[hay] 386_10_79_
![Download [HAY] 386_10_79_](https://vdocuments.us/public/t1/desktop/images/cloud_download_icon2.png)
TRANSCRIPT
-
8/10/2019 [HAY] 386_10_79_
1/99l
i h ycle atigue o
Welded
Bridge Detail
THE
PREDICTION
OF
F TIGUE
STRENGTH
OF
WELDED DET ILS
FRITZENG N EER\NG
L O R T O F ~ V L fBR RY
Nicholas Zettlemoye
John
W Fishe
eport No 386 19
-
8/10/2019 [HAY] 386_10_79_
2/99
COMMONWE LTH OF PENNSYLV NI
Department of Transportation
Bureau of Materials,
Te s tin g
and Research
Leo
D
Sandvig
D i r ect or
Wade L. Gramling
Research
Engineer
Kenneth
L Heilman
Research Coordinator
P ro je c t 72-3:
High
Cycle
Fatigue
of
Welded Bridge
D etails
THE
PREDICTION
OF FATIGUE STRENGTH
OF W L DET ILS
by
N i ch o la s Z e tt le m oy e r
John W Fi sher
Prepared in
c oo pe ra ti on w i th
the Pennsylvania Department
of Tr anspor t at i on an d th e U S . D ep ar tm en t o f Tr anspor t at i on,
F e d e r a l Highway
Administration.
The
c ont e nt s
o f
t h i s r epor t r e
f lec t
the
views of th e
authors who
a re r es po ns ib le for
the
facts
an d th e
accuracy of the
dat a presented
herein.
The contents
d o
n o t
necessar i ly r e f l ec t the off ic ia l
views
o r .p o li ci es o f
th e
Pennsylvania Department of
Tr anspor t at i on,
the
U
S. Department
of
Tr anspor t at i on,
Federal
Highway A d mi ni st ra ti on , o r
th e Rein
forced Concrete
Research Council.
This
report does
no t
c o n s t i
tu te
a st andar d,
s p e c i f i c a t i o n
o r
regulation.
LEHIGH
UNIVERSITY
O f f i c e
of
Research
Bethlehem,
e ~ n s y l v n i
J a n u a r y
1979
ritz E n g in e e ri n g L a b o ra t o ry Report
No
386-10 79)
-
8/10/2019 [HAY] 386_10_79_
3/99
TABLE OF CONTENTS
ABSTRACT
I
INTRODUCTION
1 1
Objectives and Scope
1 2 Fatigue Life Pr edi ct i on
1 3
F Evaluation
Approaches
1 4 Previous Work on the Stress Concentration/
Stress I n t e n s i t y Relationship
iv
2
5
7
2
STRESS
CONCENTRATION
EFFECTS
9
2 1 Crack Free Stress Analysis
9
2 1 1
Anal yt i cal Models
9
2 1 2 S tre ss C o nc e nt ra t io n R e s ul ts
2
2 2 Stress Gradient C o rr ec ti on F ac to r
4
2 2 1 Green s
Function
4
2 2 2
Typical
Results
6
2 2 3
Prediction
6
3
STRESS INTENSITY
3 1
Other
Correction
Factors
2
3 1 1
Crack
Shape Correction
e
3 1 2 Front Free Surface
Correction
F
22
s
3 1 3
Back
Free S u rf ac e C o r re c ti o n
F
23
w
3 2
Crack Shape Effects on
F
7
g
3 3
Crack
Shape
Var i at i ons During Growth
7
-
8/10/2019 [HAY] 386_10_79_
4/99
3.4 Total
Stress Intensity
3.4.1
Through
Crack
lb
=
0
3.4.2
Half-Circular
Crack lb
3.4.3 In te rp ola tio n
for
Half-Ell ipt ical Cracks
0 l
3.4.4
F
Variation
at
y p i ~ l Details
s
4 FATIGUE LIFE CORRELATIONS
4.1
Transverse
Stiffeners
Fillet-Welded to Flanges
4.2 Cover
Plates
with Transverse End Welds
5. SUMMARY AND CONCLUSIONS
.6 . TABLES
7 FIGURES
8 NOMENCLATURE
REFERENCES
9
3
35
38
39
4
43
46
49
5
58
81
-
8/10/2019 [HAY] 386_10_79_
5/99
STR CT
This report provides
a means
of estim ating the s t ress concentration
effects
when
predicting the fatigue l i f of several welded detai ls .
The
resul ts
of an analytical study of the fatigue behavior of welded
s t i f f -
eners and
cover plates were
compared with
th e
t s t dat a r epor ted in
NCHRP
Reports 2 and
147.
The comparison indicated that the varia t ion in tes.t
data
cou ld be accoun ted
for
considering
the
probable variation in
in i t i l crack s izes
and crack
growth
rates .
The
s tress
gradient
co rr ec tion fac to rs developed for s t iffeners
and
cover plates welded to beam flanges provide the necessary analytical tools
for
est ima ti ng the
applicable s tress
intensity
factors.
In this study a
lower bound crack
shape
relationship was uti l ized which was derived from
cracks that
formed t th e weld
toes
of
ful l
size
cover plated beams.
iv
-
8/10/2019 [HAY] 386_10_79_
6/99
INTRODU TION
1.1
Objectives and Scope
I t
is well
known tha t geometric
s t ress
concentration plays a s igni
f icant role in fatigue
crack
growth
a t
welded deta i ls In fact the
categories
of detai ls
established in
the
SHTO
Code 2 primarily ref lec t
differences in s tress concent ra t ion cond i tions .
Hence,
in order to
analyt ical ly
predict
the fat igue l i f e of a
deta i l
one must account
fo r
the
local
s t ress
dist r ibut ion
as
determined
by
the
sudden
change
in
geometry.
The primary objective of th is report is
the
inc lusion
of s tress
concentration
effec ts in the
predict ion of
fatigue l i f e
The
coverage
includes
f i l le t -welded
deta i ls of the
s t i f fener
and
cover
pla te type.
However, the basic techniques employed in the study could have been
extended to
any
type of
de t a i l
even those which
dontt
involve
welding.
Besides
s t ress
concentration,
fatigue l i f e
predict ion
also
neces-
s i t a t es an understanding of the
influence
of crack
shape
on growth
ra tes
A secondary
objective
of th is
repor t i s c la rif ic a tio n of the
relat ionship between
s tress
concentration and crack
shape
ef fec ts Also,
there is
some
inves t igat ion of crack shape variat ion during
crack
growth.
One implici t assumption made
throughout
the report is tha t the
nominal s t ress range
a t
the
deta i l
i s known Clearly,
inclusion of
-
8/10/2019 [HAY] 386_10_79_
7/99
stress concentration
in
the
analysis
is of l i t t l use i f the
basic
stress range
is seriously
in
error.
1.2 Fatigue Life Prediction
Recent years
have
seen
great
strides in the development of techniques
to analy tical ly predic t fatigue
l i fe . Fatigue crack growth
per cycle,
dafdN,
can be
empirically related to
s tress intensi ty factor,
K,
from
linear
fracture mechanics, as
follows: 27
da _ n
dN
C M
1
where i
is
the
range
of s tress intens ity
factor
and C and n
are
based
on material properties.
By
rearranging Eq 1 and integrating between
=
C
LiK n
da
f
the stress range i s
included in
the express ion, i t takes
on
the
following form:
-2-
-
8/10/2019 [HAY] 386_10_79_
8/99
~ ]
S n =
r
r
3)
F i s h e r e x p e r i m e n t a l l y found tha t p ar am eter A changed
value
fo r
d i f f e r e n t types
o f welded deta i l s . 10 )
The
slope o f
th e S -N
curves,
r
- n , is approximately
const ant
a t -3. 0
f o r
a l l c a t e g o r i e s . 1 0 ,1 1 ) C
ca n
be
taken
as constant fo r t y p i c a l bridge s t e e l s A36 A441
A514).
10,11)
In i t i a l crack
s i z e ,
a .
is known approximately fo r some
welded
d e t a i l s 3 1 ,3 6 )
1.
an d
a
f
b ein g
much
larger
than
a i
i s
usual l y
of l i t t l e
consequence.
e n c ~ fat i gue l i fe p r e d i c t i o n fo r cr ack propagation res ts pri m ari l y on
th e
e va lu at io n o f
th e s tress
in tens i ty
f a c t o r range a t th e de ta i l .
The range o f s tr es s i nt en si ty f a c t o r is oft en e xp re sse d a s 8K
for .
a c e n t r a l through crack
in
an in f in i te
plate
under
u n i a x i a l
s t ress
adjusted
by
numerous superimposed) correction
f a c t o r s .
