Thermal and mechanical analysis of weld

7/26/2019 Thermal and mechanical analysis of weld

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N U C L E A R E N G I N E E R I N G A N D D E S IG N 3 2 ( 1 9 7 5 ) 1 1 0 - 1 2 0 . N O R T H -H O L L A N D P U B L IS H I N G C OM P A N Y

T H E R M A L A N D M E C H A N I C A L A N A L Y S I S O F W E L D E D S T R U C T U R E S

R .E . N I C K E L L t

Division of Engineering Brown University Providence Rhode Island 02912 USA

a n d

H . D . H I B B I T T

MarcAnalysis Research Corporation ProvMence Rhode Island 02906 USA

R e c e i v e d 1 0 S e p t e m b e r 1 9 7 3

T h e p r e d i c t i o n o f t h e r e s i d u al s t a t e o f s t re s s a n d d e f o r m a t i o n i n a w e l d e d s t r u c t u r e i s o n e o f t h e m o s t i n t e r e s t i n g ,

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

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

m o d e l s p r o v i d e s t h e c o m p l e x i t y . E v e n i f it is a s s u m e d t h a t r e a s o n a b l e p r e d i c t i o n o f t h e t r a n s i e n t t e m p e r a t u r e f i e ld i n

t h e s t r u c t u r e i s p o s s i b le , d e t e r m i n i n g t h e r e s i d u a l m e c h a n i c a l s t a t e i s b o t h a d i f f i c u l t a n d a n e x p e n s i v e t a s k . T h e a n a -

l y si s i n e v i t a b l y in v o l v e s t e m p e r a t u r e - d e p e n d e n t m e c h a n i c a l p r o p e r t i e s a n d , i n a d d i t i o n , t h e s e v e re t h e r m a l g r a d i e n t s

a n d h i g h t e m p e r a t u r e s g e n e r i c t o t h e w e l d i n g p r o c e s s i n d u c e i r r e c o v e r a b l e i n e la s t ic c r e e p a n d p l a s t i c i ty i n t h e s t r u c t u re .

I n s p i t e o f t h i s, t h e s t r e s s a n a ly s i s i s n o w c o n s i d e r e d t o b e a s t r a i g h t f o r w a r d a p p l i c a t i o n f o r g e n e r a l p u r p o s e , n o n -

l inea r f in i t e e lement s t ruc tu ra l p rogr : ims . A few spec ia l f ea tu res o f such ana lyses , however , w i l l be d i scussed : (1 ) the

e g l t im a c y o f t i m e - d e p e n d e n t p l a s t i c i t y t h e o r i e s f o r t r e a t i n g t h e r e s i d u a l s tr e s s p r o b l e m ; ( 2 ) c r i t e ri a f o r c h o o s i n g

p l a n e s t re s s , p l a n e s t r a i n , g e n e ra l i z e d p l a n e s t r a in , o r f u l ly t h r e e - d i m e n s i o n a l m o d e l s ; ( 3 ) m e t h o d s f o r c o p i n g w i t h

p o s s ib l e f l o a t i n g so l id r e g i o n s d u r i n g c o o li n g ; a n d ( 4 ) t h e u s e o f l i n e a r c o n s t r a i n t s t o t r e a t w e l d m e t a l d e p o s i t i o n a n d

i n t e r m i t t e n t c o n t a c t .

S i n c e th e m o s t i m p o r t a n t p a r a m e t e r s i n t h e w e l d i n g p r o c e s s t h a t p e r t a i n t o t h e s t r e s s a n a l y si s ar e t h e c o o l i n g r a t e

a n d t h e w e l d i n g t o r c h e f f i c i e n c y , t h e h e a t t r a n s f e r p r o b l e m s e e m s t o r e q u i r e a c r i ti c a l l o o k . T h e d o m i n a n t f e a t u r e s

o f t h i s p r o b l e m a r e : ( 1 ) s o u r c e ( t o r c h ) c h a r a c t e r i z a t i o n ; ( 2 ) r a d i a t i o n f r o m s u r f a c e s t h a t a r e h e a t e d t o h i g h t e m p e r a t u r e s ;

( 3 ) l a t e n t h e a t e f f e c t s ; a n d ( 4 ) s u b s i d i ar y c o n s i d e r a t i o n s , s u c h a s e n f o r c e d c o n v e c t i o n h e a t t r a n s f e r m o d e s t h a t a r e

d e s i g n e d t o c o n t r o l t h e c o o l i n g r a t e . M o t i o n o f t h e w e l d i n g t o r c h , e v e n a t l o w s p e e d s , i s n o t u s u a l ly a c r i t i c a l f a c t o r

i n d e t e r m i n i n g t h e r e s i d u a l m e c h a n i c a l s t a t e . A g a i n, f i n i t e e l e m e n t a n a l y s i s is a p p l ic a b l e , p r o v i d e d t h a t t h e s o l u t i o n

a c c u r a c y c a n b e a d e q u a t e l y e s t i m a t e d . S e v e ra l a l t e r n a t i v e , tw o - s t e p , i m p l i c i t t i m e i n t e g r a t i o n s c h e m e s w i l l b e c o m -

p a r e d , e s p e c i al l y w i t h r e g a r d t o a c c u r a c y a n d n u m e r i c a l s t a b i l i ty f o r w e l d i n g - t y p e p r o b l e m s . T h e e f f i c a c y o f ' f l u x

c o r r e c t i o n ' w i ll a ls o b e d i s c u s s ed a n d t h e a p p l i c a t i o n o f t h e s e i d e a s t o t y p i c a l i n d u s t r i a l w e l d in g p r o b l e m s w i l l t h e n

b e o u t l i n e d .

1 I n t r o d u c t i o n

T h e f a b r i c a t i o n a n d i n t e g r i t y o f la r g e -s c a le s t r u c t u r e s ,

s u c h a s b u i l d i n g s , b r i d g e s , s h i p s , p r e s s u r e v e s s e l s, a n d

o t h e r c o m p l e x c o n f i g u r a t i o n s , d e p e n d s u p o n t h e us e o f

r e li a bl e w e l di n g t e c h n i q u e s. A l t h o u g h m a n y o f th e s e

t e c h n i q u e s h a v e b e e n i n u se f o r d e c a d e s , t h e q u a n t i t a -

* I n v i t e d p a p e r F 6 / 1 " p r e s e n t e d a t t h e S e c o n d I n t e r n a t i o n a l

C o n f e r e n c e o n S t r u c t u r a l M e c h a n i c s i n R e a c t o r T e c h n o l o g y ,

B e r l in , G e r m a n y , 1 0 - 1 4 S e p t e m b e r , 1 9 7 3 .

t P resen t address : A ppl ied M echan ics I D iv i s ion , Sand ia Lab-

o r a t o r i e s , A l b u q u e r q u e , N e w M e x i c o 8 7 1 1 5 , U S A .

t i ve a s p e c ts o f t h e m e c h a n i c s o f w e l d i n g h a v e o n l y

r e c e n t l y r e c e i v e d a t t e n t i o n f r o m a n a l y s t s . P e r h a p s t h e

m o s t s o p h i s t i c a t e d w o r k i s t h a t o f B j o r h o v d e e t a l. [ 1 ]

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

( e. g . a b u i l d i n g c o l u m n ) f r o m t h e r e s i d u a l s t r es s s t at e s

i n e d g e w e l d e d a n d c e n t e r w e l d e d t h i c k p l a t e s . A n

e a r li e r a n a l y s is [ 2 ] u s e d a r u d i m e n t a r y f i n i t e e l e m e n t

p r o g r a m f o r c o m p a r i s o n w i t h e x p e r i m e n t a l r e s u lt s f or

t h i c k s e c t i o n s .

