 0

S TR E SS IN T E N S I T Y F A C T O R S F O R T H R E A D E DC O N N E C T I O N S

F . P. B R E N N A N a n d W . D . D O V E RN D E C e n t r e , D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r in g , U n i v e rs i t y C o l l e g e L o n d o n , T o r r i n g t o n P l a c e ,L o n d o n W C I E 7 J E, U . K .A b s t r a c t - - A g e n e r i c s tr e s s in t e n s i ty f a c t o r ( S I F ) s o l u t i o n f o r t h r e a d e d c o n n e c t i o n s i s p r e s e n t e d i n th i sp a p e r . T h i s i s b a s e d o n m u l t i p l e r e f e re n c e s ta t e w e i g h t f u n c t i o n t h e o r y , m a k i n g i t a p p l i c a b l e t o a n y m o d eI c r a c k e m a n a t i n g f r o m a t h r e a d r o o t . T h e r e f e r e n c e S I F s r e q u i r e d f o r t h e s o l u t i o n a r e a c o m b i n a t i o no f p u b l i s h e d r e s u l ts r e p r e s e n t i n g g e o m e t r i c a l f e a t u r e s e n c o u n t e r e d i n t h r e a d e d c o m p o n e n t s . T h e s ec o n s t i t u t iv e c o m p o n e n t s o f th e r e f e re n c e s o l u ti o n s c a n e a s i ly b e r e p l a c e d i n t h e e v e n t o f m o r ec o m p r e h e n s i v e o r a p p r o p r i a t e s o l u t i o n s b e c o m i n g a v ai l ab l e . F o r t h i s re a s o n i t is c o n s i d e r e d p r u d e n t t op r e s e n t t h e s t r u c t u r e o f t h e s o l u t i o n r a t h e r t h a n a r i g i d s e t o f p a r a m e t r i c e x p r e s s i o n s s o t h a t s p e c if ics o l u t i o n s c a n b e f a s h i o n e d f o r p a r t i c u l a r a p p l i c a t i o n s . T h e n e w s o l u t i o n is r i g o ro u s l y e x a m i n e d , c o m p a r i n gi t w i t h p u b l i s h e d S I F s o l u t i o n s f o r t h r e a d e d a n d r e l a t e d c o m p o n e n t s .

N O M E N C L A T U R EadDf~(a,x)P,hhK ,K osgRodKo~

C r a c k d e p t h K ,C r a c k h a l f l e n g t h MM i n o r d i a m e t e r o f t h re a d N AD i a m e t e r o f a r o d rW e i g h t f u n c t i o n o v e r a l o a d i n g s y s t e m r i" m "N o r m a l i s e d S I F tS t e p o r t o o t h h e i g h t xM o d e I s t r e s s i n t e n s it y f a c t o r ( S I F ) YS I F f o r a n e d g e c r a c k e m a n a t i n g f r o m a na n g u l a r c o r n e r " 0 " i n a f in i te t h i c k n e s s 0s t r i p pS I F f o r a s u r f a c e c r a c k i n a r o d ~rBS I F f o r a s u r f a c e c r a c k e m a n a t i n g f r o m ra n a n g u l a r c o r n e r " 0 " i n a r o d

T h e o r e t i c a l s t r e s s c o n c e n t r a t i o n f a c t o rB e n d in g m o m e n tN e u t r a l a x i sA v e r a g e t u b e r a d i u sI n t e r n a l t u b e r a d i u sT u b e w a l l t h i c k n e s sD i s t a n c e f r o m n o t c h t i p in " x " d i r e c t i o nG e o m e t r i c s t r e s s i n te n s i t y c o r r e c t i o n

f a c t o rT h r e a d f l a n k a n g l eN o t c h / t h r e a d r o o t r a d i u sB e n d i n g s t r e s sN o m i n a l s t r es sK s e S I F f o r a n e d g e c r a c k e m a n a t i n g f r o m a n t r ( x ) S t r e ss d i s t r ib u t i o n i n u n c r a c k e da n g u l a r c o r n e r " 0 " i n a s e m i - fi n it e s t r ip p l a n e .