1,22,28,29,37)
a
= CF
S
=
F F F F .
-;t
S jna
r s w e
g
r
4)
CF i s
th e
combined cor r ec t ion f a c t o r as a f unc tion o f c ra ck
l e n g t h - p l a t e
w id th
ra t io
afw,
crack shape
ra t io lb
an d
geometry.
F is t h e c a r
s
r e c t i o n
a s s oc ia te d
w i t h a f r e e
s ur f a c e a t th e
crack
origin th e f r o n t
f re e s ur fa ce ),
F
accounts
fo r
a
free
s ur f a c e
a t
some
f in i te
le n g th o f
. w
crack
growth a ls o c alle d th e back s ur f a c e o r
f ini te
width
c o r r e c t i o n ) ,
- 3 -
-
8/10/2019 [HAY] 386_10_79_
9/99
and F
adjus ts for
shape
of
crack
front
often assumed to
be e l l ip t i c
with maJor
semidiameter b
and
minor semidiameter a .
While
in fracture
problems
s tr es s i nt en s it y
often
includes
a
plas t ic
zone
correc t ion
f ac tor
F it is
usual ly
disregarded in
fa t igue
analyses since
small
p
s t ress
ranges and reversed
yielding cause the
crack t ip plas t ic zone to
b
11
22,23,30,37
e sma .
F is
the
factor
which accounts for
e i ther
a
nonuniform applied
g
s t ress such as bending or a s t ress
concentrat ion ~ u s e the deta i l
geometry.
This
s t ress gradient
should not be confused with
t ha t which
always
occurs a t
the crack t i p .
F
co rre cts fo r
a
more
g loba l cond it ion
than exists
a t
the
crack
t ip .
Yet,
for stress concentrat ion
s i tuat ions
F corrects
for
a more loca l condit ion than
the
nominal s t ress s t rength
o f ma te ri al s
type a t the
de ta i l .
Whether th e a pp lie d
s t ress
is non-
uniform or a s t ress concentration exi ts S represents an
arb i t rar i ly
r
se lec ted
s t ress
range
usual ly the
nominal
maximum
value
a t
the
de ta i l .
Therefore, F i s
inseparably
l inked
to
the
choice
of S .
r
Values of the various
c o rr ec ti on f ac to r s
are
dependent
on the
specif ic
overal l geometry, crack shape, and dis t r ibu t ion of
applied
s t ress . Solutions
for
of many ideal ized problems are ava i l -
b l
23,32,34,35
a
e .
However, prac t ica l
bridge
deta i l s present d is t inc t
analyt ical
di f f icu l t ies
p a rt ic u la rl y i n evaluating
Cracks normally
emanate from weld toes 9,lO near which
varying degrees
of s tress concen-
t r a t ion exis t . For some geometries the
crack quickly
grows out
of
this
concen t ra t ion reg ion ;
fo r o t ~ geometries
the
ef fec t of
concentration
-4-
-
8/10/2019 [HAY] 386_10_79_
10/99
is
sustained
over a broad
range of crack sizes . Fillet-welded
connections
have the added d if fi cu lt y t ha t th ey p re sent
a
theoretically s ingular
s t ress condi ti on neg lect ing
yielding
r igh t
a t
the weld
toe.
Therefore,
F has
been
impossible
to
obtain in a closed-form fashion.
Numerical
techniques
such
as
the
f in i te
element
method normally
must be employed.
1.3
F
Evaluation Approaches
Based on
Eq.
l i fe predict ion involves summing the cycle l ives
for
increments
of
crack
growth.
be
known for each inc rement.
F as
well as other corr ect ion f acto rs must
Three approaches are
avai lable for
deter-
mining
F
for
varying
crack
s ize;
a l l approaches
involve
f in i te
element
analyses.
The
f i r s t normally.
termed a
compliance
analysis ,
n e e ~ s i t t e s
analyzing the deta i l for dif fe ren t lengths
of
embedded crack. The
s t ra in
energy
release ra te it
is
found as
a
function of the
slope
of
the
compliance-crack
length
curve.
18
Irwin
showed
that
there
is
a
direc t
re la t ionship e t w e e n ~ n d
the
s tr es s i nt en si ty factor .
19,20
Thus,
K
can
be
found and
compared, i f desired, with some
base
K for a
specimen lacking
the
in flu en ce o f
a
s tress
gradient
i . e . F
1.0 .
g
The rat io
of the
two
s tr es s i nt en si ty f ac to rs y ie ld s
F
for the actual
deta i l under invest igat ion.
A
second re la t ive ly
new
approach incorporates special elements
with
inver/se s ingular i ty for
modeling of
the
crack t ip .
These elements
permit
accurate
resolution
of the displacements
and
stresses
in
the
5
-
8/10/2019 [HAY] 386_10_79_
11/99
cr ack t i p
r e g i o n .
Both d is p lacem en t an d s t r e s s a re d i r e c t l y r e l a t e d to
s tr e ss i nt en s it y
and,
as b e f o r e ,
F i s found d i v i d i n g some base K
Once
a g a i n ,
a
s e pa r a te a n a l y s i s
i s
required fo r
each
cr ack depth.
Both o f th e above
methods
can be very expensive
an d
r e quir e e x t e n si v e
t im e. A
compliance a na ly si s a ls o p re se nt s a c c u r a c y
d i f f i c u l t i e s
a t v e r y
small
an d
very
l a r g e c ra ck d ep th s. A more r eas o n ab le a l t e r n a t i v e
e x i s t s
which r e quir e s
only on e
s t r e s s a n aly sis f o r
a given d e t a i l
geometry.
Bueckner a nd Hayes
showed
t h a t l t can be found from th e s t r e s s e s i n th e
cr ack f r e e
body
which a c t
on
th e p lan e
where
t h e cr ack
i s
to
l a t e r
e x i s t .
8, 16) . I r w in
im p lied t h i s when
he
foundU by c on s id er in g t he
energy needed to
r e c los e
a
crack.
19) Based on t h i s concept
o f t e n
c a l l e d
superposition o r
d i f f e r e n c e s t a t e
i s p o ~ s i l to d e sc r i b e
known s tr es s i nt en s it y s olu tio ns fo r s t r e s s e s o r
c o n c e n t r a t e d loads
a p p l i e d
d i r e c t l y
to
cr ack
su r f a c e s
as Green s
functions
f or loading
remote
from
the
cr ack .
28,34,35)
The
s t r e s s
or
concentrated
load
i s
simply a d j u st e d to s u i t th e s tr es s d is tr ib u ti on along t h a t p lan e w i t h
t h e c r a c k a b s e n t .
A
c o n c e n t r a t e d
load i s r e p r e se n t e d s t r e s s a p p l i e d
to an in cr em en tal a r e a .
Through
i n t e g r a t i o n , numerical o r clo s ed - f o r m ,
an y s t r e s s d i s t r i b u t i o n
can
be r e p r e s e n t e d . )
Depending
on which
Green
t s
f u n c t i o n i s u s e d , F
alo n e
o r a
combination o f
c o r r e c t i o n . f a c t o r s can
be
e v a l u a t e d .
Th e
o nl y r eq ui re me nt
i s
t h a t
th e
cr ack
p a t h
be
known
Since
7 9
10
12)
a c t u a l t e s t s have
prov1ded
1nformat10n on cr ack pa tha ,
method
t h r ~ i s employed i n th e
study.
-6-
-
8/10/2019 [HAY] 386_10_79_
12/99
1.4 Previous ork on the
Stress
Concentration/Stress Intensi ty
Relationship
Accurate a na ly sis o f the influence o f stre ss concentration on
s tr es s i nt en si ty
stems from Bowie s
~ r
on cracks emanating from
circular
hol es. 6)
Since
then good estimates
of
K have
been determined fo r
cracks
growing from
e l l ip t i ca l holes, rectangular cutouts, and
a l l sorts
h
24,26,32,34) h d Id d
o no c
es .
t regar
to
f i
et-we
e connect1ons,
Frank s work on cruciform jo ints marks an
early
intensive e f for t
to
deve lo p an
expression
f o r
F
12 )
A s i m i l a r
.study
was
pursued
by
Hayes
g
an d Maddox
s h o r t l y
t h e r e a f t e r . 15 ) Unfortunately,
th e n u me ri ca l c on cl u-
s ians of these two investigations weren t in agreement. F ur ther ,
the
accuracy of
each
was ques t ionable at very small crack s izes Gurney
later tried
to resolve the differences and d id succe ed in producing
s ev e ra l h el pf ul graphs in vo lv in g th e
geometric v a r i a b l e s . 14)
However
no general formulas
were
developed and
the
accuracy at very small crack
s i ze s was not
known.