T w o f u n d a m e n t a l s h o r t c o m i n g s a re i n h e r e n t i n t h i s

w o r k : f i rs t , n o a t t e m p t w a s m a d e t o q u a n t i f y t h e t h e r m a l

p a r t o f t h e p r o b l e m ( m e l t i n g , s o l i d i f ic a t i o n , a n d c o o l i n g


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R.E. Nickel l H.D. Hibbi t t Analys i s o f welded s t ructures 111

r a te ) ; and second, the r e s t r a in t exe r ted by ad jacent

me m b e r s d u r in g t h e w e ld in g p ro c e s s is e x t r e m e ly imp o r -

tan t in de te rmining the r e s idua l s t r e sses and de forma-

t ions in a bu i l t -up s t ruc ture . A more r ea l i s t ic approach

f o r s u c h w e ld in g p r o b l e m s is t h a t o f H ib b i t t a n d

Ma r c a l [ 3 ] , w h o a n a ly z e b o th t h e t h e r ma l a n d t h e

me c h a n i c a l p a r t s o f th e w e ld in g p r o b l e m w i th f i n it e

e l e me n t me th o d s . R a th e r t h a n a d o p t t h e l i n e - s o u r c e

s o lu t i o n t o t h e h e a t t r a n s f e r p r o b l e m [ 4 ] , a n o n l i n e a r

t r ans ien t f in i te e lement so lu t ion , inc luding la ten t hea t

e f f e ct s , ma t e r i a l p r o p e r ty t e m p e r a tu r e d e p e n d e n c e ,

a n d t h e r ma l r a d i a t io n b o u n d a r y c o n d i t i o n s , i s o b t a in e d .

T h i s t ra n s i e n t t e m p e r a tu r e d i s t r ib u t i o n i s t h e n u s e d a s

a forc ing func t ion for a fin i te e lem ent s t r e ss ana lys is ,

u s in g in c r e me n ta l p l a s t ic i t y t h e o r y w i th t e m p e r a tu r e -

d e p e n d e n t me c h a n i c a l p r o p e rt i e s .

The f ind ings in r e f . [3 ] , a l though in conf l ic t wi th

p u b l i s h e d e x p e r ime n ta l re s u l ts [ 5 ] , c o n f i r me d t h e u s u a l

not ions of r e s idua l s t r e ss s ta te s in we lded pa r t s [6] .

F i r s t, the r e s idua l s t re ss s ta te for a th in we lded sec -

t ion (he re de f ined to be a fu l ly fus ion-pene t ra ted sec -

t ion) exhib i ted p lane s t r e ss cha rac te r i s t ic s , wi th the

s t r e ss pa ra l le l to the we ld ing pa th domina t ing . In the

fus ion zone th is s t r e ss was tens i le , wi th magni tudes

a p p r o a c h in g t h e r o o m t e mp e r a tu r e y i e ld s t r e n g th .

Subs tan t ia l bending was obse rved in the sec t ion .

Second, s imi la r s t r e ss pa t te rns were pred ic ted for a

t h i c k w e ld e d s e ct i o n (p a r t ia l f u s io n - p e n e t r a t i o n ) ,

except tha t the p lane s t r e ss behavior was no longer

va l id . High s t r e sses cor responding to hydros ta t ic tens ion

occur red a t the ro o t of the we ld . These au tho rs a l so

mo d e l e d p o s t - w e ld h e a t t r e a tme n t b y n o n l i n e a r c r e e p

a n d o u t l i n e d a me th o d o lo g y f o r a n a ly ti c a l l y t re a t i n g

weld dressing.

F r o m th e p o in t o f v i ew o f t h e d e s ig ne r t r y in g t o

cope wi th void nuc lea t ion and f rac ture in i t ia t ion in

we ldments , the s tanda rd guide l ines a re thus seen to be

v a li d , b u t o f t e n imp o s s ib l e t o im p le me n t . F o r i n s t a n c e ,

the we ld ing sequence can be p lanned , in most cases ,

so tha t no addi t iona l tens i le s t r e ss i s imposed on the

res idua l s t r e ss s ta te ; however , for inde te rmina te s t ruc -

t u r e s , th e s i t u a t i o n i s s o c o m p le x t h a t a n e x p e n s iv e

ana lys is may be necessa ry . P rehea t ing of th ick sec t ions ,

a s r e c o m me n d e d b y t h e A W S S t ru c tu r a l W e ld in g C o d e ,

a n d p o s t - w e ld h e a t t r e a t i n g a re e f f e c t iv e me a n s o f

reduc ing th e r e s idual s t r e sses (un less s ign i f ican t load

i s t r a n s f e r r e d t o a d j a c e n t s t r u c tu r e ) , b u t t h e y a r e t ime

c o n s u m in g a n d e x p e n s iv e . S t re s s c o n c e n t r a t i o n d e s ig n

fea tures , such a s no tches , should be avoided nea r we lds

a s fa r as p o s sib l e, b u t m a n y c o n t e m p o r a r y d e s ig n s d o

not a l low th is luxury . Conse rva t i sm, u t i l i ty , and cos t

would seem to be pul l ing the des igne r in d i f f e ren t

d i rec t ions ; the re fore , sophis t ica ted ana lyses to de te r -

min e t h e s t ru c tu r a l i n t e g r it y o f t h e w e ld m e n t m a y b e

the only r easonable a l te rna t ive .

As show n in r e f. [3 ] , f ind ing the r e s idua l mechanica l

s ta te , a l though expens ive , can be cons ide red a s a c i r -

c u m s p e c t a p p l i c a t i o n f o r g e n e r al p u r p o s e , n o n l i n e a r

f i n it e e l e me n t s t r u c tu r a l p r o g r a ms . A n i n c r e me n ta l ,

t ime - in d e p e n d e n t p l a s t i c i t y t h e o r y w i th t e mp e r a tu r e -

d e p e n d e n t me c h a n i c a l p r o p e r t i e s se e ms t o b e a n a d e -

qua te cons t i tu t ive r epresenta t ion dur ing the we ld ing

cyc le , s ince the dominant s t r e sses and d is tor t ions a re

in d u c e d a t t e mp e r a tu r e s a n d o v e r a t ime s c a le t h a t

inh ib i t s s ign i f ican t c reep . On the o the r hand , c reep

d e f o r m a t io n a n d s t re s s re l a x a t i o n c l e ar l y d o min a t e

p las t ic i ty e f fec ts dur ing pos t -we ld hea t t r ea tment . De-

f o r m a t io n a r is i ng d u r in g t h e m o l t e n s t a t e i s c o m mo n ly

ignored .

T e mp e r a tu r e i n c r e me n t s f r o m th e w e ld in g t h e r ma l

ana lys is must be se lec ted wi th some caut ion in tha t

la rge inc rementa l changes in the mechanica l s ta te may

resu l t , wi th cor respondingly inaccura te in tegra t ion of

t h e e l a s t i c - p l a s t i c r a t e e q u a t i o n s . C o m p u ta t i o n a l

a c c u r a c y is imp r o v e d ( a n d , t h e r e f o r e , s o m e w h a t l a r g er

t e mp e r a tu r e i n c r e me n t s ma y b e c h o s e n ) b y i n c o r p o r -

a t in g me a n s t i ff n e s s a n d me a n n o r m a l c o n c e p t s [ 7 ] ,

a n d b y a d d in g a n o u t - o f -e q u i l i b r iu m r e s id u a l l o a d

cor rec t ion [3] to successive inc rem ents .

The u l t ima te va l id i ty of the mechanica l ana lys is

wi l l thus depend, in la rge measure , upon the va l id i ty

o f t h e s o lu t io n t o t h e t h e r ma l p r o b l e m; t h e c o o l i n g

ra te and the we ld ing a rc e f f ic iency wi l l de te rmine the

e x t e n t o f t h e f u s io n z o n e a n d t h e ma g n i tu d e o f t h e

the rma l grad ien ts in the ad jacent hea t -a f fec ted zone ,

for example . Cons ide r ing the c r i t ica l na ture of the

th e r ma l a n a ly s is , it i s n o t c l e a r t h a t t h e c o m p u ta t i o n a l

t e c h n iq u e s t h a t a r e i n g e n er a l u se t o d a y a d e q u a t e ly

t rea t th is c la ss of nonl inea r , t r ans ien t h ea t t r ansfe r

p r o b l e m .

