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Stress G ra d ie n t C o r re c t io n Fac tor fo r

Stress In te n s i ty a t W e l d e d G usset P la tes

Stress gradient correction factor is determ ined for gusse t plates

with circular transitions and groove welded to flange tips

B Y N . Z E T T L E M O Y E R A N D J . W . F I S H E R

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

r o p a g a t i o n is t h e r a n g e o f s t re s s

i t y f a c t o r , A K . T h i s p a r a m e t e r i s

s u p e r i m p o s i n g v a r

c o r r e c t i o n f a c t o r s o n A K f o r a

t h r o u g h c r a c k in a n i n f i n i t e

t e n s i o n . ' '

1

' -

1

n e o f t h e c o r r e c t i o n f a c t o r s , F

e

, is th e

r es s g r a d i e n t c o r r e c t i o n f a c t o r a n d i s

n d e d to a c c o u n t f o r e i t h e r a n o n -

o r st re s s c o n c e n t r a t i o n c a u s e d b y

T h e a u t h o r s ' e v a l u a t e d

F

e

f o r s t i f f

e d u r e e m p l o y e d is o f t e n c a l l e d

G r e e n ' s f u n c t i o n o r s u p e r p o s i t i o n

i 1 7

I n t h i s a p p r o a c h t h e

in t h e c r a ck f r ee bo d y a r e used

a t e s t re s s i n t e n s i t y b y m e a n s o f

a p p r o p r i a t e G r e e n ' s f u n c t i o n . T h e

v a n t a g e is t h a t o n l y o n e st r es s a n a l

e n t t e c h n i q u e ) n e e d s t o b e m a d e

e a c h j o i n t g e o m e t r y . T h e o n l y

u i r e m e n t i s t h a t t h e c r a c k p l a n e b e

o w n in a d v a n c e .

T h e o b j e c t i v e s o f t h e s u b j e c t s t u d y

1. T o d e t e r m i n e F

E

f o r gu ss e t p l a t e s

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

2. T o p r o v i d e f o r t h e a u t o m a t i c

r e d i c t i o n o f

F

K

f o r a n y c r a c k l e n g t h

r a r y d e t a i l g e o m e t r y .

I n g e n e r a l , t h e p r o c e d u r e p r e v i o u s l y

t h e a u t h o r s ' w a s f o l

l o w e d ;

t h e o n l y m a j o r d i f f e r e n c e w a s

h e d e t a i l i n v o l v e d .

N. ZETTLEMOYER is Research Engineer,

Exxon Production Research Co., Houston,

Texas; /. W. FISHER is Professor ol Civil

Engineering, Fritz Engineering Laboratory,

Lehigh University, Bethlehem, Pennsylvan

ia.

C r a c k F r e e S t r e ss A n a l y s i s

A na l y t i ca l Mode l

T h e b a s i c g e o m e t r y f o r t h e g u s s e t

p l a t e i n v e s t i g a t i o n i s s h o w n i n F ig . 1 .

S y m m e t r y w a s a s s u m e d a b o u t t h e

g u s se t m i d l e n g t h a n d th e w e b c e n t e r -

l i n e . ( T h e i m p l i e d s y m m e t r i c a l g u s s e t s

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

b r i d g e s a n d a l s o m a x i m i z e s t re s s

concen t ra t i on .

7

) F u l l p e n e t r a t i o n o f

t h e w e l d w a s t h e o n l y c a s e c o n s i d

e r e d ;

t h e w e l d d e p t h w a s t a k e n a s

c o n s t a n t a n d e q u a l t o t h e g u s s e t p l a t e

t h i c k n e s s . T h e t e r m i n a t i o n o f t h e w e l d

a t t h e p o i n t o f t r a n s i t i o n t a n g e n c y w a s

a s s u m e d t o b e g r o u n d s m o o t h t o

m a i n t a i n t h e i n t e n d e d c o n t o u r . T h e

w e l d t h i c k n e s s i n t h e p l a n e o f t h e

f l a n g e a n d g u s s e t w a s t a k e n a s p a r t o f

t h e g u s s e t w i d t h .