Moreover,
there was
no assurance
tha t graphs or
expressions
developed any of the
three
studies could be applied to
other, more complex, f i l let-welded bridge detai ls .
The
analytical
approach used to
find
F in Refs.
12, 14,
and 5 was
g
that
of
a compliance a n a l y s i s
based
on
f ini te
element
d is cr e ti za ti on o f
th e jo n t
The
Green s
f ~ t i o n
t ~ i ~
was
e a r l y u se d by K oba ya shi , 2 1 )
28
35)
w o
successfully estimated Bowie s
results , and
others
Later,
Albrecht developed approach
to
F
for
fi l let-welded j Q,ints using a
g
Green s
function. 1)
However no parametric
study
was conducted.
-7-
-
8/10/2019 [HAY] 386_10_79_
13/99
s
of
f in i te
elements
with
inverse singular i ty to analyze welded
det i ls has not
yet
appeared in the l i t er tu re although th is approach
h as p romise
fo r the
fu tu re
One cur ren t
need
i s
the
development
of
three dimensional
elements
with
the
s i n u l r i ~ y
condition
8
-
8/10/2019 [HAY] 386_10_79_
14/99
2
STRESS
ON ENTR TION
EFFE TS
2.1
Crack
Free
Stress
Analysis
2.1.1
Analytical
Models
The overal l
det i l
geometries
for the
s t i f fener and cover
pl te
investigations
are
shown
in Figs.
1
and
2,
respect ively. In
b oth cases
the
det i l was assumed
to
exis t on both
sides
of
the
beam web
which,
therefore ,
marks
a
plane of
symmetry. Uniform
s tress
was
input
to
the
det i l s
t a
posi t ion
far
enough removed
from
the f i l l e t weld so as to exceed the
l imi t
of
s tress concentra
t io n e ff ec ts
In
both det i ls the flange width was held constant.
The
f i l l e t weld angle
was
se t
t 45
-degrees.
The
th ic kn ess o f
a s t i f fener i s
generally
small compared
to
the
flange thickness, T
Hence, the
s t i f fener
i t s e l f was omitted
from Fig.
1. The
variable under
investigation was
the weld leg,
Z
Z
can be expected to range between 0 .25
T
and
1.00
T
The
three
values of Z
selected
for
th is
study
were
0.32 T
f
0.64 T
f
and
0.96
T
A
s tress concentration
analysis
was c.ompleted
for each
value.
A pi lo t study
on t he cover -p la ted det i l
Fig.
2
revealed
th t
s tress
concentration
reaches a plateau
with
increasing
at tach
ment
length. Hence, the length of cover pl te
was
se t so
as
to
-9-
-
8/10/2019 [HAY] 386_10_79_
15/99
ensure maximum c o n c en t ra t io n c o n d it io n s . Th e width o f
c ove r
pla te
was held c o n st a n t an d th e
weld
s i z e
was assumed c o n st a n t
a long
the
cover pla te edge. Th e
v a r i a b l e s
st u d i e d were t h e weld
l e g ,
Z, an d
cover p la te thickness, T Th e
w eld
leg s i zes considered
w er e
th e
cp
same as f o r th e s ti ff e ne r d e ta il an d th e cover pla te
t h i c k n e ss
was
taken as 0 .6 4 T
1.44
T
f
and
2.0
T
f
.
Both of th e e t i l ~
were
i n i t i a l l y analyzed
t h r e e -
dimensionally.
However, in
o r d e r to
reduce
c o s t s
th is
f i r s t
level
o f
inves t igat ion was
only o f
suf f ic ien t accuracy
to
pr ovi de
reason-
a b l e
i n p u t to a more loca l two-dime nsiona l s t ress a n a l y s i s . F or
th e tw o deta i l s s tu di ed , f at ig ue c ra c ks n or ma ll y o r i g i n a t e along
.
9
10 )
th e
weld
to e and pr opa ga t e through th e flang e th ic k n e s s .
Hence, th e pl a ne o f in te r e s t f o r
two-dime nsiona l
s t ress a n a l y s i s
was th e sec t ion shown in F ig s . 1 an d 2. Pi lo t s t u d i e s
showed
t h a t
th e s t ress c o n c e nt ra t io n i n c re a s e s
s l igh t ly
as th e
s e c t i o n
g e t s
n e a r e r th e web
l ine
T he r e f or e , th e
speci f ic s e c t i o n
f o r
two-
dimensional a n a ly s is was t a ke n
r ight
a t th e web l ine
An
exis t ing
f in i te
e le me nt
c o mp ut er p r og r am ,
S P
IV , 4) was
used f o r th e ent i re ~ t s s a n a l y s i s
ef for t
Th e
program
is intended
f o r
e las t ic
a n a ly s is
only;
Young s
modulus
was
se t a t
29,600
k s i
an d P o i s s o n s
ra t io
was t a ke n to be 0. 30. A t hr e e - di m e ns i ona l
coarse mesh, using
br ick e l e m e n t s ,
was e st ab li sh ed f or each deta i l
an d
su b j e c t e d
t o
uniform
s t ress Nodal displacement out put from
th e D
mesh
was then i n p u t
to
a
two-dime nsiona l
f i n e mesh. Nodal
- 10-
-
8/10/2019 [HAY] 386_10_79_
16/99
displacement output
from the
fine
mesh
was
subsequently inpu t to an
u l t r a fine
mesh which
was very local
to the weld
toe . Final ly, the
element s t ress
concentrat ions were extrapolated
to give a maximum
s t ress
concentrat ion
factor , SCF r igh t a t
the
weld toe .
Figures
and
4 show
the
coarse
meshes
used fo r
each
deta i l .
The
dashed
l ines are
intended to
indicate how
the
var iable
cover
plate
thickness
and/or weld
leg
dimension were effected . In a l l
cases the flange
discre t iza t ion
and
actual
thickness
0.78 in .
were held cons tant . Also the f i r s t
l ine
of
weld
elements adjacent
to the
overal l weld
toe had constant s ize . For
both models dis
placements perpendicular
to l ines
of
symmetry were prevented.
Displacement
perpendicular
to each
plan view
was
prevented along
the web l ine and in
the
s t i f fener case also
along the
weld
l ine
o f symmetry to simulate the s t i f fener .
Figures
5 and 6 show
the
fine
and
ul t ra f ine
meshes
which
were
common to both deta i l inves t igat ions . Planar
elements o f c on sta nt
thickness were u sed
throughout .
The
heavy l ines
denote
the
outl ine
of
previous mesh elements.
Displacements a t common mesh
nodes
along
th e b ord er
were
known from
the
pr ior analysis ; displacements
a t newly created nodes a t the boundary were
found
by l inear in te r -
polat ion
between known values .
The need fo r an ul t ra
f ine
mesh deserves
fur ther
explanation.
The
assumed geometry a t the weld toe
creates-
an e la s t i c s t ress
-
8/10/2019 [HAY] 386_10_79_
17/99
singular i ty
condition. Hence, a decrease
in mesh
s ize a dja ce nt to
the toe yields
ever
higher s t ress values.
However, the s t resses
somewhat removed
from
the toe
become
stabi l ized
and
the distance to
s t ab i l i za t i on dec re as es w i th decreas ing mesh s ize .
Since the
analyst i s
i nt er es te d i n
accuracy
of
s t resses beyond the i n i t i a l
crack s ize seemed reasonable to
ensure the
mesh
s ize
was
a t
leas t as small
as the
i n i t i a l
crack
s ize . For the assumed
flange
thickness 0.78
i n .
the minimum mesh size was 0.001 in . Several
invest igat ions
have establ ished th is as a
lower
l imi t of i n i t i a l
crack size .
31,36
hypothetical
maximum
s tress
concentration
factor was obtained
f i t t ing
a f ou rth o rd er polynomial
through
the averaged concentra-
t ion
values on
e i ther side
of
the
node
l ine
down from the weld
to e
in
th e
u l t r a fine mesh.
2 .1 .2 S tre ss Concentration
Resu l t s
he
S
values
for each s t i f fener and cover pla te geometry
analyzed are
p lo tte d in Fig. 7.
S increases
f or i nc re as in g
s t i f fener weld
s ize but
decreases
f or i nc re as in g
cover
plate
weld
size.
These
trends are
simi lar
to
Gurneyfs
f indings
for
non-load-
14
carrYlng
an oa
-carrylng
cruel
arm JOlnts.