2 N o n l i n e a r h e a t t r a n s fe r

T h e f i n i te e l e me n t m e th o d ( F E M ) h a s b e e n u s e d , f o r

ma n y y e a r s , to f i n d a p p r o x im a te s o lu t i o n s t o l i n e a r

d i f f u sio n p r o b l e ms . F o r mu la t i o n s h a v e b e e n b a s e d o n


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112 R.E. Nickel l , H.D. Hibb i t t , Ana lys is o f welded s t ructure s

seve ral a l te rna t ive g loba l e r ror d is t r ibu t ion pr inc ip le s:

( 1 ) th e c o n c e p t o f ' re s t r i c t e d v a r ia t i o n ' [ 8 ] , d e v e lo p e d

wi th FEM b y Beck e t and Pa r r [9] ; (2 ) conjug a te f ie ld

concep ts [ 10] tha t have neve r ach ieved a grea t dea l o f

p o p u l a r i t y ; ( 3 ) t h e t r a n s f o r me d v a r i a t i o n a l p r i n cip l e

o f G u r t i n [ 11 ] , d e v e lo p e d w i th F E M b y W i l s o n a n d

Nicke l l [12] ;and (4) the Ga le rk in me thod, a subc lass

o f t h e m e th o d o f w e ig h t e d r e sid u al s (MW R ) , d e v e lo p e d

w i th F E M b y Z i e n k i e w ic z a n d P a r e k h [ 1 3 ] . A n e x-

c e l le n t s u m ma r y o f t h e v a r io u s a p p r o a c h e s i s c o n -

ta ined in r e f . [14] , where the i r equiva lence for l inea r

p r o b l e ms i s p o in t e d o u t .

A s c o m p u ta t i o n a l u s e o f s o f tw a r e b a s e d o n t h e se

me th o d s h a s i n c re a s e d , t h e n e e d t o e x t e n d p r o g r a m

capabi l i ty in to the nonl inea r r eg ime has become evi -

dent . The nonl inea r i ty mani fes ts i t se l f in th ree ways :

( a ) t h e r ma l p r o p e r ty v a r i a t io n w i th t e m p e r a tu r e ; ( b )

nonl inea r bou nd ary condi t ion s , such a s those a ssoc ia ted

wi th r ad ia t ive hea t t r ansfe r ; and (c ) phase t r ans i t ions ,

such as me l t ing and so l id i f ica tion . T he la t te r i s o rd i -

na r i ly t r ea ted a s e i the r a r ap id ch ange in e f fec t ive

spec i f ic hea t be tween so l idus and l iqu idus ( a type (a )

n o n l i n e a r i t y ) o r as a n i n te r n a l b o u n d a r y c o n d i t i o n

th a t d e p e n d s u p o n t h e t e mp e r a tu r e d i s t r i b u t i o n ( a

type (b) nonl inea r i ty ) .

F r o m th e v e r y o u t s e t , f i n it e e l e me n t h e a t t r a n s f e r

p r o g r a ms t h a t w e r e b a s e d u p o n l i n e a r f o r mu la t i o n s

were used to so lve nonl inea r problems, even though,

s t r ic t ly speaking , on ly the Ga le rk in me thod i s eas i ly

extended to inc lude nonl inea r behavior . Usua l ly the

th e r ma l p r o p e r t i e s a n d t h e b o u n d a r y c o n d i t i o n s a r e

e s t ima t e d o n t h e b a s is o f t h e mo s t r e c e n tl y c o mp u te d

tempera tures ( see r e f . [151 for an appl ica t ion to the r -

ma l r ad ia t ion) , a procedure tha t i s bes t te rmed 'quas i -

l i n e a ri z a t io n ' . T h e o n ly a p p a r e n t d r a w b a c k t o t h e s e

a d h o c

extens ions in to the r ea lm of nonl inea r ana lys is

was an occas iona l numer ica l ins tab i l i ty , even though

impl ic i t , uncondi t iona l ly s tab le t ime in tegra t ion

s c h e me s (e .g . C r a n k - N ic o l s o n ) w e r e b e in g u s e d. T h e

common recourse was a sma l le r t ime s tep . Never the le ss ,

a sizable bo dy of l i tera tu re (e .g. refs [ 16] and [ 17] ) ,

c i t ing good compar ison wi th exper imenta l r e su l t s , has

been accum ula ted thro ugh the use of quas i - l inea r iza t ion

procedures .

Mo r e re c e n t l y , s o me e f f o r t h a s b e e n d i r e c t e d t o w a r d

a r igorous unders tanding of the convergence and s tab i -

l i ty cha rac te r i s t ic s of the t ime in tegra t ion schemes

used in nonl inea r , t r ans ien t hea t t r ansfe r ana lys is . An

a t t e m p t t o imp r o v e o n t h e q u a s i - l in e a r iz a t io n p r o c e d u r e

f o r n o n l i n e a r b o u n d a r y c o n d i t i o n s w a s r e p o r t e d [ 1 8 ] ,

where an ' ind i rec t i te ra t ion ' a lgor i thm was used . A

c lass ica l pap e r by Douglas and Du po nt [19 ] , how ever ,

provides the pro pe r f r am ew ork for an a ll - inc lus ive

approach to the ques t ion . In the fo l lowing , the i r

approach i s r ecas t in f in i te e lement form for c la r i ty .

A s s u min g t h a t a G a l e rk in a p p r o x im a t io n h a s b e e n

used to expl ic i t ly e l imina te spa t ia l dependence , in

favor of a f in i te de gree -of - f r eedo m vec tor o f t ime- .

d e p e n d e n t mo d a l p o in t t e mp e r a tu r e s ,

O t ) ,

then a

c o n -

t i n u o u s t i m e G a l e r k i n

r e p r e s e n t a t i o n o f t h e n o n l i n e a r

hea t t r ansfe r problem is g iven by

M ( 0 ) O t ) + K ( 0 ) O t ) = Q(0, t) , (1)

where M(0), K(0) , and Q a re the tem pera ture -d epe nde nt

h e a t c a p a c i t y ma t r i x , c o n d u c t i v i t y ma t r i x , a n d f l u x

vec tor , r e spec t ive ly ; t i s the t ime ; and a supe rposed dot

represents a pa r t ia l t ime de r iva tive . I f the p rob lem

happened to be l inea r , fu l l o r pa r t ia l e igenva lue ex t rac -

t ion (M is pos i t ive de f in i te ; K i s pos i t ive semi-de f in i te ;

the re fore , a ll o f the e igenva lues a re r ea l ) wo uld enable

eq . (1) to be in tegra ted prec ise ly [20] . Such a pro-

c e d u r e h a s b e e n u s e d b y G u y m o n [ 2 1 ] f o r t h e li n e ar ,

c o n v e c t iv e d i f f u s io n f i n i t e e l e me n t f o r mu la t i o n . T h e r e

is a l so some ev idence tha t th is p rocedure i s equa l ly

a p p l ic a b l e t o t h e n o n l i n e a r p r o b l e m [ 2 2 ] .

T h e

C r a n k N i c o l s o n G a l e r ki n

r e p r e s e n t a t i o n i s

found by so lv ing eq . (1) a t the middle of the t ime

interval t n , tn + l ) and by eva lua t ing the t ime de r iva t ive

w i th a c e n te r e d d i f f e r en c e a p p r o x ima t io n

O = O ( t a ) = (O n + l - - O n ) / a t , ( 2 )

wh ere t a = ( tn + tn+l a n d A t = t n + l - - t n . T h e r e f o r e .