I N T R O D U C T I O ND ES PIT E t h e w i d e u s e o f t h r e a d e d f a s t e n e rs i n t h e a s s e m b l y o f s t r u c t u r a l a n d m a c h i n e c o m p o n e n t s ,s u r p r i s i n g l y l it t le o f t h e p u b l i s h e d m a t e r i a l t h a t e x i s t s a d d r e s s e s f a t i g u e c r a c k g r o w t h i n s c r e wt h r e a d s . T h i s i s p a r t l y d u e t o t h e c o m p l e x i t y o f t h e i n t e r n a l s t re s s d i s t ri b u t i o n i n t h r e a d s , a n d t h ee n o r m o u s v a r i a ti o n o f c o n n e c t i o n t y p es , m a t e ri a ls , q u a l i t y o f m a n u f a c t u r e a n d m a i n t e n a n c e .O b s e r v a t i o n o f f a t i g u e c r a c k g r o w t h i s a l s o d i ff ic u lt a s t h e c r a c k i s o f te n h i d d e n w i t h i n t h ec o n n e c t i o n u n t i l it e m e r g e s a s a t h r o u g h c r a c k . F a t i g u e c r a c k s i n it i a te m o s t f r e q u e n t l y i n a t h r e a dr o o t . T h e s i m i l a r it y b e tw e e n t h is s i t u a t i o n a n d t h a t o f c r a c k s in c i r c u m f e r e n t i a ll y n o t c h e d b a r s h a sl e d t o t h e u s e o f t h e H a r r i s [ 1] s o l u t i o n f o r e x t e r n a l c i r c u m f e r e n t i a l c r a c k s i n a t u b e u n d e r u n i f o r ma x i a l t e n s i le s t re s s o r t o r s i o n . T h i s i s a u s e f u l s o l u t i o n , h o w e v e r i t is n o t d i r e c t l y r e l e v a n t t o t h r e a d e dc o n n e c t i o n s . S o l u t i o n s d o e x i s t f o r r e l a t e d g e o m e t r i c a l s h a p e s a n d s o m e S I F s o l u t i o n s h a v e b e e np u b l i s h e d f o r s p e c i f ic t h r e a d e d c o n n e c t i o n s [ 2, 3 , 4 ]. J a m e s a n d M i l ls [5 ] p r o d u c e d p r o b a b l y t h em o s t u s e f u l, r e vi e w a n d i n n o v a t i v e s y n t h e s i s o f p os s i b le s o l u t io n s f o r b o l t s . T h i s is m a i n l y a na s s e s s m e n t o f S I F s o l u t i o n s in r o d s a n d i n c lu d e s in t e r e s ti n g o b s e r v a t io n s o n h o w t h e s e cr a c k s g r o wi n r e a l i t y . H o w e v e r , n o c o n c l u s i v e s o l u t i o n i s f o r m e d , a n d t h e v a r i a t i o n s s u c h a s t h r e a d f o r m ,h o l l o w t u b e a n d i n d e e d i n t e r n a l c r a c k s a r e n o t e v e n c o n s i d e r e d .F o r t h e s e r e a s o n s t h e d e v e l o p m e n t o f a g e n e ri c s o l u t io n f o r f a t i g u e c r a c k g r o w t h f r o m a t h r e a dr o o t v a l id u n d e r a n y s y m m e t r i c a l st re s s s y s te m , i s a t t e m p t e d i n th i s p a p e r. T h e s o l u t i o n e x c lu d e s

54 5


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5 46 F . P . B R E N N A N a n d W . D . D O V E R

\

T F i g . 1 . E d g e c r a c k i n a t h r e a d r o o t .

Y

a calculation for the load distribution within the entire connection such as those by Sopwith,Patterson and Kenny, or Bluhm and Flanagan [6-8]. Geometric features remote from the threads(such as Stress Relief Groove Pin and Bore Back Box in drill tools) have become commonplaceas stress relief mechanisms in highly critical fatigue areas within connections. For this reason it isconsidered that the load distribution is best analysed separately using finite element or photoelasticmethods, and that once known the SIF solution proposed in this paper can be implemented.Weight function solutions provide relatively simple stress intensity factor solutions forarbitrary symmetric loading systems. Finite element methods are invariably computat ionallyinefficient, especially where a number of growth scenarios are to be considered. These are, howeverwhen used with full scale test results useful validation techniques. The weight function method ismathematically exact, errors in a final solution can be solely attributed to inaccurate mechanicalinterpretation or simplification of the problem being addressed.The proposed solution uses a recent development in weight function theory [9] whicheliminates differentiations in the calculation, making the routine far more stable and computation-ally efficient. The solution is a combination of published solutions each of which themselves weighthe influence of several geometrical features. The solution is structured modularly so that it canevolve easily with the advent of more comprehensive sub-solutions. The background, developmentand details of the SIF solution are described in the following sections.

jP3 i

P l

F i g . 2. A p p l i e d l o a d s c o n t r i b u t i n g t o t h r e a d r o o t s t r e ss .


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SIFs for threaded connections 547C I

t

Fig. 3. E xternal surface crack in a tube.W E I G H T F U N C T I O N F O R T H R E A D E D C O N N E C T I O N S

2 D s o l u t i o nT h e g e o m e t r y a n d n o t a t i o n o f a lo a d e d t h r e a d t o o t h - r o o t i s s h o w n i n F i g . 1 . T r a d i t i o n a l l y