F o u r v a r i a b l e s w e r e s t u d i e d . T h e y

w e r e t h e t r a n s i t i o n r a d i u s R , t h e

a t t a c h m e n t l e n g t h L , t h e g u s s e t p l a t e

w i d t h W

g p

, a n d t h e g u s s e t p l a t e t h i c k

ness T

gp

. T a b l e 1 l i st s t h e p a r a m e t r i c

c o m b i n a t i o n s i n v e s t i g a t e d . E a c h o f t h e

v a r i a b le s w a s n o n d i m e n s i o n a l i z e d b y

e i t h e r f l a n g e w i d t h o r t h i c k n e s s . T h e

f l a n g e w i d t h w a s t y p i c a l l y 1 2 i n . ( 30 5

m m ) .

A ll p r o b l e m s w e r e a n a l y z e d t w o -

L / 2

d i m e n s i o n a l l y ; h e n c e , o n l y t h e r e l a t iv e

p l a t e t h i c k n e s s e s w e r e i m p o r t a n t . E c

c e n t r i c i t y o f t h e g u s se t p la t e ' s c e n t r o i -

dal p l a n e a t m id th i ckness r e l a t i v e t o

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

g l e c t e d .

P l a n e s t re s s e l e m e n t s f r o m a n e x i s t

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

S A P I V ' , w e r e u s e d f o r t h e s t re s s a n a l y

sis.

Y o u n g ' s m o d u l u s w a s s e t a t 2 9 ,6 0 0

ks i ( 2 x 1 0 ' M P a ) , an d P o i ss on ' s r a t i o

was t aken t o be 0 .30 . T he f ac t t ha t a

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

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

s i n g u l a r i t y e x i s t e d a t t h e d e t a i l . T h u s ,

t h e v e r y f i n e m e s h s i z e s u s e d i n t h e

p r e v i o u s s t i ff e n e r a n d c o v e r p l a t e

a n a l y s e s ' w e r e u n n e c e s s a r y . S t i l l , as

t h e ra d i u s d e c r e a s e d c o n c e n t r a t i o n

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

s i z e a l s o i n c r e a s e d . T h e m e s h s i z e h a d

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

t o c a p t u r e t h e p e a k c o n c e n t r a t i o n .

U n i f o r m s t re s s w a s i n p u t t o a t w o -

d i m e n s i o n a l c o ar s e m e s h w h i c h e m

p l o y e d p l a n e s t re s s e l e m e n t s . F or

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

( R / W , > 0 . 5 ), t h e e l e m e n t w i t h t h e

h i g h e s t s tr e s s w a s l o c a t e d a n d t h a t

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

m a x i m u m s tr es s c o n c e n t r a t i o n f a c t o r ,

2.42W

f

/2

gp

W

f

/2

Gusset Plate

r Groove Weld

X Flange Tip

Region of Interest

Flange

Point of Transition

Tangency

x;

Symmetry *- Web (

Fig.

1Detail

geome try for gusset plate investigation

WELDING RESEARCH SUPPLEMENT

I

57-s

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2 1667 W

f

4@O I667Wf

3@0.0833W

f

0.04l7W

f

6

()0.0208W

f

2 0.0417 Wf

2

a)

0.0833W

f

O.I667W

f

l . 2083W

f

R/W

f

= 0 .25

L/W = 4.33

W

g p

/W

f

=

1.00

3 10

3 @ '

0.0833Wf 0.0208W

f

0.0833Wf

Fig. 2Sample coarse mesh for gusset plate investigation

SCF.

For gussets wi th smal l t ransi t ion

rad i i ( R / W , < 0.5), i t was also neces

sary to emp loy a f i ne mesh wh ich used

the d i sp lacemen t ou tpu t o f the coa rse

mesh as input . SCF was der iv ed f ro m

the max imum cen t ro ida l s t ress found

in any o f the f ine e lements.

A sample coarse mesh is given in Fig.