{ 2 M ( 0 a ) + K ( 0 a ) } O n + , = 2 Q a ( 0 a )

+ M ( 0 a ) - K ( 0 a ) 0 n , ( 3 )

where

Q a ( 0 a ) = O ( 0 a , t a ) . ( 4 )

I t s h o u ld b e n o t e d t h a t t h e s e p a r a t io n o f t h e f l u x

v e c to r i n to c o n t r i b u t i o n s a t t n and t n+ 1 , i.e.

o a = O n + Q n + l , ( 5 )


4/11

R.E. Nickell, H.D. Hibbitt, Analysis o f we lded structures 113

i s va l id , in gene ra l , on ly for l inea r boundary condi t ions

and so urce te rms. Also , s ince ne i the r 0 a nor

On+ 1

are

known, eq . (3) i s a nonl inea r , a lgebra ic sys tem of

equa t ions . Quas i - l inea r iza t ion would involve the eva lua -

t i o n o f M, K a n d Q a , u s in g th e a p p r o x im a t io n 0 a =

On.

A genera l ized

C r a n k - N i c o l s o n - G a l e r k i n

represen-

t a t i o n c a n b e f o u n d b y u s in g th e s a me c e n t e r e d d i f-

f e r e n ce a p p r o x im a t io n f o r t h e t ime d e r iv a ti v e b u t ,

i n s t ea d , s e e k in g th e so lu t i o n a t so me o p t im u m p o in t

in the interval ,

0 = (1 + 7 ) 0 n + 1 + ( 1 - - 7 ) 0

n ,

(6)

w h e r e T is t h e o p t im iz a t i o n p a r a me te r .

In this case

{ ~ t t M + ( 1 + 7 ) K } ' 0 n + I = 2 Q

+ M - - ( 1 - - 7 ) K 0 n , (7 )

where M, K and QO are eva lua ted a t 0 . I f the hea t

c a p a c i t y a n d c o n d u c t iv i t y d o n o t d e p e n d u p o n t e m-

p e r a tu r e a n d t h e b o u n d a r y c o n d i t i o n s a r e n o n l in ea r ,

then 7 = 1 represents the ind i rec t i te ra t ion scheme

[ 18] . There w ere a lso ind ica t ions in r e f . [ 18] tha t

7 = 0 .7 r e p r e se n t e d a n o p t im u m p o in t , a t l e a s t f o r th e

c la ss of problems be ing inves t iga ted . Note tha t the

imp l i c i t c o e ff i c i e n t ma t r i x ( 2 /A t ) M + (1 + 7 ) K ,

n e e d b e f a c to r e d o n ly o n c e , u n le s s t h e t ime s t e p

changes , whi le seve ra l i te ra t ions may be requi red a t

e a c h t ime s t e p f o r t h e so lu t i o n v e c to r t o c o n v e r g e .

I n o r d e r t o a v o id mo r e t h a n o n e i t e r a t i o n f o r t h e

f u l l y n o n l in e a r p r o b l e m, a p r e d i c to r - c o r r e c to r s c h e me

c a n be e m p l o y e d w i t h t h e g e n er al iz e d C r a n k - N i c o l s o n -

G a le r k in r e p r e se n t a t i o n . A n in t e r me d ia t e t e mp e r a tu r e

field,

On+ 1,

can f i r s t be found f rom eq . (7) wi th M, K

and Q eva lua ted a t

On;

t h e n ,

On+ 1

i s found, in the

c o r r e c to r s t e p , f r o m e q . ( 7 ) w i th t h e p r o p e r t i e s a n d

bou nda ry co ndi t ion s eva lua ted a t (1 7)0 'n*+ 1 +

(1 --

T ) O n .

T h i s s c h e me r e q u ir e s tw o f a c to r i z a t i o n s

p e r t ime s t e p - a n e x t r e me ly e x p e n s iv e p r i c e to p a y

for accuracy .

A n o th e r s c h e me , w h ic h h a s b e e n imp le m e n te d i n a

g e n e ra l p u r p o se f i ni t e e l e me n t h e a t t r a n s f e r p r o g r a m

[3] , i s the

C r a n k - N i c o l s o n - G a l e r k i n e x t r a p o l a t i o n .

Here eq . (7) i s so lved wi th M, K and Q be ing eva lua ted

a t

~O n - O n- 1

(a s l igh t modif ica t ion i s r equi red when

the t im e s tep i s changed) . No c or rec to r i s used .

D o u g las a n d D u p o n t [ 1 9 ] a l so r e c o mm e n d a pr e -

d i c to r - c o r r e c to r s c h e me th a t r e q u i r e s b u t o n e f a c to r i -

z a t i o n p e r t ime s t e p , ma k in g i t n o min a l ly c o mp e t i t i v e

w i th t h e e x t r a p o l a t i o n s c h e me . T h e c o r r e c to r f o r o n e

t ime s t e p c o r r e sp o n d s t o t h e p r e d i c to r f o r t h e n e x t

t ime s tep . For s im pl ic i ty , the scheme wi l l be desc r ibed

here w i th T = ~ . F i r s t , an in te rme dia te f ie ld , 0 % l , is

c o m p u te d f r o m e q . ( 7 ) w i th M, K a n d Q e v a lu a t e d a t

1

~ O n + O n - 1) ;

then , the cor re c tor f ie ld 0n + l i s com-

p u te d f r o m e q . (7 ) w i th M, K a n d Q e v a lu a t e d a t

1

~ ( 0 ,, , 1 + 0 , ) .

3 . A mo d i f i e d o p e r a to r

A m o d i fi e d C r a n k - N i c o l s o n - G a l e r k i n s c h e m e c an b e

d e r iv e d b y so lv in g f o r t h e t e m p e r a tu r e a t t h e mid d l e

of the t ime in te rva l f rom eq . (3) , bu t

n o t

e x t e n d in g

the so lu t ion to the end o f the in te rva l. Then ,

2 M ( 0 a ) + K ( 0 a ) . 0 a = Q a ( 0 a ) + ~ M ( 0 a)

On.

8 )

The s im pl ic i ty of eq . (8) is obviou s and , i t should b e

n o te d , a ll o f t h e p r e v io u s ly me n t io n e d p r e d i c to r -

cor rec tor and i te ra t ive ru le s s t i l l apply . This modif ied

o p e r a to r , s imi la r t o t h e o r igin a l C r a n k - N ic o l so n s c h e me ,

c a n b e sh o w n to b e u n c o n d i t i o n a l l y s t a b le f o r l i n e ar

p r o b l e ms .

For compar ison , wi th rega rd to osc i l la t ion cha rac -

te r i st ic s and s tab i l i ty l imi ts , a one -d im ensiona l , non-

l inea r example was so lved wi th both eqs (3) and (8) ,

u s in g e x t r a p o l a t i o n t o e s t ima te t h e f o u r th - p o w e r

6 0 0

5 0 0

u~

o

u J 4 0 0

Q:-

Q_

I 0 0

0

? , ; ' ~

' f

t

, - - , A~ 5 0 s e

~ t = 2 0 0 s e

h

5 0 I 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0

T I M E , s e c

Fig. 1. Crank-N icolson-Galerkin heat transfer examp le, no

iteration, no correction, ,xt = 1000 sec, diverged imm ediately.


5/11

114

R.E. Nickell, H.D. Hibbitt, Analysis of welded structures

6

5 0 0 . ~

4 O 0

F

< 3 0 0

rr

~

2 n t :

200sec.

| A /~:~

o 2 4 6

O 5 I 1 5 1 ~ 0 1 0 0 2 5 1 3 1

T I M E , s e c

Fig. 2 . Modified Cra nk- Nic olso n-G aler kin heat t ransfer ex-

ample, no i te ra t ion, no correc t ion.

r a d i a t io n b o u n d a r y c o n d i t i o n s . F o r t h e o r i g i na l C r a n k -

N i c o l so n o p e r a t o r w i t h l u m p e d h e a t c a p a c i t y , s o l u t io n s

w e r e a t t e m p t e d f o r t h r ee d i f f e r e n t c o n s t a n t t i m e s t ep

s i z es : A t = 5 0 s e c , A t = 2 0 0 s e c , a n d A t = 1 0 0 0 s e c .