w e i g h t f u n c t i o n S I F s o l u t i o n s r e q u i r e d a r e f e r en c e s tr e ss i n t e n s i ty f a c t o r u n d e r a n y m o d e I l o a d i n gs y s t e m , th e c o r r e s p o n d i n g s t re s s f ie l d a n d t h e a p p r o p r i a t e c r a c k o p e n i n g d i s p l a c e m e n t p r o fi le t ob e k n o w n . I n t h i s c a s e i n s t e a d o f u s i n g a n a s s u m e d c r a c k p r o fi le , t w o r e f e re n c e s tr e ss s y s t e m s a r eu s e d , i .e . u n i a x i a l t e n s io n a n d p u r e b e n d i n g . B y v i rt u e o f t h e d e f i n i t i o n o f t h e w e i g h t f u n c t i o n b e i n ga u n i q u e p r o p e r t y o f g e o m e t r y , th e i n f lu e n c e s i t re p r e s e nt s c a n b e i s o l a t e d a n d c o m b i n e d . T h i s w a se f fe c t iv e l y a c h i e v e d b y N i u a n d G l i n k a [ 10 ] a n d b y M a t t h e c k e t a l . [ 1 1 ] a f t e r I m p e l l i z z e r i a n dR i c h [ 1 2] u s e d w h a t t h e y c a l le d G e o m e t r y C o r r e c t i o n F a c t o r s f o r t h e i n f l u e n c e o f g e o m e t r i ca n o m a l i e s o n t h e s t r es s in t e n s i t y f a c t o r . T h e l o a d i n g o n a t o o t h i s u n i q u e i n t h a t u n l i k e m a n ys o l u t i o n s d e a l i n g w i t h a " n o t c h - t y p e " g e o m e t r y , o n l y o n e f a c e o f t h e " n o t c h " r e l a y s a t e n s il e o rc r a c k o p e n i n g s t re s s " P I " , s ee F i g . 2 . T h i s is w h y t h e s o l u t i o n c a n b e i s o l a t e d i n t o t h e s i m p l i fi e df o r m o f F i g . 1 . T h i s i g n o r e s a n y e f f e c t o f t h e c o m p r e s s i v e t o o t h b e n d i n g s t r es s es d u e t o l o a d o nt h e p r e c e d i n g t o o t h . T h e s t re s s i n t e n s i t y f a c t o r s o l u t i o n a l s o i g n o r e s t h e e f fe c t o f t h e r o o t r a d i u s" p " , s in c e a n y f r a c t u r e m e c h a n i c s s o l u t i o n r e q u ir e s t h a t a c r a c k i s a l re a d y p r e s e n t, a n d o u t s i d et h e e f fe c t o f " p " o n t h e m a g n i t u d e a n d p o s i t i o n o f t h e m a x i m u m r o o t s t r es s h a s l i tt le ef f ec t o nc r a c k p r o p a g a t i o n .T h e a n a l y ti c a l c o n f o r m a l m a p p i n g s o l u ti o n s o f H a s e b e e t a l . [ 13 ], a n d H a s e b e a n d U e d a [ 14 ]c a n b e u s e d t o d e v e l o p a r e l a t io n s h i p f o r t h e s tr e ss i n t e n s i ty f a c t o r f o r a t w o - d i m e n s i o n a l c ra c kg r o w i n g f r o m a t h r e a d r o o t w i t h r e s p e c t t o f la n k a n g le , s h a n k ( o r t u b e w a l l) th i c k n e s s a n d t o o t hh e i g h t . T h e s e s o l u t i o n s a r e f o r s tr i p s w i t h a n g u l a r s t e ps . S o l u t i o n s a r e a v a i la b l e f o r " U - n o t c h " a n d" V - n o t c h " g e o m e t r ie s w h i c h m a y r e se m b l e a t h r e a d g e o m e t r y in a p p e a r a n c e b u t a s e x p l a in e d a b o v eo n l y o n e f a c e i s l o a d e d m a k i n g t h e s t e p o r s h o u l d e r s i t u a t i o n m o r e a p p r o p r i a t e . H a s e b e ' s s o l u t i o n

Z . . . .

Fig. 4. Internal surface crack in a tube.


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548 F .P . BREN NAN and W. D. DOVE Rf o r a c r a c k o r i g i n a t i n g f r o m a n a n g u l a r c o r n e r o f a se m i - i n f in i t e p l a t e w i t h a s t e p [1 4] w a s a n a l y s e da s a p l a n e e l a st ic p r o b l e m , u n i f o r m t e n s i o n b e i n g t h e b o u n d a r y c o n d i t io n . S t re s s i n t e n s it y f a c t o r sw e r e g i v e n i n t e r m s o f

r s v = F , h a o W / '- ~ . (1 )T o n o r m a l i s e th i s i n t o a m o r e s t a n d a r d f o r m , " F ~ h " w a s s i m p l y m u l t i p l ie d b y ~ / - ~ .T h e f i n i te s t r i p S I F s o l u t i o n s o f r e f . [ 1 3] ( s t ri p w i t h a s t e p 0 = 0 ) w e r e n o r m a l i s e d a s ,

F : h ( t - - a ) 1"5 K I (2 )K - tao"

D i v i d i n g b y

g i v es t h e s t a n d a r d Y f a c t o r f o r te n s i o n a n dK~t 2

f o r b e n d i n g .

(3 )

(4 )

4

2 . 5!21 .5

0 . 5

0 0

". +x | e

I I ! I0.1 0 .2 0 .3 0 .4 0 .5

a ~

a 2ac

M20 UNC (Weight function solution)Edge Crack

O

+

X

Daoud, Car twright & Carny (Com pliance)[19]Daoud, Car twright & Carey (FE) [19]Blackburn [20]Bush [21]

Fig. 5. Edge crack in a roun d bar under tension.


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SIFs for threaded connections 549H a s e b e n o t e d t h a t f o r s h o r t c r a c k s t h e s t r e s s i n t e n s i t y f a c t o r w a s m o r e a f f e c t e d b y t h ed e c r e a s in g v a l u e o f th e f l a n k a n g l e . H o w e v e r , w i t h i n c r e as i n g c r a c k d e p t h t h e i n fl u e nc e o f " 0 "l e ssen s b u t t h e ex i s t en ce o f t h e s t ep c o n s id e rab ly a f fec t s th e s t re s s i n t en s i ty fa c to r t o t h e l a s t .A s s u m i n g t h a t t h e e f f e c t o f g e o m e t r ic a l f e a t u r e s o n t h e s t r e ss i n t en s i ty f a c t o r s c a n b e w e i g h t e do r s e p a r a t e d

K ~ s r K ~ sK O r K O s (5 )

= ~ K ~ s = K e s r K sK O F (6)H a s e b e ' s t a b u l a t e d s o l u t i o n s w e r e c u r v e fi t te d g i vi n g t h e f o l l o w i n g r e l a t io n s f o r u n i a x i a l t e n s i o n

2Ksr = 1 . 3 2 5 ( ~ ( -0 . ,867+0.0223 ,o~(~) )_ F

a a 2F = e (no+ n , ' ~ (~ ) + n2 'o~(~) )H 0 = - 0 . 1 3 8 x 10 6 0 3 + 0 . 0 0 6 4 x 1 0 - 6 0 4 - 0 . 0 0 0 1 x 1 0- 60 5H 1 = 0 .3 6 0 3 x 1 0 - 5 0 2 - 0 .0 2 7 5 x 1 0 - 5 0 3 + 0 .0 0 0 5 x 1 0 - 5 0 *

/ -/2 = 0 .17 35 x 1 0 - 5 0 2 - - 0 .0 0 5 6 x 1 0 - 5 0 3 ( 7 )

, . 0> -

4

3 . 5

3

2 . 5

2

1 .5

1q

0 . 5

00

0 0 u 0

i I I I0 1 0 . 2 0 .3 0 . 4 0 . 5

,/o

a 2a_ m

M20 UNC (Weight function solution)

Edge Crack

E F M 5 0 / 4 - - 1

B u s h [ 2 2 ]Fig. 6. Edge crack in a round bar under bending.