Apparently, a
t rans i t ion
from a non-load-carrying t o load-ca rrying condit ion
occurs as
the
attachment
length along
the beam increases.
-12-
-
8/10/2019 [HAY] 386_10_79_
18/99
Regression
analyses for the stiffener and
cover
plate data
38
suggest
the followlng equatlons:
Stiffeners:
Cover
Plates:
S F
=
1.621
log
3.963
5
S F
= -3.539
log i 1.981 log
~ c p
5.798
6
The
standard errors of
estimate,
s , fo r
Eqs and 6
are 0.0019
and
0.0922, respectively.
Stress
concentration
factor decay curves for sample
st i f fener
and
cover plate details are
p lo tted in Fig.
8.
Each K
t
curve
eventually drops below
1.0
due
to the equil ibrium requirements.
The
cover
plate
seen
to
cause
mu
more
disruption
to
stress
flow than does
the
stiffener. In both
cases
the stress concentra-
tion most pronounced at small crack sizes.
Unfortunately,
most
of
the fatigue
l i
is expended in the same range. 9
-13-
-
8/10/2019 [HAY] 386_10_79_
19/99
2.2
Stress Gradient
Correction
Factor
2.2.1
G re en s F u nc ti on
The
Green s fu nc tio n or superposition approach
makes us e of
s t r e s s
i n t e n s i t y
s olu ti on s f or
loading
directly
on the
crack surface.
Use
of the Greents function suggested by Albrecht an d amada leads
to
the
following stress i n t e n s i t y expression: 1
a
K c r ~ ~
o
d
7
where is the
nominal
stress on the section and K
t
is
the crack
free stress
concentration
factor
a t
position
Q.
Since
is
the s tr es s i nt en si ty fo r
a through
crack
under
uniform s t r e s s , th e
stress gradient
correction
factor,
F a t
crack
length a i s :
g
a
F
=
2 /
g J
o
K
t
dQ
2
-
l
i
8)
crack
length, a,
and distance, are
both
nondimensionalized by
the
flange
thickness,
T
f
Eq. 8 becomes:
F
=
g
IT
a
/
o
K
t
d
_ 2
9)
-14-
-
8/10/2019 [HAY] 386_10_79_
20/99
where
= a/T
f
Occasionally
is
p o s s i b l e to express K
t
by a polynomial
e q u a t i o n such as:
10 )
where A, B, C, an d D
a re dimensionless c ons ta nts .
For such a
r e p r e s e n t a t i o n
of
s t ress c onc e ntr a tion, Eq. 9
can
be solved i n a
closed-form
manner. 37,38)
F
--8-
=
SCF
1
A
B 2
4C
3
D a
4
u
a
3rr
a
8
11
E q u a t i o n 11 c a n be a pplie d to
polynomials
o f l e s s e r or de r by merely
equating
the
a ppr opr ia te decay c o n st a n t s to z e r o .
t i s
di f f icu l t
to make use o f
Eq .
11 fo r s t i f feners an d cover
plates
s inc e
the typical c onc e ntr a tion curves
Fig.
8) a r e n ot w ell
s ui te d to p o l y n o m i a l representa t ion Hence,
Eq. 8
usually c a n t
be solved in
a closed-form manner. However, a numerical so lu t ion
1
i s
easi ly
d ev is ed as
suggested
by
Albrecht.
F
=
g
2
[arCS in - arcs in :j ]
-15-
12)
-
8/10/2019 [HAY] 386_10_79_
21/99
in
which
K
i s
the
s t ress
concentrat ion
factor in
element
j
of
th e
f in i te element discre t iza t ion
or the
average
of
two adjacent elements,
both
of
equal
distance
along
opposi te
sides of
the crack
path.
Limit
m
is the number of elements crack length a.
2.2.2
Typical Results
Figure
shows the
predicted
t rend of the
s t ress
gradient
correct ion in r ela tio n to s t ress
concentrat ion
a t
typ ica l
welded
Each decay s to
eta i l s
Both
the F
g
and
K
t
curves begin a t
SeF
l eve l below 1.0 al though the F decay is less rapid . In
general ,
the
separation
of the two curves
a t any p oin t o th er
than the or igin
depends
on deta i l geometry.
Figure 10 presents F decay curves F /SCF for sample s t i f fener
and cover pla te deta i ls Usually,
the
curves for cover plates
are
below those
for s t if feners since the S F values
are
higher.
t
i s
also apparent tha t deta i l s with s imilar S F often have different
decay ra tes
This f a c t
means t ha t
the F curve i s
related
to
a
di f fe ren t mix
of
geometrical parameters than is
S F
2.2.3 Predic t ion
t
i s highly
advantageous
to
have
a
means
of obtaining F for
d if fe re n t d e ta il s without individual
f in i te
element s tudies One
approximate
method is to
represent
the
F
e quati on a s:
-16-
-
8/10/2019 [HAY] 386_10_79_
22/99
F
S F
13
For
a
typical
s t i f fener
geometry,
Table 1 shows constants d and q
can
be taken as 0.3602 and
0.2487,
r s p ~ t i v l y n average cover
plate
deta i l
is reasonably
represented
values
of 0.1473 and
0.4348.
A second, more
precise method of
obtaining F i s also available.
g
I t
can
be
seen
tha t
the
character is t ics
of the
F
curve
Fig.
9
g
are q uite
s imilar
to fe atu re s of s t ress concentration decay
based
on
gross
sect ion
s tress
from the
end of an
el l ip t ica l
hole in an
inf ini te uniaxially
stressed
plate . 5,25 Each curve begins a t
the maximum
concentration
factor , SCF and decays
to
a
value
near
one. The asymptote for the e l l ip t i ca l hole
is
exactly 1 . There-
fore ,
is
p os sib le to corre la te
a
hypothet ica l
hole
to
actual F
g
curves
Fig. 10
such
t ha t
any
F
curve can be
est imated
from the
g
appropriate hole shape
an d
s ize.
The
e l l ip t ica l
hole shape determines
the
maximum s t ress concen-
t r a t ion factor .
S F
1
2 ~
h
-17-
14
-
8/10/2019 [HAY] 386_10_79_
23/99
in which
g
an d
h a re th e hole major an d minor semidiameters,
r e s p e c t i v e l y .
C on ve rs el y, g iv en th e
maximum c onc e ntr a tion a t a
d e t a i l
Egs.
5
an d
6 ),
th e c or re la te d
hole
shape
can
b e d et er mi ne d.
Figure 11 shows th e s t ress c onc e ntr a tion
factor decay
from an
e l l ip t i ca l
hole i s
dependent
on both
th e
hole shape an d i t s
a bs olute
s i z e .
A
smaller hole has a more r ap id c o nc en tr at io n decay. In
order to p r e d i c t F
g
from the e l l ip t i ca l hole
K
t
curve, is neces
s a r y to
establish th e p r o p e r
e l l ipse
s i z e - semidiameter g. Such
c o r r e l a t i o n is here based on
eq u al l i fe
p re di ct io n f or crack
growth
through
th e
flange t hi ckness.
S t r e s s c onc e ntr a tion decay along th e
major
a x is o f an el l ip t ical
hole
u n i a x i a l tens10n in minor axis
d i r e c t i o n )
can be e xp re sse d
a8: 5
15)
sinh 2f1) {COSh 2
cosh 2) ) - ] / cosh 2f1)-1
in which
is
th e g en eral e l l ip t i c
coordinate
an d y is th e s p e c i f i c
v a l u e
o f a ss o ci at ed w ith the
hole pe r im e te r .
The
el l ip t ic
c o o r d i n a t e ca n r e a d i l y
be
evaluated.
-18-
-
8/10/2019 [HAY] 386_10_79_
24/99
cosh y
=
16
cosh I1
=
{I
i}
cosh y
17
----
For any given geometry and assumed value of
crack
length,
a, the
stress
concentration factor,
K
s
known
g
s
known
Further,
i f g has been cor re la ted to equate K
t
and F
g
, then
the
stress
gradient
correction
factor
s also established.
A correlation
study
for th e
geometries
investigated
related
the
optimum
el l ipse size to the var ious geometr ical parameters n
in i t ia l
crack size, a . . The
regression curves
which resulted are: 38
Stiffeners:
2
t- =
-0.002755
0.1103 i -
0.02580
i
f
f
18
2
O. 6305 ;i - 7. 165 ;i
f f
Cover
Plates:
= 0.2679
0.07530
i - 0.08013
if 2
f f
19
T .