2. T h e b o u n d a r y co n d i t i o n s p r e ve n t

noda l d i sp lacemen ts a t and pe rpen

d icu lar to l ines o f symmetryFig. 1. In

genera l , the f lange d iscre t iza t io n is

una f fec ted by the va r ious pa rame t r i c

changesTable

1. In fact , the d iscre t i

zat ion of the gusset plate is also

basica l ly constant except the extent

var ies wi th the parametr ic va lues.

D iscre t i za t i ons fo r o the r geomet r i es

can be deve loped by ske tch ing the

per im eter o f the gusset on F ig. 2 and

obse rv ing the mesh pa t te rn w i th in the

boundar ies . I f ex tens ion o f the w id th is

r e q u i r e d ,

the e lement s izes are

i d e n

t ica l to those in the current outer row.

In the t ransi t ion zone the mesh for a

smal l rad ius is fo un d by s imp ly

extend ing the g iven mesh l ines.

The coa rse mesh does con ta in some

er ro r due to the i naccu ra te rep resen

ta t i on o f the c i r cu la r t rans i t i on w i th

s t ra igh t l i nes (cho rds) be tween nodes.

The nodes themse lves a re pos i t i oned

d i rec t l y on the cu rve . One way to

measure the geometr ica l er ror is by the

largest dev ia t i on o f any cho rd f rom the

curve , as a perc ent o f the rad ius -

Figure 3 shows this error can be

est ima ted f rom the cho rd l eng th and

the cu rve rad ius . The max imu m cho rd

length var ies s ign i f ican t ly f rom rad ius

to rad ius s ince the la rger rad i i reach

the larger mesh sizes. However, the

largest error is fo un d for the s ma l lest

radius and is under 5%.

In the tangency reg ion the error is

a lways less than 1 % . Such error in

geom et ry is cons ide re d to have a

negl ig ib le e f fect on resu l tspart icu

lar ly s ince the e lement 's cent ro ida l

stress was used for SCF w i t ho ut ext ra p

o la t ion. Theoret ica l ly, s ingu lar st ress

cond i t i ons do ex i s t a t the skewed

in tersect ions o f chords, but the ang le

d i f fe rence be tween cho rds i s a lways

ve ry s l i gh t and the i n te rsec t i ons them

se lves rece ive no specia l f in i te e lement

t rea tmen t .

The reg ion o f in terest fo r h ighest

s tress con cen t ra t i o n is in the v i c i n i t y o f

the po in t o f t rans i t i on tangency-F ig .

1. Based on pho toe la st ic stud ies ma ny

re fe rences suggest tha t the max imum

concen t ra t i on occu rs p rec i se l y a t the

po in t o f tangency .

1 3

-

1 S , K

H o w e ve r ,

the f ind ings o f th is study ind icate the

wors t con d i t i o n is s l i gh t l y rem oved

f rom th i s po in t . The dev ia t i on o f the

p o i n t o f ma x i mu m co n ce n t r a t i o n f r o m

the po in t o f t rans i t i on tangency cou ld

be caused by the cho rd app rox ima t ion

o f the smoo th cu rve .

In order to examine th is premise, the

resu l ts f rom two d i f fe rent coarse mesh

d i sc r e t i za t i o n s w e r e co mp a r e d . O n e

mes h size was eq ual to th at in Fig. 2,

whi le the o ther was twice as la rge in

the reg ion o f in terest . The resu l t ing

po si t io n o f SCF f ro m the po in t o f

tangency was found a t rough ly R /5 for

both discretizations. (Th is app rox im a

t ion seems to be reasonable for any

rad ius no mat ter what the gusset p la te

l e n g t h ,

w id th , o r th i ckness. ) H ence ,

Transition Curve

% Error= 100[-=-] 100 [l -cos(S)] = 100 [l

J\

- ( ^ - )

Fig. 3Geometric error due to approxima

tion of transition curve by chords

0 0 0 4 0 W

f

I9O0002I

W

f

5

0 0

0042 W.

5 @0 . 0 2 0 8

W

f

Max. Stress Concentrat ion

/-Point Of Tangency

2 0 )

0.0208 W,

Fig. 4Sample fine mesh for gusset plate investigation

5 8 -s I F E B R U A R Y 19 78

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he chord approx imat ion d id no t

appear to cause the shi f t in maximum

onc en t ra t i o n l oc a t i on .