T h e r e s u l t s a r e s h o w n i n f i g . 1 . N o s i g n i f i c a n t o s c i l l a -

t i o n s ar e n o t e d f o r A t = 5 0 s ec , i n d i c a t i n g c o n v e r g e n c e

t o t h e c o r r e c t s o l u t i o n ; f o r A t = 2 0 0 s e c , s o m e e a r l y

o s c i l l a ti o n s a re d e t e c t e d t h a t g r a d u a l l y d i e o u t a s s t e a d y

s t a t e i s r e a c h e d ; f o r A t = 1 0 0 0 s e c , t h e s o l u t i o n d i v e r g e s

a h n o s t i m m e d i a t e l y . T h e s am e r a d i a t i o n p r o b l e m w a s

s o l v e d w i t h l u m p e d h e a t c a p a c i t y a n d i d e n t i c a l t i m e

s t e p s iz e s u s i n g th e m o d i f i e d o p e r a t o r . T h e r e s u l ts a r e

s h o w n i n f i g . 2 . A n e s s e n t i a l l y id e n t i c a l s o l u t i o n w a s

o b t a i n e d f o r A t = 5 0 s e c ; f o r A t = 2 0 0 s e c , t h e s o l u t i o n

d o e s n o t o s c i l l a te a n d i s b o u n d e d f r o m a b o v e b y t h e

c o n v e r g e n t s o l u t i o n ; f o r A t = 1 0 0 0 s e c , t h e s o l u t i o n

d i s p l a y s s l o w d i v e r g e n c e ( s e e i n s e t i n f i g. 2 ) b u t r e m a i n s

b o u n d e d d u r i n g t h e c a lc u l a ti o n s . T h e m o d i f i e d o p e r a t o r

w o u l d a p p e a r , o n t h e b a s is o f th i s l i m i t e d s t u d y , t o

p o s s es s s o m e d i s t i n c t a d v a n t a g e s o v e r th e c o n v e n t i o n a l

o p e r a t o r .

4 . W e l d i n g h e a t t r a n s f e r e x a m p l e

I n o r d e r t o t e s t t h e v a r i o u s t i m e i n t e g r a t i o n s c h e m e s

o n a t y p i c a l w e l d i n g h e a t t r a n s f e r p r o b l e m , t h e t h r e e

p o t e n t i a l l y a t t r a c t i v e c a n d i d a t e s - q u a s i - l i n e a r i z a t io n ,

e x t r a p o la t io n , a n d th e D o u g l a s - D u p o n t p r e d i c t o r -

c o r r e c t o r - w e r e u s e d t o s o lv e a o n e - d i m e n s i o n a l

e x a m p l e i n v o l v in g t e m p e r a t u r e - d e p e n d e n t p r o p e r ti e s ,

n o n l i n e a r b o u n d a r y c o n d i t i o n s , a n d p h a s e t r a n s i ti o n .

A t i m e - d e p e n d e n t s u r f a c e h e a t f l u x w a s a p p l i e d t o a

s e m i - i n f i n i te so l i d . T h e m a g n i t u d e o f t h e f l u x w a s d e -

t e r m i n e d f r o m t h e p a r a m e t e r s f o r a h i g h - s p e e d w e l d i n g

p a s s ( 1 4 . 5 V , 3 0 0 A , 1 5 i n . / m i n ) s u f f i c i e n t t o c r e a t e a

p a r t i a l f u s i o n z o n e a n d b r i n g a b o u t s u r f a c e r a d i a t i o n ,

i n a d d i t i o n t o c o n v e c t i v e c o o l i n g . T h e w e l d i n g e n e r g y

r a t e is , t h e r e f o r e , a b o u t 4 B t u / s e c a n d 4 s e c a r e r e q u i r e d

t o t r a v e l 1 i n . ( t h e h i g h e n e r g y r a t e a n d w e l d i n g s p e e d

a r e d e s i g n e d t o a c c e n t u a t e d i f f e r e n c e s in t h e t i m e i n -

t e g r a t i o n s c h e m e s ) .

T h e s u r f a c e h e a t f l u x f r o m t h e w e l d i n g a r c i s

a s s u m e d t o b e u n i f o r m l y d i s t r i b u t e d i n s p a c e ( o v e r a

w i d t h e q u a l t o a n e l e c t ro d e d i a m e t e r ) an d n o r m a l l y

d i s t r i b u t e d i n t i m e :

q t) = qo

e x p [ - - { ( t- - t o ) / r } 2 ] , ( 9 )

w h e r e q o i s t h e p e a k h e a t f l u x a n d r i s a c h a r a c t e r i s t i c

t i m e a p p r o x i m a t e l y e q u a l t o t h e t i m e r e q u i r e d f o r t he

b e a d t o t r a v e l o n e e l e c t r o d e r a d i u s. T h e p a r a m e t e r t o

i s c h o s e n t o b e l a r g e e n o u g h t h a t , e s s e n t i a l l y , al l o f t h e

h e a t f l u x i s a c c o u n t e d f o r b y a n a n a l y s i s s ta r t i n g a t

t i m e t = 0 . F o r t h i s ca s e , t h e e l e c t r o d e d i a m e t e r w a s

t a k e n t o b e 0 . 0 5 i n . ; t h e r e f o r e , r = 0 .1 s e c . F o r t o l a r g e

e n o u g h , t h e t o t a l f l u x p e r u n i t l e n g t h o f w e l d i s

= 0 .0 5 (Tr/2) w2 qo ~-. (1 0)

F o r a w e l d e f f i c i e n c y o f 4 0 % , t h i s w o u l d y i e l d a p e a k

h e a t f l u x o f a b o u t 8 0 0 B t u / i n . s e c.

T e n f i r s t - o r d e r , p l a n e i s o p a r a m e t r i c e l e m e n t s w e r e

u s e d in t h e a n a l y s e s . E a c h o f th e e l e m e n t s h a d d i m e n -

s i o n s e q u a l t o a n e l e c t r o d e r a d i u s. T h e t h e r m a l p r o -

p e r t i e s f o r t h e m a t e r i a l a r e g i v e n i n f i g . 3 . I n a d d i t i o n

t o t h e v a r i a t i o n i n c o n d u c t i v i t y a n d s p e c i f i c h e a t w i t h

t e m p e r a t u r e , t h e d e n s i t y i s 4 0 9 l b / f t 3 a n d t w o l a t e n t

h e a t r a n g e s w e r e g iv e n : 1 8 . 2 B t u / l b a t 1 6 0 0 F a n d

1 0 8 .7 B t u / l b a t 3 3 0 0 F . F o r t h is p r o b l e m , t h e l a t e n t

h e a t a t 1 6 0 0 F w a s t r e a t e d a s a l o c a l p e r t u r b a t i o n o f

t h e s p e c i f i c h e a t v e r s u s t e m p e r a t u r e c u r v e . T h e c r o s s-

h a t c h e d a r e a o f f ig . 3 w as c a l c u l a t e d t o b e e q u a l t o

1 8 . 2 B t u / l b . T h e l a t e n t h e a t a t 3 3 0 0 F w a s t r e a t e d i n

t h e c o n v e n t i o n a l m a n n e r , w i t h t h e s o l id u s a n d l i q u i d u s

a s s u m e d t o b e 3 3 0 0 a n d 3 4 0 0 F , r e s p e c t i v e l y . T h e

e m i s s i v i t y u s e d i n t h e t h e r m a l r a d i a t i o n c a l c u l a t i o n s

a n d t h e c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t w e r e

s e l e c t e d a s 0 . 9 a n d 5 B t u / f t 2 h r , r e s p e c t i v e l y .

F o r t h e f i rs t c o m p a r i s o n q o , r , a n d t o w e r e c h o s e n

t o b e 8 0 0 B t u / i n . 2 s ec , 0. 1 s e c , a n d 0 . 3 s e c , r e s p e c t i v e l y .

T h e s u r f a c e t e m p e r a t u r e r e s u lt s , fo r a ti m e s t e p o f


6/11

R .E . N i c k el l H .D . H i b b i t t An a l y si s o f weM ed s tr u c t u r e s

1 1 5

0 .005 se c ; ar e show n in f ig . 4 . In a l l thr e e ana lyse s , the

r a d i a t io n c o o l i n g w a s t r e a t e d i m p l i c i t l y ; i . e. a s a c o n -

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

b o u n d a r y l a y e r h e a t t r a n sf er c o e f f i c i e n t ,

h Z )

e o l T + T . ] [ T 2 + T 2 ] , 1 1 )

w h e r e e , o , a n d T a r e t h e e m i s s iv i t y , S t e f a n - B o l t z -

m a r m c o n s t a n t , a n d s i n k t e m p e r a t u r e , r e s p e c t i v e ly .