2
0.2002
log
Tc
p
L39 1 ; i -
11.74 ;i
f
f
f
-19-
-
8/10/2019 [HAY] 386_10_79_
25/99
The standard erro rs of estimates s for Egs. 5 and
6 are 0.0041
and 0.0055 respectively.
Figure 2
compares
the
K
t
curve
at
sample cover
p late d e t a i l
with the
actual F curve from
the Green s
function
CEq
2 and
th e
g
F
curve
from
the
correlated
ellipse.
is apparent
the
two F
g g
curves are in close
agreement
and cross over each other twice.
Such i s
also common
for
s t i f f e n e r
d e t a i l s .
-20-
-
8/10/2019 [HAY] 386_10_79_
26/99
3. STR SS INT NSITY
3.1 Other C o rr ec ti o n F a ct o rs
3.1.1
Crack
Shape Correction - F
e
F
is based
upon Green and Sneddon s solution
for
the
opening
e
displacement of an embedded el l ip t ical crack under perpendicular
st r e ss. 13 )
Irwin 29)
made
use
of
Westergaard s relationship
between crack opening displacement and s tr es s i nt en si ty . The
resulting F for an y position along
the
crack front, described
e
angle to the major
axis,
is :
1 [ 2
4
e =E(k) 1-k cos
20 )
where k i s 1.0 minus the ratio
alb
squared
and E k) i s the complete
ell ipt ic i n t e g r a l of the
second
kind. I n t e r e s t is usually directed
to
the minor axis end
of
the e l l i p s e where =n 2
For
t h i s
p a r t i c u l a r
position:
e
1
=
E k)
-21-
21 )
-
8/10/2019 [HAY] 386_10_79_
27/99
Equation
was derived on
the
basis
of
uniform tension
across
the
crack
surface.
While
may
be argued
that
variable
tension
will
modify
the
resul t ,
such
is
taken
into
account
the
F
g
correction la ter described. Hence F
is
maintained in i ts original
e
form for s tr es s int ensi ty estimates.
3.1.2 Front
Free
Surface Correction F
s
F is
generally
necessary
for
edge
cracks
since
stress,
not
s
displacement, is zero on the free boundary. A girder web as
well
as
attachments l ike s ti ff ene rs and
cover plates can provide
some
restr ic t ion to displacement
at
weld
toes or
terminations.
The
magnitude of
such restr ic t ion is
no t
known
to
any specific degree
although
i t
is
estimated to
be
modest. Furthermore, inclusion
of
an
F other than 1.0 usually leads to a lower bound or conservative
s
fatigue l i fe estimate.
Hence front
fr ee surf ace displacement
restr ic t ion by attachments is disregarded
in
this
report.
Tada and
Irwin
have tabulated
variabi l i ty
of F
s
with
the
distribution
of stress applied to
the crack.
34,35) Table 2 shows
this
variabi l i ty for
the
types of crack shapes and stress
distri
butions
of
in te re st a t welded
st if fener
and
cover
plate details .
I f
a
through crack
exists and
the stress is uniform over the
crack
length, F
is
1.122.
I f the stress
varies
linearly
to zero a t th e
. s
crack
t ip F is 1.210. And i f a concentrated load
exists
a t the
s
-22-
-
8/10/2019 [HAY] 386_10_79_
28/99
crack o rig in F is 1 .3 00 . Obviously i f the s tr e ss d i st ri bu t io n
s
decreases
from the crack
or i gi n
more rapidly than
the
linear
con-
dition F must have a value between 1.210 and 1.300.
s
Reference also d ir ec ts a tt en ti on to the half-circular
crack. For
a
uniform s t r e s s over
the
e n t i r e crack
p lan area F a t
s
the crack
t ip
is 1.025.
For
a str ess which
var i es
l i n e a r l y to
zero
a t
the crack t ip F
a t the
same location
is
1.085. The solution
s
for
a concentrated
load
a t the crack o rig in is not known.
F
for
s
t h i s
condition
i s
estimated
t
1.145
which
incorporates
twice
the
increment
increase exhibited
changing from a
uniform
to a linear
str ess p attern .
3.1.3 Back F re e S ur fa ce
Correction
F
w
The
s olu tio ns f or
F
consider
i n fi n i t e
h alf
spaces.
When
the
s
space
is not i n f i n i t e
thought must be
given to
the
back
surface
correction Once again
the
form
of
the correction depends on
w
s t re s s d i st ri b ut io n
and
crack
shape.
However F also is quite
w
sen sitiv e to whether or not
the
section i s
permitted
to bend as
crack growth occurs. The
bending tendency is
n atu ral for any
s t ri p
in which crack growth i s not
symmetrical
with respect to the s t r i p
cen terlin e.
-23-
-
8/10/2019 [HAY] 386_10_79_
29/99
The l i terature often
cites
two forms of F
almost interchange
w
ably
for the
symmetrical
crack cases
presented
in Fig.
13.
28,32,34,35
These two
expressions are
also app li cable to
nonsymmetrical
crack
configurations
where bending is prevented imposed boundary
conditions.
Hence,
the
str ips in Fig. 14 are
comparable
to those
in Fig. 13. At real structural detai ls
the
rol ler supports might
be provided a web, st i ffener,
and/or cover
plate.
Bending amplifies
the back surface correction
-
particular ly
a t
high
vaiues of
a/w where more
bending
occurs. f
the rollers
on
either
strip
of Fig.
14
are
removed,
the
back surface correct ion
takes on the following form:
34,35
[ ] 1/2
F
=
Q
tan
22
w
0.37 [I-sin
]
3
0.752
2.02a
where
Q
=
1.122
cos
a
a
=
w
The coefficient,
Q
?y which
the tangent
correction is
modified i s
p lo tted in Fig.
15.
-24-
-
8/10/2019 [HAY] 386_10_79_
30/99
Table gives
th e
back
s ur f a c e c or re cti on s f or
t hr ou gh c ra ck s
with an d w i t h o u t bending
pe r m itte d.
Without bending
th e
fam i l i ar
s e c a nt
c o r r e c t i o n i s
used f o r
a
uniform
s tress while
the secant
is
amplified as s t ress
c onc e ntr a tion
occurs t th e crack o r i g i n .
This
am pl i fi cat i on has
maximum
va lue 1.297 ~ n f f o r
a
concentrated
load
t
th e
crack o r i g i n an d
=
1 . 0 . Both no bending sol ut i ons
stem
from a f in i te width pl te
w ith
a
c e n t r a l through
crack. 34) A
l ine r ly varying s tress case i s assumed to be th e average
o f
the
two
extremes.
A
s ha r pe r s tress decay h as
a c o r r e c t i o n somewhere
between
the
concentrated load
an d
the
l ine r ly
varying
s tress
val ues.
The back
s ur f a c e
c o rr e ct io n s a s so c ia te d ~ t bending show
s ignif ic nt a m p li fi ca ti o n o f the sec a nt c o r r e c t i o n . Since th e
ta nge nt
an d
s e c a nt correct i ons are
s i m i l a r ,
Fig.
5 gives a good
i n d i c a t i o n
o f
th e a m plif ic atio n f or both
cases when uniform
s tress
i s a pplie d.
The
am pl i fi cat i on f c to r fo r
a
concent r ted
load
i s
much
highe r
tha n
th t
f o r a uniform s t ress For example, t a = 0 .6
the uniform
s t ress am pl i fi cat i on
is 1 .9 7 w hile
th e
concentrated
load
am pl i fi cat i on
is
6.07 Table 2 ). These back s ur fa ce s s o lu ti on s
are
directly l inke d
t o fr o n t
s ur f a c e
corre ctions si nce
bending
demands l ack
o f
symmetry. The combined
correct i on
f a c t o r s for
t hr ou gh c ra ck s
found in
Refs.
34 and 34 were divided
th e
asso
c ia te d f ro nt
s ur f a c e correct i ons T able 1)
to
i so l te th e back
s ur f a c e c o r r e c t i o n
f a c t o r s .
-25-
-
8/10/2019 [HAY] 386_10_79_
31/99
The
choice
between bending
and
no bending
depends
on the
st ructural deta i l as well as how the crack is growing.
Fatigue
crack
growth a t
welded
s t i f feners
cover
pla tes ,
and
other
common
girder
attachments is rare ly symmetrical.. Yet, bending is
usually
l imited by vir tue of the
girder web
and/or the attachment i t se l f . 7)
Hence, the no bending corrections are considered to be most
appl i
cable in
typ ica l
bridge structures .
th is point a l l discussion of
back free
surface
correct ion
factors
has
centered on the
through
crack
configuration lb
0).
23)
Maddox
recent ly
condensed the work of numerous researchers
and
est imated
how F var ies fo r crack shape rat ios
and
a
values
between
w
zero
and
1.0 .