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550 F.P . BRENNAN and W. D. DOVERKs 1 . 4 _ 2 . 4 ( t ) + 10 .1 ( t ) 2 + 3 7 . 6 ( 7- 3 0 1 . 3 ( 7) S + 9 8 1 . 7 ( t ) 1 3 _ 1 5 4 5 . 7 ( t ) ' 7 + l 1 4 0 . 5 ( t a ) 2 3 ( 8 )

crKrx/~ = 1.325 ( h ) ( - ' 1 8 6 7 + ' 2 2 3 o g e ( ~ ) . (9 )The bending so lu t ions a re a s sumed the same a s the ax ia l for KF and K r , s ince in an inf in i t e lyth ick p la te

o'B = a o (1 x )NA_o~ ,~ ao (10)_ _ 1 6 .1 8 8 3 + 3 6 .6 tll l o ~ ( t ) + , 4 6 .8 2 4 1 1 o ~ ( t )+ 2 7 .9 9 9 3 1 O g e ( t) +1 . 0 8 3 3 1 o g ( 7 )6M ~/~-~ = 46.118

t 2 + 2"8668 ge(a 6 + 0"4646 ~ t)7 + 0'04271gt) 8 + 0'0017 ge( )9 . (11)S u b s t i t u t i n g t h e a b o v e s o l u t i o n s i n t o e q . ( 6 ) g i v e s a r e f er e n c e s t r es s i n te n s i t y f a c t o r f o r a 2 D c r a c kg r o w i n g f r o m a " t h r e a d - l i k e " n o t c h .

3

2 .5

,~ 1.5>. 1 t \ :.5

0 i ' ' i0 1 0 . 2 0 . 3 0 4 0 . 5

M20 UNC (W eight unction solution)

Circular Crack [23]

o Daoud & Cartwright [24]+ Wilhem [25]* Forman & Shivakumar[23]

Fig. 7. Circular-arc surface crack in a rou nd bar under tension.


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SlF s for threaded connect ions 551Th e tw o so l u t i on s ar e ide a l for u se a s re fe re n ce s ta te s in O jdr ov i c R as ko a n d P e tr osk i ' s w e i gh t

f u n c t i o n s o l u t i o n [9 ] g iv e n b e l o w .f , , ,( a , x ) = 2 t r ( x ) x / / - ~ C f l - x - ( 1 2 )K ( a ) j = o \ a

w h e r eCj = q-Lw ~

w ij = a i ( x 1 - - d xa

K ,O" ~ / ~

B o t h a x i a l a n d b e n d i n g s o l u t io n s f o r Ks su b s t i t u t ed i n to eq. (12) y ie ld a n a c c u r a t e s i m p l e w e i g htf u n c t i o n f o r t h a t g e o m e t r y , k n o w i n g t h e s tr e ss d i s t r i b u t i o n i n th e u n c r a c k e d p l a n e .

1 .5

0 . 5

0 !o 0 11 0 : 2 o h 0 . , 0 . 5

c

M20 UNC (Weight funct ion solution)

Circular Crack [23]

o Daoud & Cartwright [24]+ Forman & Shivakumar [23]

Fig. 8 . Circular -arc sur face crack in a round bar unde r bending.


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552 F.P . BREN NAN and W. D. DOVER3D so lu tion

M o s t t h r e a d s a r e t h i c k f l a t , a n g u l a r o r t u b u l a r b o d i e s , w h e r e p l a i n s t r a i n c o n d i t i o n s p r e v a i l .H o w e v e r , o n c e t h e i n fl u en c e o f th e t o o t h i s k n o w n i n a 2 D s e n s e i t i s c o m m o n p r a c t i c e t os u p e r i m p o s e t h i s in f lu e n c e o n t o a n y 3 D b o d y [ 11 , 1 5 ]. C o m p o n e n t s o f t h r e a d e d c o n n e c t i o n s a r et u b u l a r b y n a t u r e , t h e m a l e s e c t io n s c a n b e s o l id r o d s o r h o l l o w e x t er n a l ly t h r e a d e d p i pe s . F e m a l es e c t io n s a r e a l m o s t a l w a y s i n t er n a l ly t h r e a d e d p i pe s . I n s t e a d o f u s i ng t h e 2 D s o l u t i o n s f r o m t h ep rev io u s sec t i o n an d i g n o r in g t h e t h read h e l ix an g l e, o n ly t h e i n fl u en ce o f t h e p ro j ec t i o n ( t o o th )i s r e q u i r e d . A s b e f o r e t h e f o l l o w i n g m u s t h o l dKOsF K~,,d (1 3 )K s/= KRo