The crack path can be assumed

perpend icu la r to the f l ange t i p o r the

i rect ion of s t ress input .

s

Hence, SCF

represents s t ress in the d i rect ion of

app l ied st ress, but not gen eral ly r ight

at the point of tangency. Nevertheless,

the nominal s t ress used to evaluate

SCF was that wh ich was inp ut

(i.e.,

that across the f lange width pr ior to

the t rans i t i on) .

Figure 4 presents a typical f ine mesh

d isc re t i za t ion . Imposed boundary d is

p lacements are e i ther taken di rect ly or

der i ved by l i near in te rpo la t ion f rom

the output of the coarse

meshFig.

2.

The mesh size is typical of f ine

d isc re t i za t ions fo r o ther rad i i . Since

the max imum s t ress concent ra t ion

fac to r is norm al l y somew hat remo ved

f rom the po in t o f tangency , the f i ne

mesh usual ly doesn' t s t raddle that

l o ca t i o n . However, the cases of

R

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3.0

SCF

2.0

Peterson

J L

0.1 0.2 0.3 0.4 0.5

W,

Fig.

5Variation

of maximum stress con

centration factor with gusset plate tran

sition radius

2 . 0 0

I W.

1.75

S CF

1.50

1.25

1.00

D a t a Point ( t yp . )

W,

0.25

0 . 5 0

P e te r son

1.0

2.0

3 .0 4 .0

5.0

Wf

Fig. 6Variation of maximu m stress concentration factor with gusset p late

length

1.75 r -

1.25

-

1.00

Data Point ( typ.)

o o

Peterson

igp

=4 . 33

=1.00

0.5

1.0

W,

gp

w

3.0

2.5

SCF

2.0

1.5-

1.5

2.0

1.0

_ o -

-a

A

_ J > '

A

__o- -

J >

R

___--

A

___

-cr

Wf

4.33

1.67

1.00

0.67

=0 . 083

'gp

1.00

0.25

0.50

T,

0.75

1.00

gp

Fig.

7Variation

ot maximu m stress concentration factor with Fig. 8Variation of maxim um stress concentration factor with gusset

gusset plate width plate thickness

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

Y a m a d a . ' T h e n u m e r i c a l f o r m o f t h e F

s

e q u a t i o n c a n b e w r i t t e n a s:

-

2 m

r

/ **

j

+1 \

K

t l

[ a r c s i n ( j J

J

1

H^ ]

(2)

i n w h i c h a is t h e c r a c k l e n g t h a n d / is

t h e d i s t a n c e a l o n g t h e c r a c k p a t h . K,j

is t h e s tr es s c o n c e n t r a t i o n i n e l e m e n t j

o f th e fi n i t e e l e m e n t d i s c r e t i z a t i o n ; /,,.,

a n d /j a r e t h e d i s t a n c e s t o o p p o s i t e

s i d e s o f e l e m e n t j . L i m i t m i s

t h e n u m b e r o f e l e m e n t s t o c r a c k

l e n g t h a .

F i g u r e 1 0 p r e s e n t s F

B

d e c a y c u r v e s

(F

g

/SCF) f o r s a m p l e g u s s e t p l a t e

d e t a i l s . A s w i t h t h e s t i f f e n e r s a n d

c o v e r p l a t e s , S C F a l o n e d o e s n o t

d i c t a t e t h e e n t i r e d e c a y c u r v e . '

1 1

T h e F

p r e d i c t i o n p r o c e s s m u s t a c c o u n t f o r a

d i f f e r e n t m i x o f g e o m e t r i c a l p a r a m e

t e r s t h a n a f f e c t s S C F a l o n e .