Uniform s tress
and
unres t r icted
bending were
assumed.
His resu l t s
essent ia l ly
agree
with Table
when
l
equals zero,
but vary nonl inear ly
to
almost 1.0 for any value when alb equals
1.0. In
other words,
F
might
well be disregarded fo r the
hal f -
w
c i rcu la r crack. The
ne t l igament on
e i ther
side of the crack
inh ib i t s
bending and
s igni f icant ly l imi ts
the crack from sensing
the upcoming
f re e s ur fa ce .
The curves Maddox produced
are
approximations
s ince few data
points
exis t
for crack
shape
ra t ios
between
zero
and
1.0.
Never-
the less i s reasonable to assign F a value of
1.0
fo r
any a i f
the
w
crack shape i s
c i rcu la r
r eg ar dl es s o f the
bending
and
s tress
dis t r ibu t ion considerations.
The
fac t
tha t
some references show an
F value sl ight ly above 1.0 when a
is la rge
and l 1 .0 is not
w
important
since most fat igue
l i f e is
expended a t small a.
-26-
-
8/10/2019 [HAY] 386_10_79_
32/99
3.2 Crack Shape Effects on F
g
The
ent i re
development
of
Chapter
2
was
based
upon
a
through
crack
conf igura t ion
lb
= 0). Implementation of each
of
the analy t i ca l
approaches to
F
usual ly involves th i s assumption
although
solut ions fo r
other crack shapes
are
actual ly possible .
Moreover,
i s worthwhile
noting F
var ies
w ith c ra ck
shape and such
variance
should be taken into
account in
the
fa t igue l i f e predict ion process.
Table
4 has been developed using
through, and
c i rcu la r crack Green s
functions for a
crack
in
in f in i te so l id . (1,35) The
t ab le
tha t the
values
for
F
diverge as the
s t r e i s
decay becomes more rapid. The
l imi t ing r a t io for a
concentrated
load
a t
the crack or igin is
es t i
mated to be
0.548.
This number represents twice the devia t ion from
1.0
recorded between the uniform and l i ne ar s tr es s cases.
3.3 Crack Shape Variat ions During Growth
A major f ac to r a ff ec tin g correc t ion
fac tors
is
the
crack shape
during growth. Gurney
has
found t ha t th e importance increases
as the
l
(14) N 11 k
s t r e s s concen t ra t10n or
e ta1
sever l ty 1ncreases . orma y, crac
s
are ideal ized
as
e l l i p t i c a l although
experimenters have recognized
tha t
k
11
1 h (29 ,33)
most
crac s
are ac tua y l r r egu
a r
In
s
ape.
Crack
shape r a t io alb,
i s
dependent
upon
severa l var iables .
One
of
them is
the proximity of free surfaces, as dis t inguished
by
the ra t io
-27-
-
8/10/2019 [HAY] 386_10_79_
33/99
a/w.
Another is th e ov era geometry
of
the de ta i l which
affects
F
and, in
g
turn,
a/b. Therefore, plates of different width or
thickness
i f crack i s
growing in
that
direct ion) can
cause different crack shapes to occur.
37)
Also, stu dy
has
shown
that c racks beg in a t several s i tes along
a
t rans
, 10
37)
verse weld toe, but e vent ua ll y
they
coalesce.
Coalescence occurs
several
times
during th e
growth process.
In order to
establish
a crack shape relationship
is
necessary to -
rely
on actual
measurements of
crack size during growth.
These
measure-
ments
can
only be per fo rmed accurately breaking
apart structural
deta i ls
at dif fe ren t crack
growth s tages .
Reference
37 has reviewed
and compared
the available crack
shape
varia t ion equations for
growth
a t
the toe
of a
t ransverse f i l l e t
weld.
References 10 and 23 prov ide equa tions re la t ing
single
crack shape
to
crack
depth
that are in
marked
disagreement with
each
other . The re la t ionship
suggested
in Ref. 23
provides
a lower
bound
conservative)
characterization
of
early
crack shapes.
This
equation
i s :
b 0.132 1.29
a
23)
where a crack depth and el l ipse minor semidiameter in .
b
half
surface length
of crack
and
el l ipse
major
semidiameter in.)
The re la t ionship
suggested
in Ref.
10
b 1.088 aO.
946
was
observed
to
provide
an upper
bound
re la t ionship.
-28-
24)
-
8/10/2019 [HAY] 386_10_79_
34/99
When coalescence
begins Eq. 23 or
4 no longer app li es s in ce
the
shape t rend is toward f l a t t e r rather than
more circular
cracks.
The
multi-
pIe
crack
data
in
Ref.
10 sugge st ed the following crack shape
relat ionship:
b = 3.284
1.241
a
25)
However, an
examination
of cracks
a t
the weld toes of
f u l l
size cover-plated
beams
suggested that
a
lower
bound for the crack
shape
durin g th e coales
cence phase
was
provided by the relat ionship.
39)
b =
5.462
1.133
26)
All four
of
the
crack
shape relat ionships
are
plot ted in
Fig. 16 and
compared with
the
data for f u l l
scale
cover-plated beam d e t a i l s . The
intersect ion of Eqs. 23 and 26
i s
near a crack
depth
of 0.055 in . These
two equations
are employed for use a t a l l transverse
welds , regardless
of
d e t a i l
geometry, pending
f u rt he r s tud ie s.
3.4
Total Stress Intensi ty
Fatigue l i f e analysis
of
typical
welded d e t a i l s
i nvolves es timat ing
the s t r e s s
intensi ty
for
a
nonuniform s t r e s s dis t r ibut ion. Figure
17 shows
the
type
,of dis t r ibut ion which is common to any s t r e s s concentra tion region .
This dis t r ibut ion
may be
separated
into
uniform
and vari able cons ti tuent s.
Since s t r e s s intensi ty i s l inear i n s t r e s s cr superpo si ti on app lie s p ro
vided the crack
displacement
mode i s unchanged. 34,35) Hence,
the
t o t a l
-29-
-
8/10/2019 [HAY] 386_10_79_
35/99
s tress intensity can be
found
by adding the
s tress in tens i t i t ies
K
and
u
K , for two
sub
s tress distributions which sum to the to ta l s tre ss d is
v
I
tr ibution. The s tress intensi ty co rr ec tion fac to rs for
the
uniform
st ress
dist r ibut ion are known while
the co rr ec tion fac to rs
applicable
to the
var iab le subdist r ibu tion can be
estimated. (37)
Consideration must
also
be
given
to crack
shape
effects on st ress
intensi ty. Thus,
an approximate procedure is
adopted
whereby
the
to ta l
s t ress intensity defined
above is est ab li shed for
the
extremes
of crack
shape lb
=
0 and alb
=
1). A l inear
interpolation
between these l imits
approximates the s tress intensi ty
for the
actual crack
shape
at any crack
depth.
Obviously,
a very impor tant input to s tress intensi ty estimates is
the
crack shape variation.
3.4.1
Through Crack alb 0)
Using the co rr ec tion fac to r s of Art. 3.1 no bending) the st ress
intensi ty for
uniform
s tress can
be
writ ten as:
[ 1Tll ]1/2
K
1.122
K
te t
sec
(27)
u
where
C t
=
a/T
f
T
f
flange
thickness
Kt t
=
s tress
concentration
factor
a t
rosition
Since this is a
through
crack, F
is 1.0. t is
apparent
that
for
e
uniform st ress the s tress gradient
correction has
constant value K
t
.
-30-
-
8/10/2019 [HAY] 386_10_79_
36/99
pevelopment
o f th e
s t ress i n t e n s i t y fo r varying
s tress is more
complex.
Table
2 shows t h a t F
is
between
1.300
an d
1.210.
The
cor
s
ree t
in te r m e d ia te
value depends on the shape of
th e
a c t u a l s t ress con-
c e nt ra ti on f a ct o r
decay
curve K
t
re la t ive
to a l inear decay l ine
Figure 18 a demonstrates t h a t
th e
proximity to a l inear condition
var i es w ith a. s a increases, F i ncr eas es from
1.210
to a value
s
n ea r 1 .3 00 .
I f
F
s
r epr es ent s the des i r ed v alu e of
F
s
then:
F
s l
= 1.300 -
1.300-1.210)
=
1.300
- 0.09 W
where = measure
of p roxim ity to
l inear i ty has value 1 .0 i f
actually linear.
ca n
be evaluated on
th e
b a s i s of Fig
18b.
r epr es ent s that
2 8)
v alue of
a t
which
th e slo pe o f th e
s t ress
c o n c e n tr a tio n
decay
curve
equals th e
s lo p e
of a hypot het i cal
s t ra igh t decay
l ine from SCF to
a
The
lower l imi t of
is zero w h i l e the
upper
l imi t is a/2.