~ K O o d _ K s eKRoa1 .1 2o r x / ~ " ( 1 4 )T h e s i g n i f ic a n c e o f n o r m a l i s i n g t o t h e v a l u e o f a n e d g e c r a c k i n a n i n fi n it e s tr i p is n o t o n l yc o n v e n i e n t b u t is e s s e n ti a l t o m a i n t a i n t h e u n i v e r s a l it y o f t h e e x p r e s s i o n u n d e r a n y l o a d s y s t e m .T h e r e s u l t i n g w e i g h t f u n c t i o n w i ll o n l y b e v a l i d f o r o n e s t r e ss s y s t e m i f a c o m p o n e n t o f t h e r e f e r e n c es t r e ss i n t e n s i t y f a c t o r i s s p e c i f ic t o t h a t s t r e s s s y s t e m . T h e s t r e s s i n t e n s i t y f a c t o r f o r a n e d g ec r a c k i n a n i n f i n i t e s t r i p i s t h e o n l y s i t u a t i o n w h e r e K t i s i d e n t i c a l f o r a n y o p e n i n g m o d e s t r e s ss y s t e m .

0 .5

0

1 .5

!0 !o o i , 0 . 2 o1 3 o i , . 5

ac0.64

0.78

a /v

M20 UNC (W eight unctionsolution)o Erjian Si [26]

- 0.78, a < ~- o . 6 4 ,

Fig. 9. S emi-elliptical surface crack in a round bar un der tension.


9/23

S IF s fo r t h r eaded conne c t i ons 553K n o w in g t h e s t re s s i n t en s i t y fac to r fo r a semi -e l l i p t i ca l c rack i n a ro d , t h e s t re s s i n t en s i t y fac to rf o r a c r a c k i n a b o l t c a n b e e v a l u a t e d . F i g u r e s 3 a n d 4 s h o w t h e n o t a t i o n f o r a n e x t e r n a l a n d i n te r n a lsu r face c rack i n a t u b e , re sp ec t i v e ly . Pu b l i sh ed so lu t i o n s fo r c rack s i n ro d s , ex t e rn a l an d i n t e rn a lt u b e s u n d e r t e n s i o n a n d b e n d i n g w e r e c u r v e fi tt e d a n d u s e d a s r e f er e n c e S l F s f o r w e i g h t f u n c ti o n s .T h e e x t e r n a l s t r e ss i n t e n si t y f a c t o r s f o r c i r c u m f e r e n t i al s u r f a c e c r a c k s p r e s e n t e d w e r e e v a l u a t e d b yR a j u a n d N e w m a n [1 6 ] u s i n g s i n g u la r i ty f in i te e l e m e n t s a l o n g t h e c r a c k f r o n t a n d l in e a r -s t ra i n

e l e m e n t s e ls e w h e r e . T h e n o d a l - f o r c e m e t h o d w a s e m p l o y e d i n t h e d e r i v a t io n o f t h e s t r es s i n te n s i tyf a c t o r s . T h e p a r t - t h r o u g h i n t e r n a l c i r c u m f e r e n t i a l c r a c k s o l u t i o n s e v a l u a t e d b y D e l a l e a n dE r d o g a n [ 1 7] u s e d a l i n e - sp r in g m o d e l . T h e r e s u lt s c o m p a r e f a v o u r a b l y w i t h f i n it e e le m e n t r e s u lt sf ro m v a r io u s s t u d i e s . In a l l c a se s o n ly t h e s t re s s i n t en s i ty fac to r a t t h e d eep e s t p o in t o f t h e c rackw as co n s id e red . W i th in t h e v a l i d i t y limi t s o f t h e co n s t i t u t i v e re fe ren ce SIFs , c rack s i n an y t h re ad edc o n n e c t i o n u n d e r a n y s y m m e t r i c a l l o a d s y s t e m c a n b e e v a l u a t e d f o r K ~ .S T R E S S D I S T R I B U T I O N

Th e p reced in g sec t i o n s d e t a i l ed t h e fo rmu la t i o n o f s t re s s i n t en s i t y fac to rs fo r t h read edc o n n e c t i o n s . R e m e m b e r i n g t h a t t h e s e s o l u t i o n s a r e o n l y r e f e r e n c e c a s e s i n t e n d e d f o r w e i g h tfu n c t i o n ca l cu l a t i o n s , t h e an a ly s i s i s n o t y e t co m p le t e . In o rd e r t h a t " C / ' i s fo u n d fo r eq . (1 2) , iti s n e c e s s a r y t o k n o w " a ( x ) " , i . e . t h e c o r r e s p o n d i n g s t r e s s d i s t r i b u t i o n a l o n g t h e p o t e n t i a l c r a c kp l an e t o t h e re fe ren ce s t re s s i n t en s i t y fac to r d i s t r i b u t i o n . Th ese w ere ca l cu l a t ed f ro m H aseb e ' so r ig in a l an a ly se s . H av in g t h e w e ig h t fu n c t i o n , t h e s t re s s d i s t r i b u t i o n f ro m a t h read ro o t i s req u i red

:I o i , 0 1 2 0 : 3 o i ,0 .5

0 .640 .78

. 8

M 2 0 U N C ( W e i g h t f u n c ti o n s o l u ti o n )o Er j ian S i [26]

~ 0 . 7 8 , a < ~ D

Fig. 10. Semi-el l ipt ical surface crack in a round bar under bending.