Co r r ec t i on F ac to r P r ed i c t i on

A n a l y s t s o f t e n d e s i r e a m e a n s o f

p r e d i c t i n g F

B

w i t h o u t f i n i t e e l e m e n t

s t u d i e s. O n e a p p r o x i m a t e f o r m u l a f o r

p r e d i c t i n g F a u t o m a t i c a l l y i s : '

SCF

1

1 + - a

d

(3)

i n w h i c h a is t h e n o n d i m e n s i o n a l i z e d

c r a c k l e n g t h , a / W

r

. F o r a t y p i c a l g u s s e t

p l a t e g e o m e t r y c o n s t a n t s d a n d q c a n

be t ak en as 1 . 158 an d 0 . 6 0 5 1 , r e s p e c

t i v e l y .

A m o r e p r e c i s e p r o c e d u r e f o r p r e

d i c t i n g t h e e n t i re F

E

c u r v e f o r a r b i t r a r y

g e o m e t r i c c o n d i t i o n s is c o m p a r a b l e t o

t h a t a d o p t e d e a r l i e r b y t h e a u t h o r s . '

T h e a c t u a l F

g

d e c a y c u r v e s a r e c o r r e

l a t e d w i t h t h e s tr es s c o n c e n t r a t i o n

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

i n f i n i t e p l a t e . S i n c e S CF is g e n e r a l l y

l es s t h a n 3 . 0, t h e h o l e i s o r i e n t e d w i t h

i ts m i n o r s e m i d i a m e t e r , h , p e r p e n d i c

u l a r t o t h e a p p l i e d s t r e s s .

F i g u r e 1 1 a s h o w s t h e e l l i p s e s h a p e i s

d i r e c t l y r e l a t e d t o S C F . K n o w i n g S C F

[ e q

( 1 ) ] ,

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

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

o f t h e f o l l o w i n g e q u a t i o n :

h

SCF = 1 + 2 -

g

(4)

F i g u r e 1 1 b d e m o n s t r a t e s t h a t t h e

r a p i d i t y o f s tr es s c o n c e n t r a t i o n d e c a y

60-s I FEBRUARY 1978

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9Stress concentration factor decay

prospective crack path across flange

at sample gusset plate detail

p e n d s o n a c t u a l h o l e s i z e , n o t j u s t

H e n c e , t o p r e d i c t F

g

f r o m t h e

K, c u r v e , s e m i d i a m e t e r

t b e k n o w n . Fo r g u s s e t p la t e s t h e

r e l a t i o n b e t w e e n t h e c u r v e s is

u p o n e q u a l l if e p r e d i c t i o n f o r

g r o w t h t o t h e w e b l i n e .

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

i n o r a xi s o f a n e l l i p t i c a l h o l e ( u n i a x -

l t e n s i o n i n m a j o r a xi s d i r e c t i o n ) c a n

as :

1

I i -j

1 + 2e

2T

-

4e-

2

+

e

-[

H

l+ 2e

+ e Y

} {

,n

| +

3

+

s i n h ( 2 n ) { c o s h ( 2 n ) + l c o s h ( 2 y ) +

} ] / ( c o s h ( 2 n ) + l )

;

(5)

w h i c h n is t h e g e n e r a l e l l i p t i c c o o r

a t e a n d y i s t h e v a l u e o f n a s s o

d w i t h t h e h o l e p e r i m e t e r . T h e

o r d i n a t e c a n r e a d i l y b e

e v a l

(6)

i nh(y ) = I y g v2

L ( h ) - 1 J

s i n h ( n ) = 1 + 5 \ sin h(y ) (7)

F o r a n y g i v e n g e o m e t r y a n d c r a c k

a , t h e s t re s s c o n c e n t r a t i o n

K, is k n o w n if h is k n o w n .