Thus,
J
is
taken
as:
a/2
a
29)
Proper r e s o l u t i o n of Eq. 29
depends
on knowledge of the concen-
t rat ion f actor decay curve
which v a r i e s
fo r each de ta i l geometry.
However,
since the change in
F
s l
is
usually small
over
the fu l l
-31-
-
8/10/2019 [HAY] 386_10_79_
37/99
range
of .va lues, r easonab le accuracy
i s at tained
by using
an
approximate
e qu atio n o f
the
following form for
a l l cases:
=
30)
where
~
=
posi t ion
in
the
crack
growth
direct ion
C, P = constants.
f
the s tress gradient results
for average
s t i f fener and cover
pla te geometries
are
used,
the
values of c and p summarized
in
Table resu l t
Equation 30 can be used to evaluate the indicated sl?pes a t
A
=
and A
~
A nonlinear t tcharacteris t ic equation resul ts
which
must be solved for
2P
P _ eP -1 = a
c
2
S c S
D
S
31
where
D =
-32-
-
8/10/2019 [HAY] 386_10_79_
38/99
The s ol ut i on o f
Eq. 31 is readily obtained
fo r
an y given
th e
b i s e c t i o n method.
Thus, can be c a lc u la te d
an d
F
s I
d e f i n e d .
is also
employed in
c a l c u l a t i n g th e appr opr i at e back f r ee
s u r f a c e c o r r e c t i o n ,
F ,
f o r
th e variable
s tress
subdistribution.
w
From A r t .
3 . 1 . 3 th e
e x p r e s s i o n
fo r F
can be
developed
i n a manner
w
s i m i l a r
t o
F
s I
.
32)
where
The
s tress
gr adi ent
cor r ect i on
f a c t o r fo r
the
subdistribution,
F , i s r e l a t e d
to
F c al cu la te d f or
th e
whole dis tr ibution
g g
A l b r e c h t s Green s f u n c tio n
Eq
7)
in
nondimensional
form y i e l d s :
where G
h
a KtA-K
t
fo r the
s u b d i s t r i b u t i o n .
33)
Hence,
34 )
-33-
-
8/10/2019 [HAY] 386_10_79_
39/99
Equation
9
y ie ld s the
following relationship:
F = F - K
g g
tCt
The
s tr es s i nt en si ty expression
for, the
variable
stress
35)
subd i s t r ibu t ion can
now
be
e s t i ~ t e
using Eqs. 28)
32 and
35 .
36)
Combining
Egs. 27 and
36
r esu l t s in the
t o t l s t r es s
in tens i ty
expression for s t if feners and
cover plates
with through cracks.
37)
F K
ga
tCt
t
is
h elp fu l to rearrange
Eq. 37
in the following
form:
K
=
[F
1.122-F
s1
C X]
38)
81
[ ] 1/2
F
t
cr
sec a .
where
X
K
t
=
F
g
-34-
-
8/10/2019 [HAY] 386_10_79_
40/99
E va lua tion
o f X
i n
Eq . 38
can be performed w ith
th e
stress
g r a d i e n t
c o r r e c t i o n provided
by the a r t i f i c i a l
el l ipse
c o r r e l a t i o n
given
in A r t. 2 . 2 . 3 . I t
is
al so possib le to use th e approximate
e qua tion fo r F w h i c h
sac r i f i ce s
l ttl a c c u r a c y .
F
S F
39)
Fo r average s t i f fener an d cover pla te geometries, d an d -q
a re
given
in
Table 1.
y
c om bi ni ng E qs .
30
an d
39,
X
becomes:
x
=
1 1
a
P
c
40)
-3.4.2 H alf-C ircular
Crack lb
=
1) .
Th e s tr es s i nt en si ty
fo r th e
uniform s t ress
s u b d i s t r i b u t i o n
on
a hal f -c i rcular crack can
be
defined
a8: 35
K = 1.025
u
41
C ra ck s ha pe
c o r r e c t i o n , F
is
r e pr e s e nte d
by
n
an d
F
is assumed
e W
to b e 1.0 .
-35-
-
8/10/2019 [HAY] 386_10_79_
41/99
Fo r
the varying
str ess condition F
s
i s represented
by
F
s2
:
F
s2
=1.145 -
0.06
i s
the
same value calculated for th e t hr ou gh c ra ck case.
42)
The str ess gradient correction associated with the circu lar
crack
front, is defined:
.
g
=F - K
g g t a
43
Article
3.2
recognized t h a t F
f
is
not
of
the
same
numerical value
g
as F calculated for the
through
c ra ck G re en s function unless the
g
applied str ess h as u ni fo rm d istrib u tio n .
Equations
42 and 43
ca n be used
to
develop
the
str ess intensity
factor for the variable str ess subdistribution.
K
=
F
F
t
K
s2 g ta
44
Adding K to K CEq. 41 gives th e t o t a l K for the half - cir cular
v u
crack shape.
~ ~ k ~
tCl It
s2
F -K ~
1
..jna
g te l 7t
J
45)
- 3 6 -
-
8/10/2019 [HAY] 386_10_79_
42/99
Rearranging Eq.
45 gives:
F
gCi
.Jna
(46)
where
F
y = J
F
g
Evaluation of
rat io
Y is
assisted by
Table 4.
However,
must
be borne
in
mind that
F and F a re eva luat ed
for
the
to ta l
g g
s tr e ss d is tr ib u ti on not
j u s t
the
variable
subdistribution.
The Y
ra t io
for
the
t o t a l
dis t r ibu t ion
is
dependent
on
the
re la t ive
in flu ence o f
the
variable
s tress
subdistribution
as
compared with
the
uniform
s tress dis t r ibut ion . Hence, Y is taken as the sum
of
two par ts
where
y =W i ::
y
+ (l-W) itC
y
Y
=
0.548
0.226
=
1.000
(47)
W
=
weighting factor
of
variable
s tress
subdistribution
re la t ive to the uniform s tress subdistribution.
Factor Wmay be
based
upon
the ra t io of the
two shaded areas
under the concentrat ion decay curv e in Fig. 19 .
w=
A
v
A
u
=
=
- 1
48)
-37-
-
8/10/2019 [HAY] 386_10_79_
43/99
Subst i tut ing Eq. 5) i nt o
23)
gives:
w
=
a
1
dJ\
1
+
l\P
c
1
+
. riP
c
- 1
4,9)
The i n te gr a l in the numerator
can
be
evaluated
numerical ly; t h e r e -
f o r e f a c t o r
W i s
e a s i l y
o b t a i n e d . However, W
decreases
w i t h
increa ing
r e l a t i v e crack length and must be
reevaluated
for
each
crack
p o s i t i o n .
For the b e n e f i t
of
l a t e r
calcula t ions
it i s
worthwhile multiplying
and dividing Eq.
46
the secan t r d i l ~
K =
F
s2
Y 1.02S-F
s2
X
[
e c ~ a
2
F
g
2
7t
50)
3 .4 .3 I n te rp o l a ti o n
f o r
H a l f - E l l i p t i c a l
Cracks
0 < l < 1)
An
intermediate ,posit ion
between
Egs. 38 and 50 i s
necessary
for h a l f - e l l i p t i c a l crack shapes. Comparison of th e
two
equat ions
reveals t h a t e a c h contains the F factor and
the
secant
r a d i c a l .
g
These
a re h en ce fo rt h termed the combined stres_s gradient and
back
free surface correct ion
f a c t o r s respect ive ly .
I t i s
also
apparent
t h a t
each
equat ion con ta in s
the appropriate , i s o l a t e d crack shape
correc t ion f a c t o r F = 1.0 i s implied in Eq.
38.) Therefore ,
e e
-38-
-
8/10/2019 [HAY] 386_10_79_
44/99
use
of the
normal
el l ip t ic l integral Art. 3.1.1 for th e combined
crack shape correction is warranted. This integral automatically
provides
a
nonlinear
interpolation.