10/23

554 F.P . BRENNA N and W. D. DOVERf o r t he c a l c u la t io n o f t he s t re s s in t ens i t y f a c t o r . K uj a ws k i ' s [1 8 ] s t re s s d i s t r ib u t io n d ue t o a no t c hwas used in th is ana lys is . This i s def ined as

t r ( x ) = - - - ~ - 1 +--p-- q- 1 h'- -~ f (15)

t a n ( 2 ~ r ) ( x- - 0 . 2 ) f o r x >~. 2 - ( 1 6 )f = l - ~ 2 .8 p p

f = 1.0 for x < 0 . 2 . ( 1 7 )PT h e n o m i n a l s t r e s s i s t a k e n a s t h e s t r e s s a t a c r o s s - s e c t i o n e q u a l t o t h e s e c t i o n f r o m t h e m i n o rd i a m e t e r o f t h e t h r e a d , i .e . th e t o o t h h e i g h t i s n o t i n c l u d e d . I n b e n d i n g , t h e n o m i n a l s t r e s s i s t h em a x i m u m b e n d i n g s t r e s s a t t h e m i n o r d i a m e t e r s o t h e s t r e s s d i s t r i b u t i o n m u s t b e m o d i f i e d b ym u l t i p l y i n g b y

I 1 2 x- - d ] ( 1 8 )

4

3 .5

3

2 .5!21 .5

1

0 .5

0 0

0oo

I I I I0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5

a/D

a 2a

M20 UNC (Weight unctionsolution)

o Benthem & Koiter [28]Fig. 11. Circumferentialcrack in a roun d bar under tension.


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55 5

2 . 5

(19)

D I S C U S S I O NV e r y f e w s t r e ss i n t e n si ty f a c t o r s o l u t i o n s f o r t h r e a d e d c o n n e c t i o n s h a v e b e e n p u b l i sh e d . O f

t h e s e m o s t a r e d i f fi c ul t t o u s e f o r c o m p a r a t i v e p u r p o s e s b e c a u s e o f o b s c u r e f e a t u r e s o f a n a ly s i s o rp r e s e n t a t io n . O t h e r s u s e u n c o n v e n t i o n a l g e o m e t r i c a l n o t a t i o n , s u c h a s t h e p r o j e c t i o n o f c o o r d i -n a t e s o n t h e c y l in d e r s u r f a c e o n t o a p l a n e , m a k i n g t h e r e p r o j e c t io n a n d s u b s e q u e n t i n te r p r e t a t io nd if fi cu lt . T h e m a in u n ce r t a in ty w i th u s in g p u b l i sh ed re su l t s o f t en l ie s w i th t h e d e f in i t i o n o f c rackasp ec t ra t i o . D esp i t e t h e se l imi t a t i o n s , t h e fo l l o w in g d i scu ss io n co mp ares t h e n ew g en e r i c w e ig h tf u n c t i o n s o l u t i o n w i t h p u b l i s h e d S I F s o l u t i o n s f o r r o d s , t u b e s a n d t h r e a d e d c o n n e c t i o n s .F i g u r e 5 s h o w s t h e n e w s o l u t i o n f o r a t h r e a d e d M 2 0 U N C b o l t p l o t t e d a g a i n s t v a r i o u s e d g ec r a c k s o l u t i o n s i n r o u n d b a r s u n d e r t e n s i o n . F r o m g e o m e t r i c a l c o n s i d e r a t i o n s t h e e x p r e s s i o n f o r

o +

o , I ,

2

1 .5

1

0 . 5

S I F s f o r t h r e a d e d c o n n e c t i o n s

f o r r o d s a n d e x t e r n a l c r a c k s i n p i p e s , a n d ,

fo r i n t e rn a l c rack s i n p ip es .T h e S C F s h o u l d b e d e t e r m i n e d s e p a r a t e l y u s i n g F E a n a l y s i s .

0 !o 0 1 , o . 2 0 1 ,, / a

0 . 5

c T . ( 2 0 ]a

I S O M 8 x 1 . 0 ( W e i g h t fu n c t i o n s o l u ti o n )

U p p e r B o u n d C i r c u l a r C r a c k [ 2 3 ]

L o w e r B o u n d C i r c u l a r C r a c k [ 23 ]

* T o f i b i o ~ = 1 . 0 [ 2 9 ]ao To r ib io i = 0 .2 [29]

+ James & M ills [5]F i g . 1 2 . C r a c k i n a n I S O M 8 x 1 . 0 b o l t u n d e r t e n s i o n ,