T h e c o m p a r i s o n o f t h e t h r e e m e t h o d s i s f a v o r a b l e ,

a l though the sur fac e t e mpe r a tur e f r om quas i - l ine ar iz a -

t i o n t e n d s t o w a n d e r a s m e l t i n g b e g in s a n d a l s o d u r i n g

t h e p e r i o d w h e n r a d i a t i o n p l a y s a n i m p o r t a n t r o l e i n

sur fac e he a t t r ansfe r. On ly s l ight d i f f e r e nc e s ar e ob-

s e r v e d b e t w e e n t h e D o u g l a s - D u p o n t p r e d i c t o r -

c o r r e c t o r a n d e x t r a p o l a t i o n p r o c e d u r e s.

T h e s a m e p r o b l e m w a s s o l v e d , u s i n g t h e p r e d i c t o r -

c or r e c tor , w i th the r ad ia t ion tr e a te d e xp l i c i t ly , i . e . a s a

pr e sc r ibe d sur fac e he a t f lux . T he dr i f t o f the so lu t io n

a w a y f r o m t h e c o n v e r g e n t r e s u l t i s s h o w n i n fi g. 5 .

T h e c o r r e c t o r d o e s n o t i m p r o v e t h e a c c u r a c y a n d , a s

no te d in r e f. [1 8 ] , se ve ra l i t e r a t ions w o uld be r e qu ir e d

f o r c o n v e r g e n c e . I f n o o t h e r n o n l i n e a r i ti e s w e r e p r e s e n t ,

i n d i r e c t i t e r a t i o n w o u l d m o s t l i k e l y b e m o r e e f f i c i e n t

t h a n a n i m p l ic i t t r ea t m e n t o f t h e b o u n d a r y c o n d i t i o n ;

how e ve r , in the pr e se nc e o f mate r ia l pr ope r ty t e mpe r a -

t u r e d e p e n d e n c e a n d p h a s e t r a n s i ti o n s , t h e i m p l i c it

treatment is preferable .

A c o m p a r i s o n b e t w e e n s o l u t i o n s f o r t h e s a m e p e a k

w e ld ing f lux , but d i f f e r e nt c har ac te r i s t i c t ime s , i s

s h o w n i n f i g . 6 . T h e e x t r a p o l a t i o n s c h e m e w a s u s e d

u.

~ 40

i--

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i.

o

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2

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0

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A T 1 6 0 0 F

S O O M p E , O O o . , T O 0

1 I I I I I I

5 0 0 I O 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0

T E M P E R A T U R E

F

F i g. 3 . T h e r m a l p r o p e r t ie s , o n e - d i m e n s i o n a l w e l d i n g e x a m p l e .

x

2 ~

m

8

~

;T-

o

4 w

14,000 , ~ : EXTRAPOLATION ~ ~ ~ LP/

OOUGLAS-OUPONT

12,OO0 - QUASI-LINEARIZATION

I 0 , 0 0 0

~ ~

8 0 0 0

6 0 0 0 i /

W 4 0 0 0

2 0 0 0

, : , ,

O 0 0 5 I 0 1 5 2 0

T I M E ,

s e c

F i g. 4 . C o m p a r i s o n o f t h r e e t e m p o r a l o p e r a t o r s , o n e - d i m e n -

s i o n a l w e l d i n g e x a m p l e , h i g h h e a t f l u x : q o = 8 0 0 B t u / s e c ;

r = 0 . 1 s e c ; a n d t o = 0 . 3 s e a I m p l i c i t r a d i a t i o n . )

o -- o. -o D O U G L A S - D U P O N T * *

( I M P L I C I T R A D I A T I O N ) +

+ * D O U G L A $ - D U P O N T + *

r,---CtL~ ( E X P L I C I

t e l

O .

LLJ

I-'-

2000

~+

J i i

0 5 A O . t 5 2 0

TIME

s ec

F i g. 5 . Im p l i c i t v e r s u s e x p l i c i t r a d i a t io n , D o u g l a s - D u p o n t

p r e d i c t o r - c o r r e e t o r : qo = 8 0 0 B t u / s e c ; r = 0 . 1 s e c ; a n d t o =

0 3

s e a


7/11

116

R .E . N i c k e l l H .D . H i b b i t t A n a l y s i s o f we l d ed s tr u c t u r e s

f o r b o t h a n a l y s e s . N o t e t h a t t h e p e a k f l u x i s 1 0 0

B t u / i n . z s e c ( a w e l d i n g e f f i c i e n c y o f l es s t h a n 1 0 )

i n o rd e r t o k e e p t h e s u r f a c e t em p e r a t u r e s d o w n t o

r e a s o n a b l e l e v e l s . T h e e f f e c t o f p h a s e t r a n s i t i o n d u r i n g

b o t h h e a t i n g a n d c o o l i n g c a n b e s e e n .

A f i n a l c o m p a r i s o n is s h o w n i n fi g. 7 . A l l t h r e e

m e t h o d s g i v e v ir t u a l ly i d e n t i c a l r e s u l t s e x c e p t d u r i n g

r a pi d p h a s e t r a n s i t io n a n d i n t h e p e r i o d n e a r p e a k s u r-

f a c e t e m p e r a t u r e s . I t i s u n l i k e l y t h a t a n y n o t i c e a b l e

d i f fe r e n c e s w o u l d o c c u r d u r i n g t h e r e m a i n d e r o f t h e

c o o l i n g p a r t o f t h e w e l d i n g c y c l e .

O n t h e b a s i s o f t h e s e r e s u l t s a n d c o m p a r i s o n s , t h e

t i m e i n t e g r a t i o n s c h e m e t h a t a p p e a r s t o b l e n d a c c u r a c y

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

m e t h o d .

5 . O m e g a s e a l w e l d p r o b l e m

T h e n u m e r i c a l a p p r o a c h e s d e s c r ib e d p r e v i o u s l y h a v e

b e e n a p p l i e d t o a n o m e g a s e a l d u r in g t h e f i r st p a s s o f a

m u l t i- p a s s w e l d i n g o p e r a t i o n i n o r d er t o e x a m i n e t h e

r e s id u a l d i s t o r t i o n a n d s t r e s s a s s o c i a t e d w i t h t h i s

f l e x i b l e g e o m e t r y . S u c h s t r u c t u r a l c o n f i g u r a t i o n s a r e

i n c o m m o n u s e f o r f o r m i n g f l e x i b l e , g a s - t i g h t s e a l s

b e t w e e n s u p p o r t s t r u c tu r e s a n d i n f r e q u e n t l y r e p la c e d

c o m p o n e n t s . T h e r e p l a c e m e n t s e q u e n c e c o n s i s t s o f

f l a m e c u t t i n g t o r e m o v e t h e o l d w e l d , j o i n t p r ep a ra -

t i o n , a n d r e w e l d i n g o f t h e n e w l y - i n s e r t e d c o m p o n e n t ;

t h e r e f o r e , p r e d i c t i o n o f w e l d d i s t o r t i o n i s r e q u i r e d .

u_

e c

W

t- -

O '

0 2 4 6

TIME s e c

F i g. 6 . C h a r a c t er i s t ic t i m e c o m p a r i s o n , e x t r a p o l a t i o n i m p l i c i t

r a d i a t i o n ) m e t h o d : q o = 1 0 0 B t u / i n . 2 s ec ; a n d t o = 0 . 3 s e c .

u _

o

m

c r

F -

n-

w

k-

- - D O U G L A S - D U P O N T

o o o E X T R A P O L A T I O N

* + * Q U A S I - L I N E A R I Z A T I O N

+ ,+

1

'

g

/

i

. 2

i i

O 0

4 6

T I M E ,

s e c

F i g. 7 . C o m p a r i s o n o f t h r e e t e m p o r a l o p e r a t o r s , o n e - d i m e n -

s i o n a l w e l d i n g e x a m p l e , l o w h e a t f l u x : q o = 1 0 0 B t u / s e c ; r

0 . 0 5 s ec ; a n d t o = 0 . 2 s ec .