r , i f h h as b e e n c o r r e l a t e d t o

K, a n d F

g

, t h en t h e s t r ess

i e n t c o r r e c t i o n f a c t o r is a l s o e s t a b

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

T a b le 1 h as r e l a t e d t he

t i m u m e l l i p s e s i ze t o t h e v a r i o u s

m e t r i c a l p a r a m e t e r s a n d i n i t i a l

T h e r e s u l t i n g r e g r e s s i o n

_

- 0 . 0 1 6 2 0 - 0 . 1 1 0 5

0 .03 3 07

(

W ,

w,

(Equation continued on

next page)

11Two steps required for ellipse

Oi 02

0 3

RELATIVE DISTANCE - i /Wf

R/Wf L/W f Wgp/Wf Tgp/Tf SC F

0 5

0.500

0.083

0.083

0.I67

0.083

4.33

0.67

0.67

8.67

4.33

I.O

I.O

I.O

2.0

I.O

I.O

0.5

I.O

I.O

I.O

1.465

1.770

2.022

2.229

2.788

Fg/SCF

0 . 2 0.3

d/Wf

Fig. 1 0 - f j . decay curves for sample g usset plate details

0 . 4 0 . 5

a Measured From Ellipse Tip

SCF = F (h/g)

(3) \ _ / XD

K, = G(SC F,a /g) = G (h /g , a /h )

v

[p-Direct ion of Applied Stress (typ.)

a. Ellipse Shape

b. Ellipse Size

WELDING RESEARCH

SUPPLEMENT

I

61-s

8/11/2019 WJ_1978_02_s57

6/6

(Equation continued from previous page)

L \

nnn

*.n,

1

0.02821

(

W

r

)

- 0 . 0 0 2 4 3 6

W

- 0 . 0 0 8 7 7 6 /

K

*,

)

-

0 .032 91

I

- ?

Y

+ 1.673

( W, )

( W

f

)

+ 0 . 004437

43.49

w T )

(

w;

)

0 .08587

(

r.

(8)

T h e s t a n d a r d e r r o r o f e s t i m a t e , s , f o r e q

(8) is 0 .01070.

T h e c r a c k s h a p e a s s u m e d i n t h e

a b o v e c o r r e l a t i o n s t u d y w a s q u a r t e r -

e l l i p t i c a l (a t t h e c o r n e r o f t h e f l a n g e

t i p ) u n t i l la r g e e n o u g h t o f o r m a

t h r o u g h ( e d g e ) c r a c k . ' H e n c e , t h e

p o i n t o f t r a n s i t i o n b e t w e e n q u a r t e r -

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

b y f l a n g e t h i c k n e s s . F or t h i c k n e s s e s u p

t o 1 i n . , h w a s f o u n d t o b e u n a f f e c t e d

b y c r a c k s h a p e c o n s i d e r a t i o n s . H o w

e v e r , l a r g e r t h i c k n e s s e s r e q u i r e d a

m o d i f i c a t i o n t o e q (8 ) .

F o r a n y f l a n g e t h i c k n e s s l a r g e r t h a n

1 i n . , e q (8 ) s h o u l d b e m u l t i p l i e d b y

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

1.0

U

3 4 .5 4

log ( W , )

(9)

w h e r e U

= 1.0

1.0 for

1 i n . < T

r

< 2 i n .

T

f

> 2 in .

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

ch a n ge i n h o f l e ss t h a n 1 5% .

C o n c l u s i o n

S tr es s c o n c e n t r a t i o n a n a l y s e s h a v e

b e e n c o n d u c t e d f o r v a r i o u s g e o m e

t r ie s o f g u s s e t p l a t e s g r o o v e - w e l d e d t o

f l a n g e t i p s . E a c h r e s u l t i n g s t r e s s c o n

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

t r a n s f o r m e d i n t o a s tr es s g r a d i e n t

c o r r e c t i o n f a c t o r f o r c r a c k t i p i n t e n s i t y

t h r o u g h u s e o f a G r e e n ' s f u n c t i o n . T h e

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

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

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

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

t i o n r e s u l t e d in a n o p t i m u m s i z e o f

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

r e c t i o n f a c t o r c u r v e . G i v e n t h e o p t i

m u m e l l i p s e s i ze , a u t o m a t i c p r e d i c t i o n

o f s tr es s c o n c e n t r a t i o n e f f e c t s o n

f a t i g u e c r a c k g r o w t h f o r a r b i t r a r y

g e o m e t r i e s i s p o s s i b l e .