12/23

5 56 F . P . B R E N N A N a n d W . D . D O V E Rc r a c k a s p e c t r a t io w a s c a l c u l a t e d i n t e rm s o f t h e n o t a t i o n g i ve n e a r li e r f o r a n e x t e r n a l c r a c k ( s e eA p p e n d ix ) . D a o u d et al.'s [1 9 ] 2D f i n it e e l emen t an d co m p l i an ce re su l t s ag ree ex ce ll en t ly w i th eacho th e r . B l ac k b u rn ' s [20 ] 3 D f i n it e e leme n t re su l t s o n ly ex t en d fo r v e ry sma l l v a lu es o f a iD b u t a r ein k eep in g w i th t h e o th e rs . F in a l l y Bu sh [21 ] ca l cu l a t ed d im en s io n l e ss s t re s s in t en s i t y fac to rs fo re d g e c r a c k s i n s o li d b a r s f r o m e x p e r i m e n t a l c o m p l i a n c e ( i n ve r s e sl o p e o f lo a d - d i s p l a c e m e n t c u r v e ) .Th e l a t t e r a re i n ex ce l l en t ag ree m en t w i th t h e n ew so lu t i o n a f t e r t h e e f fec t o f th e t h rea d h as ab a t e d .Th e n ew so lu t i o n i s s li g h t ly o n t h e co n se rv a t i v e s i d e o f a l l t h e re su l t s u n t i l t h e c rack d e p tha p p r o a c h e s h a l f t h e d i a m e t e r o f t h e b a r. A g a i n i t s h o u l d b e n o t e d t h a t t h e n e w s o l u t i o n is m o d e l li n ga semi -e l li p t ica l c rack h av in g t h e sam e c rack d ep th an d h a l f l en g th o f an eq u iv a l en t ed g e c rack .T h i s m o d e l b e c o m e s in c r e a si n g ly i m p r a c t ic a l a s t h e c r a c k a p p r o a c h e s h a l f t h ic k n e s s o f t h e b a r .T h e s a m e c r a c k a s p e c t r a t io m o d e l w a s u s e d i n F i g . 6 w h e r e t h e b e n d i n g s o l u t io n i s c o m p a r e dt o t h e c o m p l i a n c e r e s u lt s o f B u s h [2 2] . A g a i n t h e i n f lu e n c e o f t h e t h r e a d i s a b n o r m a l l y h i g h d u eto t h e c rack a sp ec t ra t i o . Th e n ew so lu t i o n i s o n ce ag a in co n se rv a t i v e u n t i l t h e h a l f t h i ck n ess p o in ti s a p p r o a c h e d .F i g u r e 7 s h o w s t h e v a r i a t io n o f t h e s t re s s in t e n s it y c a l i b r a ti o n f a c t o r ( Y ) w i t h n o n - d i m e n s i o n a lc r a c k d e p t h ( a / D ) fo r a c i rcu l a r -a rc su r face c rack g ro w in g i n th e c i rcu mfe ren ce o f a so l id ro d i nt e n si o n . T h e c r a c k a s p e c t r a ti o f o r t h e n e w s o l u ti o n is d e f i ne d b y F o r m a n a n d S h i v a k u m a r ' s u p p e rb o u n d re l a t i o n sh ip fo r a c i rcu l a r c rack [23 ] . Th e i r s t re s s i n t en s i t y fac to r so lu t i o n i s h o w ev e r , n o tb a s e d o n e x p e r i m e n t a l r e s u l t s b u t i s b a s i c a l l y T a d a ' s r e c t a n g u l a r b a r s o l u t i o n s m u l t i p l i e d b y a na r b i t r a r y c o r r e c t i o n f a c t o r t h o u g h t a p p r o p r i a t e . T h e s e a r e c le a r ly o v e r c o n s e r v a ti v e . M o r e

>-

1110 " 5

0 i i i i0 0 . 1 0 . 2 0 .3 0 . 4 0 . 5. 8

a_=_..L__Do

I S O M 8 x 1 . 0 ( W e i g h t f u n c t i o n s o l u t io n )

U p p e r B o u n d C i r c u l a r C r a c k [ 2 3 ]

o J a m e s & M i l l s [ 5 ]F i g . 1 3 . C r a c k i n a n I S O M8 1 . 0 b o l t u n d e r b e n d i n g .

L o w e r B o u n d C i r c u l a r C r a c k [ 2 3 ]


13/23

SIFs for threaded connections 557s u b s t a n c e i s g i v en t o t h e i r c r a c k s h a p e r e l a ti o n s h i p s . T h e f in i te e le m e n t r e s u l t s o f D a o u d a n dC a r t w r i g h t [2 4 ] a n d t h e e x p e r i m e n t a l f i n d i n g s o f W i l h e m et al. [ 25 ] c o m p a r e f a v o u r a b l y w i t h t h ep r e d i c t e d b e h a v i o u r o f t h e M 2 0 b o l t a w a y f r o m t h e i n f lu e n c e o f th e t h r e a d . A s i m i l a r c o m p a r i s o nf o r b e n d i n g ( F i g . 8) s h o w s t h a t D a o u d a n d C a r t w r i g h t ' s [ 24 ] r e s u lt s c o r r e la t e e x c e l le n t ly w i t h t h en e w s o l u t i o n , F o r m a n a n d S h i v a k u m a r o n c e m o r e o v e r e s t i m a t i n g t h e e v e n t .S i [ 2 6] r e v ie w e d m a n y s u r f a ce c r a c k s o l u t i o n s i n r o u n d b a r s . A s o l u t i o n w a s p u b l i s h e da v e r a g i n g t h e r e s u l t s f o u n d d u r i n g t h e r e v ie w , b u t w a s p r e s e n t e d a s a f u n c t i o n o f v a g u e l y d e f i n e dc r a c k a s p e c t r a t i o . T h e s e a r e s h o w n i n F i g . 9 , a n d a r e f o r a s e m i - e l l i p t i c a l c r a c k w i t h t h e s t a t e ds h a p e c h a r a c t e ri s t ic s . I n a d d i t i o n t h e n e w s o l u t i o n is g i v en a s t h e s o l i d l in e s f o r t w o c r a c k a s p e c tr a t io s . T h e t w o c o n s t a n t a s p e c t r a t i o c u r v e s s h o w g o o d a g r e e m e n t w i t h t h e r e v ie w r e su l ts . P e r h a p se v e n m o r e r e v e a li n g t o h o w S i 's s h a p e e x p r e s s i o n e v o lv e s c a n b e s e e n w h e n c o m p a r i n g h i s re s u lt sw i t h t h e n e w s o l u t i o n f o r b e n d i n g ( F i g . 1 0 ) . M a y b e a/c = 0 . 6 4 a t s o m e p o i n t a p p r o a c h i n g D/2,b u t t h e i n d i c a t i o n i s t h a t a/c = 0 . 5 a t t h e p o i n t D/2. I n t h e c o m p a n i o n p a p e r [ 2 7 ] , g r a p h i c a lr e p r e s e n t a t i o n i s m a d e o f w h a t t h e s t re s s i n te n s i t y fa c t o r s f o r t h r e a d e d b a r s s h o u l d l o o k l i k e. T h i s,d e s p i t e n o r i g o r o u s e x p e r i m e n t a l o r a n a l y t i c a l a n a l y s i s o f a n y p a r t i c u l a r t h r e a d t y p e s h o w s ab e h a v i o u r s i m i l a r t o t h e r e s u l t s o f t h e n e w s o l u t i o n , i .e . th e r a p i d l y d i p p i n g s t r e s s i n t e n s i t y f a c t o rd u e t o t h e t h r e a d a n d m o r e g r a d u a l e l e v a t i n g a s t h e c r a c k g r o w s t h r o u g h t h e b a r .