5.141

' 1 , - r

W E L D

8 , i ~ 0 . 8 5

/

.115

b L I

. 0 6 5

W E L D N S E R T

. 0 4 7

.074

D E T I L

F i g. 8 . O m e g a s e a l g e o m e t r y a n d t h e r m o c o u p l e p l a c e m e n t .


8/11

R .E . N i c k e l l, H .D . H i b b i t t , A n a l y s i s o f w e l d e d s t r u c t u r e s 1 1 7

T h e g e o m e t r y i n t h e c a s e a n a l y z e d is a x i s y m m e t r i c ; a

r a d i a l c r o s s s e c t i o n o f t h e s e a l a n d i t s w e l d i n s e r t is

show n in f ig . 8 .

5 . 1 . T h e r m a l a n a l y s i s

D u e t o t h e m a s s o f t h e p a r t s a d j a c e n t t o t h e o m e g a ~= 2 o

s e a l, o n l y t h e v o l u m e i n a n d n e a r t h e s e al i t s e l f w a s

m o d e l e d . P e r f e c t k i n e m a t i c r e s t r a i n t i s a s s u m e d t o b e ,~ ~

p r o v i d e d t o t h e s e al b y t h e s e p a r t s; i n a d d i t i o n , t h e

l a r g e h e a t s i n k i s a s s u m e d t o a c t a s a s e m i - i n f i n i t e

o

m e d i u m . T h e w e l d i n s e r t g e o m e t r y s h o w n is v a li d o

p r i o r t o t h e f i r s t w e l d i n g p a s s .

F o r t h i s a n a l y s i s , o n l y t h e f i r s t p a s s o f t h e s e q u e n c e

i s b e i n g t r e a t e d . T h e w e l d i n g p a r a m e t e r s f o r t h i s p a s s

a r e 9 V , 7 4 A , a n d a sp e e d o f 4 . 7 5 i n . / m i n ( t h e o t h e r

t h r e e p a s s e s h a v e t w i c e t h e s p e e d , t h e s a m e v o l t a g e ,

a n d a c u r r e n t o f 1 1 0 A ) . T h e m o d e l o f t h e w e l d i n g

p r o c e s s f o r t h i s f i r s t p a s s i s t a k e n t o b e a x i s y m m e t r i c .

C o r r i g an [ 5 ] h a s d i s cu s s e d t h e v a l i d i t y o f th i s a p p r o a c h

a n d h a s j u s t i f i e d i t e x p e r i m e n t a l l y . H e r e a s o n s t h a t ,

e x c e p t f o r t h e s t a r t / s t o p p o i n t , e a c h r a d ia l s e c t io n o f

t h e c o n f i g u r a t i o n w i ll e x p e r ie n c e t h e s a m e t h e r m a l

h i s t o r y a n d t h a t t h e t h e r m a l d i f f u s i o n a h e a d o f th e

t o r c h i s n o t a c o n t r o l l i n g f a c t o r i n c o m p a r i s o n w i t h

t h e h e a t f l u x s u p p l ie d b y t h e t o r c h i t s el f. I f t h e a n o -

m a l y o f t h e s t a r t / st o p p o i n t i s i g n o r e d , t h e s p e e d o f

t h e f i r s t p a s s i s s u f f i c i e n t l y h i g h , i n c o m p a r i s o n t o t h e

t h e r m a l d i f f u si v i t y o f m e t a l s, t o j u s t i f y t h e a x i s y m -

m e t r i c m o d e l .

T h e c i r c u m f e r e n t i a l l y u n i f o r m f l u x t h a t i s a p p l i e d

t o t h e m o d e l i s t h e n a s s u m e d t o b e a n o r m a l d i s tr ib u -

t i o n i n s p a c e a n d t i m e :

q r , t ) = q o

e x p [ - { ( r - -

r o ) / l } 2 ]

e x p

[ - - ~

{ ( t - t o ) / r } 2 ] , ( 12 )

w h e r e q o i s th e p e a k h e a t f l u x f o r p o i n t s o n t h e w e l d

c e n t e r l i n e , r o r e p r e s e n t s t h e r a d i u s o f t h e w e l d c e n t e r -

l i n e, a n d l i s a c h a r a c t e r i s t i c l e n g t h a p p r o x i m a t e l y e q u a l F ig . 1 0.

t o a n e l e c t r o d e r a d i u s. F o r t h i s p r o b l e m t h e t o t a l h e a t

a p p l i e d t o t h e s t r u c t u r e d u r i n g t h e f i r s t p a s s i s a b o u t

2 5 8 B t u , so t h a t t h e p e a k f l u x c a n b e f o u n d i f t h e

e f f i c ie n c y o f t h e w e l d i n g p r o c e ss c a n b e e s t i m a t e d .

T h e e l e c t r o d e d i a m e t e r i s 0 . 1 1 0 i n . w h i c h i m p l i e s t h a t

a c h a r a c t e r i s t i c l e n g t h m i g h t b e ~ i n . a n d , w i t h t h e

a r c s p e e d o f 4 . 7 5 i n . / m i n , a c h a r a c te r i s t ic t i m e w o u l d

b e a b o u t ] s e c . I f t h e e f f i c i e n c y i s a s s u m e d t o b e 7 5 ,

t h e n q o i s a b o u t 2 0 B t u / i n . 2 se c .

~ ~ 0

m

I I I

5 0 0 I O O O 1 5 o 0

T E M P E R A T U R E O F

. 2 0

0 1 5

a _

o

_--0

0 1 0

C O

i

0 0 5 ~ D

C . _

u~

o

i 2 5 1 0 0

2 0 O O

Fig. 9. Inconel thermal properties versus temperature.

4 0

30

~ 2

4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0 2 4 0 0

T E M P E R A T U R E F )

4 0

3 o

~ _ o 2 0

~ I

o

1 I I I I X

o 4 0 o B o o m o o m o o 2 o o o 2 4 o o

, ~ T E M P E R A T U R E F )

_~

C _z

O I I I | I

b J 0 4 0 0 B O O 1 2 0 0 1 6 0 0 2 0 0 0 2 4 0 0

T E M P E R A T U R E * F )

lnconel mechanical properties versus temperature.

THIRD P SS SECOND P SS

Fig. 11. Observed melt regions (from photograph) omega seal.


9/11

1 1 8 R. E . N ic k e l l H . D . H i bb i t t Ana l y s i s o f we l de d st r uc t u r e s

0 0

O

. P

O

O

d -

O

O

O

t

X

N ~

I

|

O

O

t

t4.88

t l

c 2

c [ 3

c [ 4

C

C

c 8

ISOTH RMS

] : 3 6 3 .

1 : 564.

] = 7 4 6 .

:

928,

]= 1110.

= 1 2 9 2 ,

1474.

] = 1656.

C [ 9 1 : 1838 .

C [ 10 ] : 2 0 1 9 .

C t 11 ] 2201 .

C 12 ] 2383

C [ 1 3 1 = 2 5 6 5 .

C [ 1 4 ] = 2 7 4 7 .

C [ 1 5 ] 2 9 2 9 .

C [ 1 6 ] : 3 1 1 1 .

C [ 1 7 ] = 329 2.

C [ 18 ] : 3474

C [ 19 ]= 3656 .

C [ 20 ]= 3838 ,

A

~ \ \ \ /

. .4 , , \ t

- - - ~

~0

/ / I

/ i

, 1 0 ' 9 ,

I I l I

- ~ t l I I

7 // / / ~

/ / ~ z

/ /

~ I

/ / / / /

\ / Y

1 1

I I I t t I

4.96 5.04 5.12 5.20 5.28 5.36 5.44

R-AX I S

F i g . 1 2 . P r e d i c t e d m e l t r e g i o n f i r s t p a s s o m e g a s e a l . T i m e = 3 . 5 s e c .