References

1.

Alb rec ht, P., and Yam ada , K.,' Ra pid

Calcu la t ion o f S t ress In tens i ty Factors ,

lournal of the Structural Division, ASCE,

V o l .

103, No. ST2, Proc. Paper 12742,

February 1977, pp. 377-389.

2.

Batcheler, R. P., Stress Co nc en tra t io n

at Gusset P la tes w i th Curve d Tra ns i t ion s,

CE 103 Repor t , Leh igh Un ivers i ty , Beth le

h e m ,

Pa., M ay 1975.

3. B athe , K.

).,

Wi lso n, E . L , and Peterson,

F. E SAP I V - A S t ruc tura l Ana lys is

Program for S ta t ic and Dynamic Response

of L inear Systems, Ear thqu ake E ng inee r ing

Research Center Report No. EERC 73-11,

Univers i ty o f Ca l i fo rn ia , Berke ley , Ca l i fo r

n ia , June 1973 (revised Apr i l 1974).

4. Biez eno , C. B., and Gr am m el, R.,

E las t ic P rob lems o f S ing le Mach ine E le

men t s , V o l . I I , Engineering Dynam ics,

Black ie & Son, L td . , London, Eng land, 1956,

p p . 84-90.

5. Boyer, K. D., Fisher, |. W ., I r w i n , C. R.,

Roberts, R., Kr ishna, G. V., Morf , U and

Slo ck bow er, R. E., Fra ctu re Analyse s of Ful l

S ize Beams wi th We lde d Latera l A t t ac h

me nts , F r itz Eng ine er ing Labora tory Re

por t N o. 399-2(76), Leh igh Un ivers i ty , Beth

l e h e m , Pa., Ap ril 1976.

6. Fisher, J. W. , Alb rec ht, P. A., Ye n, B. T

K l i nge r man , D . I., and McNamee , B . M . ,

Fat igue S t rength o f S tee l Beams wi th

W e lded S t i f f ene r s and A t t achmen ts ,

NCHRP Repor t No. 147 , T ranspor ta t ion

Research Board , Nat io na l Research Cou nc i l ,

W ash ing ton , D . C , 1 97 4 .

7. Gurney, T. R., Fatigue of Welded Struc

tures, Ca mb r idge Un ive rs i ty P ress, Lo ndo n,

E n g l a n d ,

1968.

8. Koba yas hi, A. S., A Simple Proc edu re

for Est imat ing Stress Intensi ty Factor in

Reg ion o f High St ress Gr ad i en t , S ign i f i

cance o f Defects in Welded S t ruc tures,

Proceedings, 1973 Japan-U.S. Seminar, U n i

versi ty of T ok yo Press, To ky o, Japan, 1974,

p. 127.

9. Law renc e, F. V., Es t im at i on o f Fa

t igue-Crack Propagat ion L i fe in But t

W e l d s , Welding lournal, 52, (5) , Ma y 1973,

Research Supp l . , p p . 212-s to 220-s.

10. M ad do x, S. J ., Assess ing the S ign i f i

cance of Flaws in Welds Subject to Fa

t i g u e ,

Welding lournal, 53 (9), Sept., 1974,

Research

Supp l . ,

p p . 401 -s to 409-s.

1.1.

N eube r , H. , Kerbspannungslehre, 2nd

e d i t i o n , Spr inger -Ver lag , Ber l in , Germany,

1958,

pp. 52-56.

12 . Paris, P. C , and S ih, G. C, Stress

Ana lys is o f Cracks, Fracture Toughness

Testingand Its Applications, STP381, A m e r

ican Soc ie ty fo r Test ing and Mater ia ls , Ph i l

adelphia, Pa., 1965, p. 30.