T h e b e h a v i o u r o f th e n e w s o l u t i o n h a s a l s o b e e n e x a m i n e d b e y o n d i t s v a l i d i ty l im i t s . F i g u r e 11s h o w s t h e p r e d i c t e d Y f a c t o r s w h e r e t h e c r a c k h a l f l e n g th i s e q u a l t o h a l f t h e c i r c u m f e r e n c e o f t h eb a r . C o m p a r e d w i t h t h e a n a l y t i c a l c ir c u m f e r e n t i a l s o l u ti o n o f B e n t h e m a n d K o i t e r [2 8], ito v e r e s t i m a t e s t h e s t re s s i n t e n s i ty f a c t o r b y a t m o s t 2 5 % . I t i s n o t s u g g e s t ed t h a t t h e s o l u t i o n b e

0 0 i ' i '2 0 .4 0 6 0 .8, / t

ri-= 1 .0tO

ISO M8 x 1.0 (Weightfunction solution)

Wilkowski& Eiber (Edge) [30]Harris (Circumferential) 1]Bush (Edge) [21]

Fig. 14. E xternal tub e crack unde r tension.


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5 58 F . P . B R E N N A N a n d W . D . D O V E Rused in such circumstances, but it is encouraging to know that the weight function solution isconservative if used outside its validity limits.

A Threaded ISO M8 x 1.0 bolt was analysed by Toribio and Sanchez-Gfilvez [2] and Toribioe t a l . [29]. Unfortunately the crack length was defined in terms of a dimension projected onto thehorizontal. Instead of trying to modify this to the more conventional notation, it was decided tocompare them to the new solution ignoring the slight difference introduced by the differentnotation. Figure 12 shows the tension solutions for the ISO M8 x 1.0 (1.0 mm pitch) using Formanand Shivakumar's upper and lower bound crack shape relations. Also plotted are the Toribio e ta l . solutions for two extreme crack aspect ratios. James and Mills' [5] hypothetical "th readed" baris also shown. Again, the upper and lower bounds give good agreement with the other data.Regrettably, Toribio's finite element results do not cover the all important thread influence region.Only one point o f James and Mills' SIF distribution is shown. Under bending, both limits are againplotted in comparison to the James and Mills solution for a circular crack in a round bar, Fig.13. The new solution appears reasonable for both tension and bending.Published stress intensity factor solutions for solid bars are rare, even more scarce are stressintensity factors for internal and external circumference cracks in tubes. Figure 14 plots the newsolution for a small crack aspect ratio against three solutions, two for edge cracks, the other fora circumferential crack in tension, this time for cracks in tubes.Wilkowski and Eiber's [30] experimentally derived stress intensity factor solutions for edgecracks have been used in fatigue analysis of threaded connections by Chen [31]. Here they are

3

2 .5

2

1 .5

i fo @

0 .1 0 .2 0 .3 0 .4 0 .5a / t

=o.1~ = 9 . 5

IS O M 8 x 1 .0 (W ei gh t func t i on so l u ti on)

Rajah (Ci rcumferent ia l ) [32]

F i g . 15. I n t e rna l c ir cumfe ren t i a l t ube c r ack und e r t ens i on .


15/23

SIFs for threaded connections 559plotted along with the Harris solution for circumferential cracks in hollow tubes [1] and Bush's [21]edge crack compliance results under tension loading. The Wilkowski and Eiber edge crack resultsagree better with the Harris circumferential crack than with Bush's edge crack. The exponentialbehaviour of ref. [30] is only explicable if r / t is extremely small, whereas the results given by Bushseem more realistic for larger values of r / t . The comparison with the new solution is inconclusivebecause of the low aspect ratio approximation, but the behaviour looks to be reasonable.

Figure 15 shows the new solution with a very low crack aspect ratio for an internal pipe crackcompared with the theoretical internal circumferential pipe solution o f Rajab [32]. Further resultsare shown in Fig. 16 for semi-elliptical internal tube cracks under tension loading. Poette andAlbaladejo's finite element results [33] compare favourably with the new solution. The results ofChen [34] for a crack emanating from the first engaged thread root of a box (female threadedsection) in a large diameter threaded connection also agree for values of a / t < 0.6. Chen noted thatthe solution was valid for a part-through-thickness crack but did not state the validity range. Thefinite element results would seem from this comparison to be limited to values of a / t < 0.6. Againno influence of the thread is represented.

Figures 17-22 show the new solutions for a threaded (unified thread form) rod, and forexternally and internally threaded tubes under tension and bending and include various crackaspect ratios. Figures 23-26 show the effect of the radius to thickness ratio on the externally andinternally threaded tube cracks under tension and bending, respectively. These show little effect ofthe radius to thickness ratio at the thread root but favourable reduction in the stress intensity factorfor larger cracks with increased values of r / t . This is less marked in bending.

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16/23

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17/23

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18/23

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Documents

Stress InStress Intensity Factors for Threaded Connectionstensity