T h e r m a l a n d m e c h a n i c a l m a t e r i a l p r o p e r t i e s f o r t h e

w e l d in s e r t a n d t h e b a s e m e t a l ( b o t h w e r e a s s u m e d t o b e

c o n s t r u c t e d o f I n c o n e l 6 0 0 ) a r e s h o w n i n fig s 9 a n d 1 0 ,

r e s p e c t i v e l y . T h e d e n s i t y i s 0 . 3 0 4 l b / i n . 3 . S in c e

n o f o r m a l d a t a o n t h e l a t e n t h e a t v a lu e s f o r I n c o n e l

6 0 0 w e r e a v a i la b l e , t h e p r o p e r t i e s o f n i c k e l w e r e u s e d

- t h e l a t e n t h e a t o f f u s i o n is a b o u t 1 3 3 B t u / l b f o r a

m a t e r ia l t h a t is 7 2 N i , 1 4 - 1 7 C r , a n d 6 - 1 0 F e ;

t h e s o l id u s a n d l i q u id u s w e r e c h o s e n t o b e 2 4 7 0 a n d

2 5 7 5 F , r e s p e ct i v e ly . I n a d d i t i o n , t h e t h e r m a l c o n -


10/11

R. E . N i c k e l l H . D . H i bb i t t Ana l y s i s o f we l de d s t r uc t u r e s 119

duc t iv i ty was a ssum ed to drop by 50 a t me l t ing 3200,

whi le the spec i f ic hea t rose by 15 .

Thermal r ad ia t ion was a lso a l lowed for on a l l ex-

2800

posed sur faces , inc luding those under the we ld torch .

T h e m o t iv a t i o n h e r e w a s t o a l l o w th o se h e a t e d su r f a c e s ~,oo

t o c o o l b y r a d i a t in g t o c o o l su r r o u n d in g s ( 7 0 F ) a f t e r

t h e w e ld t o r c h h a d p a s sed . T h e r e ma in in g b o u n d a r y ~

~ooo

c o n d i t i o n c o n c e r n e d t h e c o n d u c t io n p a th t o t h e ma s -

s ive hea t s ink . A s imple one -d im ensiona l con duc t ion ~- ,6oo

b o u n d a r y l a y e r mo d e l w a s u se d .

The the rmal ana lys is of the omeg a sea l was com par ed ~_ ,200

w i th tw o p i e ce s o f e x p e r ime n ta l e v id en c e . T h e r m o -

couple da ta were ava i lab le a t the loca t ions shown in 600

f ig . 8 , and in addi t ion , a photograph of a sea l sec t ion

( repro duc ed in f ig . 1 1) gave ev idence of the ex te n t of

400

me l t i n g . P r e d i c t e d i so th e r ms a t ma x imu m e x t e n t o f

me l t ing (3 .5 sec a f te r onse t o f hea t ing) a re show n in o

f ig . 1 2 f o r p u r p o se s o f c o mp a r i so n w i th t h e p h o to -

g r a p h ; th e c o m p a r i so n i s f av o r a b l e . A n o th e r f a v o r ab l e

c o mp a r i so n , b e tw e e n t h e p r e d i c t e d t h e r ma l h i s to r y

o ~ - * o - - - o T H E R M O C O U P L E T E M P E R A T U R E S

- - C O M P U T E D T E M P E R A T U R E S ,

I I I

8 1 6 2 J 4 3 J 2 4 1 0 4 1 8 5 6

T I M E , s e c

Fig. 13. Therm ocouple compa rison, heat-affected zone.

~

X

?-

INEREMENT I

HO@P TRE

6 E [ I ) : 2 2 6 9 7 . E [ 9 ] = 6 7 7 q 8 .

C 2 ] = 2 8 3 2 8 . C [ I O ) = 7 3 3 8 0 .

C 3 ] = 3 3 9 6 0 .

c [ q ] = 3 9 5 9 1 .

C [ 5 ) = 4 5 2 2 3 .

C ( 6 ] = 5 0 8 5 q.

C ( 7 ) = 5 6 4 8 5.

C [ 8 ] = 6 2 1 1 7 .

~ . B O

q l . B B I I

q . ~ 5 . 0 ~ 5 . 1 2 5 .L :~ ~ . 2 8 ~ . 3 6

R - A ] 5

F i g . 14 . R e s i d u a l c i r c u m f e r e n t i a l s tr e s s c o n t o u r s .


11/11

1 2 0

R.E. Nickell H.D. Hibbitt Analysis of weMed structures

a n d t h e t h e r m a l h is t o r y a n d t h e t h e r m o c o u p l e m e a s u r e-

m e n t a t l o c a t i o n 5 ( f ig . 8 ) , is s h o w n i n f ig . 1 3 .

5.2. Stress analysis

F o r t h e s t r e ss a n a ly s i s , t h e s a m e m e s h w a s u s e d a s f o r

t h e t h e r m a l an a ly s is . C l a m p e d b o u n d a r y c o n d i t i o n s

w e r e a p p l i e d t o t h e e x t r e m i t i e s o f t h e o m e g a s e a l.

A f t e r c o o l i n g i s c o m p l e t e , t h e r e s i d u a l h o o p s t r es s in

a n d n e a r t h e h e a t - a f f e c t e d z o n e is a b o u t 4 0 - 5 0 k s i.

T h e p e a k h o o p s t r e s se s o c c u r a t a s o m e w h a t h i g h e r

t e m p e r a t u r e a n d a r e s h o w n i n fi g. 1 4 . A t t h i s t e m p e r a -

t u r e , t h e y i e l d s t r e s s i s a b o u t 4 0 k s i . P r e d i c t e d r e s i d u a l

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

m e n t b e c a u s e a f l e x i b l e p o r t i o n o f th e o m e g a s e al w a s

o m i t t e d i n t h e m e c h a n i c a l a n a l ys i s .

6 . A c k n o w l e d g e m e n t s

O n e o f t h e a u t h o r s ( R o b e r t E . N ic k e ll ) w a s s u p p o r t e d

b y t h e O f f ic e o f N a v a l R e s e a r c h u n d e r C o n t r a c t

N 0 0 0 1 4 - 6 7 - A - 0 1 9 1 - 0 0 2 5 d u r in g t h e c o u r s e o f t h e s e

i n v e s t i g a t i o n s .

R e f e r e n c e s

[ 1 ] R . B jo rhovd e , J . Bro zze t t i , G .A. Alps ten and L . Ta ll ,

Res idua l s t r e sses in th ick we lded p la tes , Weld . J . 51

( 1 9 72 ) 3 9 2 - 4 0 5 .

[2 ] J . Brozze t t i , G .A. Alps ten and L . Ta l l , Weld ing pa ramete r s ,

t h i c k p l a t e s , a n d c o l u m n s t r e n g t h , W e ld , J. 5 0 ( 1 9 7 1 )

331 342 .

[3 ] H .D. Hibb i t t and P .V . Marca l, A num er ica l the rm o-

m e c h a n i c a l m o d e l f o r t h e w e l d i n g a n d s u b s e q u e n t l o a d-

i n g o f a f a b r i c a t e d s t r u c t u r e , C o m p u t . S t r u c t . ( i n p re s s ).

[4 ] K .E . Dorschu , C ont ro l o f coo l ing ra tes in s tee l we ld meta l ,

W e l d . J . 4 7 ( 1 9 6 8 ) 4 9 - 6 2 .

[ 5 ] D .A. C o r r i g a n , T h e r m o m e c h a n i c a l e ff e c t s i n f u s io n w e l d -

ing o f h igh s t reng th s tee l s , Ph .D . Disse r ta t ion , MIT (1966) .

[ 6 ] K . M a s u b u c h i, C o n t r o l o f d i s t o r t i o n a n d s h r i n k a g e i n

weld ing , Weld ing Research Counc i l Bu l le t in , 149 , New

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