13 .

Peterson, R. E., Stress Concen tration

Factors, J. W ile y & Sons, Ne w Y ork, N. Y.,

1974.

14. Ra ndal l , P. N., Sev er i ty of Na tural

Flaws as Fracture Or igi ns , and a Study of the

S u r f ace -Cr acked S pec im en , A F ML- T R- 66 -

204,

August 1966 (pub l ishe d in AST M STP

410,

p. 88).

15 .

Seely , F. B., an d Sm ith , ).

O.,

Advance d M echanics of Materials, 2nd

e d i t i o n ,

W i le y , N e w Y o r k / L o n d o n / S y d n e y /

Toronto , 1952.

16 . Tad a, H., Paris, P. C , an d

I r w i n ,

G. R.,

The Stress Analysis o f Cracks Handbo ok,

Del Research Corpora t ion , He l le r town, Pa. ,

1973.

17 . Tada, H. , and I r w i n , G. R., K-Value

Analys is fo r Cracks in Br idge S t ruc tures,

Fr i tz Eng ineer ing Labora tory Repor t No.

399.1, Leh igh Un ivers i ty , Beth lehem, Pa. ,

June 1975.

18 . T imoshenko , S . , Strength of Materi

als, Part IIAdvanced Theory and Problems,

3 r d ed i t i on , D . V an Nos t r and , P r i nce ton /

N e w Y o r k / T o r o n t o / L o n d o n , 1 9 5 6 .

19 . Ze tt le mo ye r, N ., and Fisher, J. W.,

Stress Gradient Correct ion Factor for Stress

In tens i ty a t We lde d S t i f feners and Co ver

P la tes , Welding lournal, 56 (12), Dec . 1977,

Research Suppf, pp. 393-s to 398-s.

A p p e n d i x

T h e f o l l o w i n g s y m b o l s a re u s e d i n

t h i s p a p e r :

a = c r ac k s i ze

D = c h o r d l e n g t h o n g u s s e t p l a t e

c i r c u l a r t r a n s i t i o n

F

B

= s tr es s g r a d i e n t c o r r e c t i o n f a c

t o r

g = m a j o r s e m i d i a m e t e r o f e l l i p

t i c a l h o l e i n a n i n f i n i t e p l a t e

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

t i c a l h o l e i n a n i n f i n i t e p l a t e

K, = s tr es s c o n c e n t r a t i o n f a c t o r

A K = r a n g e o f s t r es s i n t e n s i t y f a c t o r

/ = d i s t a n c e a l o n g c r a c k p a t h f r o m

o r i g i n

l o g = l o g a r i t h m t o b a s e 1 0

L = a t t a c h m e n t l e n g t h

m

= n u m b e r o f f i n i t e e l e m e n t s t o

c r a c k l e n g t h a; m a x i m u m d i s

t a n c e b e t w e e n g u s s e t p l a t e

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

a p p r o x i m a t i o n

R = r a d i u s o f c i r c u l a r t r a n s i t i o n a t

e n d o f g r o o v e - w e l d e d g u s s et

p l a t e

s = s t a n d a r d e r r o r o f e s t i m a t e

S CF = m a x i m u m s tr es s c o n c e n t r a t i o n

f a c t o r a t t h e c r a c k o r i g i n

T, = f l a n g e t h i c k n e s s

T

r a

= g u s s e t p l a t e t h i c k n e s s

W , = f l a n g e w i d t h

W

g p

= g u s s e t p l a t e w i d t h

o = n o n d i m e n s i o n a l i z e d c r a c k

l e n g t h , a / W ,

y = v a l u e o f e l l i p t i c c o o r d i n a t e n

r e p r e s e n t i n g e l l i p t i c a l h o l e p e

r i m e t e r

n = e l l i p t i c c o o r d i n a t e

6 = h a l f o f a n g l e d e l i n e a t i n g c h o r d

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

t r a n s i t i o n

6 2 - s l F E B R U A R Y 1 9 7 8