Only t he l eading b racket ed expres sions in each equation remain
to be
adju ste d fo r h l f e l l ip t ic l crack
shapes. While
not precise,
the s imple st approach is a l inear interpolation
based
on the crack
shape rat io,
a/b. The interpolated value
r ep resent s the
combined
front
f re e sur fa ce correction, F .
s
a .. .
b
{F C
1.122-F /
-
8/10/2019 [HAY] 386_10_79_
45/99
I t i s apparent tha t
the
crack shape is of
major
importance to
the value
of
for s t i f feners
he more
half c ircular
the
crack
s
shape the lower However even
with
a constant shape
s s
decays as
increases .
he decay i s more rapid for shapes nearer
the half c i rc le These character is t ics are also
val id
fo r
cover
plate
de ta i ls
40
-
8/10/2019 [HAY] 386_10_79_
46/99
4. FATIGUE
LIFE CORRELATIONS
The c o rr e ct io n f ac to r s
developed
in Chapter 3 Table 5 can be used
in
fatigue l i fe predictions. A f te r r ep la ci ng
s t re ss
0
with s t r e s s range,
S
in
the s t r e s s i n t e n s i t y
expressions,
the r esul t i ng range of s t r e s s
r
i n t e n s i t y is inser-ted
in
Eq . 2 . I t
i s
r ar e
tha t th i s
e q u a t i o n
can b e
s ol ve d c lo se d- fo rm
- p a r t i c u l a r l y when
the
combined correction f act or , CF
is
a
complicated function of crack len g th , a . T herefo re , th e cycle
l i fe
is commonly estimated on the. b a s i s
of the following numerical
i n t e g r a t i o n :
m
N
1
1
t a
52)
-
C
l\Kj n
J
l
2.0*10-
10
. 11/2
where
C
1
1.u.
Refs. 10,
17
mean
3
kip cycles
3.6*10-
10
in .
11
/
2
C
2
=
u p p e r
bound
Ref.
39 )
kip
3
cycles
M
=
range
of
s t r e s s
i n t e n s i t y ,
k si
l in
a
=
crack
length
increment,
in .
A sense o f the
r e l a t i v e
importance
of s t r e s s
range an d in i t i a l
an d
f i n a l crack si zes in l i fe estimates may be developed a ga in c o ns id e ri ng
Eq.
2. I f
the
combined
c o rr e ct io n f a ct or
is
assumed
fo r
th is
exercise)
to be
constant
th e c y c le
l i f e
i s
p re d ic ta b le i n c l o s e d - f o r m
fashion .
-41-
-
8/10/2019 [HAY] 386_10_79_
47/99
where
a. = i n i t i a l crack
s ize
a
f
= f ina l crack size
[
-1 /2 -1/2 ]
a
i
a
f
53
Since s t ress range
is cubed
and b oth crac k
lengths have a
square root
sign attached, erro r in
the
nominal s t ress range is
much
more important
than
errors in the i n i t i a l and f ina l crack
s izes .
i k ~ w i s e
er ror
in
the correc t ion
f ac tor
CF
i s
more
important
than
crack
s ize.
Even
though CF in rea l i ty
depends
on crack size
added
importance is
obviously
a tt ac he d t o
establishment
of the correct
form of
each individual
cor-
r ec ti on f ac to r.
The
negative
radical associated with each crack length typical ly
places the weighted importance on i n i t i a l crack s ize. Experimental
work
out l ined in Refs. 9 and 1 used an excessive deflect ion net
sec t ion
yie lding cr i te r ion
for
fa t igue
fa i lure and th e e sta blis hmen t o f a
f
A
more recent
study terminated
fa t igue l i f e with
f r ac ture
although th is
l i fe
was
close
to
that
found using a generalized
yielding
condition.
7
Both def ini t ions of fai lure general ly cause a
f
to be much larger than
a
i
The greater
the
dif ference
between a
i
and a f the
greater
the
i m p o r ~ a n c e
of
:I n l i gh t of
the re la t ive
importance of
a . i s indeed unfortunate
1
tha t
a
i
is much more d i f f i cu l t to est imate than af. Such
i s
even
t rue
-42-
-
8/10/2019 [HAY] 386_10_79_
48/99
for a
t e s t
specimen where the crack surface is exposed
af ter
fai lure a
i s usually clearly evident) . Nevertheless, s eve ra l i nve st iga ti ons have
established
lower
and upper l imits
of
in i t i a l crack size for weld toes of
a s t i f fener or cruciform jo in t a t abou t 0.0 03 in . and 0.02 in . respec
t ively. 31,36) The
average a.
is
between .003 in .
and .004
in .
Given
the
expression
for stress i ~ t n s i t y and i n o r ~ t i o n
on i n i t i a l
crack sizes the analyst is in a
,posi t ion to
make fatigue l i fe cycle)
estimates.
Several
sample detai ls are subsequently investigated and
their
l ives
are
compared
with
those
found
under
a ct ua l f at ig ue
t e s t
condi-
tiona.. Since
s t i f fener and cover-plated beams
had welds that
were
perpen-
d icu lar to the s t ress
f ie ld,
was assumed th e fatigue cracks
coalesced. Hence Eqs. 23 and 26 were
used
to d esc rib e the crack
shape
relat ionship for
the
analysis
described
hereafter .
4.1 Transverse Stiffeners
Fillet-Welded
to Flanges
Reference 1
provides.
a
broad exper imental base
for
fatigue fai lure
a t s ti ff en er s. St i ff ene rs f il le t -we lded
to flanges
a re t he re in designated
Type
3. One
part icular
series in
this
case
the
SG S combined series)
i s s ele cte d fo r invest igat ion
and values
of
the
crucia l geometric variables
a re tabu la te d
in Table 6.
Appendix
E of Ref.
10 notes
tha t
the
actual
weld
s ize was closer
to
0.25
in . ra ther
than
the
0.1875
in .
dimension
speci-
fied.) The
objective
is to assess how accurately the
resu l t of
the es t i
mate
~ p p r o c h Nest
predicts
the
actual
cycle
l i fe N
act
All l ives are
also recorded
in
Table 6.
-43-
-
8/10/2019 [HAY] 386_10_79_
49/99
The a c t u a l
cycle l i fe Table
6 r e l a t e s to
t he l og a ri th m ic
average of
dat a p o in t s f o r
th e given
ser i es and
s t r e s s
range
T abl eE -3 of R e f .
10).
L o ga ri th m ic a ve ra ge
means
t h a t
v a r i a b l e N
is assumed log-normally
di s t r i
buted base 10 an d
the
mean is
approximated by
the
average
logarithm
of
the data p o i n t s . For t hi s p ar ti cu la r s er ie s an d
stress
range, e i g h t dat a
poi nt s
are
a v a i l a b l e .
On
average, f a i l u r e occurred
af ter
the crack ha d
f u l l y p en etr ate d th e flange and was growing
in
a through crack
c onf igur a -
t i o n .
Since the l i fe e stim ate s a re
based
on cycle l i fe fo r growth through
th e f la ng e, 96
pe r c e nt
of the a c t u a l l i fe
found
by t he l og ar it hm ic average
is recorded.
Four estimated l i v e s were derived from
the
s t ress i n te n s it y r e la t io n -
ships Table 5 ) . They r e p r e s e n t a pp ro xi ma te a v er a ge
an d maximum i n i t i a l
crack s i z e s an d crack
growth
rates
In comparison with N average)
in
ac
Tab le 5, -it is seen t h a t the t h e o r e t i c a l r e s u l t s a l l provide l i fe estimates
t h a t a re c ons e r va tive . The r s ~ t s
a r e
al so p lo tt e d i n
Fig.
21
an d
com
pared with the resul ts of
regressi on
analyses of a l l of
th e
tes t dat a. I t
is probable t h a t Eq. represents a more reasonable
crack
shape
r e l a t i o n -
ship f or t ra ns ve rs e s t if fen e r d e ta il s The use of Eq . 26 provides a more
se v e r e condition.
Reference
10 provides two regressi on equations
which
ca n
be
used to
estimate l i fe Th e mean equation fo r a l l st i ffeners not jus t those con-
nected to the flange) i s as follows:
log N
=
10.0852
3.097 log S
-44-
54)
-
8/10/2019 [HAY] 386_10_79_
50/99
Since 96 percent
of
N
is
to
be
used for comparison purposes,
Eq.
54 can
be modified accordingly.
log
.96N
10.0675
3.097
log S
r
5 5
The mean equation for Type 3 s t if feners
only a l l
ser ies i s :
log N
10.5949
3.505
log S 56
r
For 96
percen t
of N
the
regression equation appears as follows:
lo g
.96N = 10.5772 3.505
log
S
r
5 7
I t
is
noted
that
nei ther regression equation has
a
slope of exactly
-3 .0 . However, the discrepancy
is
minor.
In
fact ,
t he e st imat es
by
Eq.
54
and
56
or
55
and
57
are
normally
quite
close
due
to
the adjustment
provided
by th e equatio n c on stan ts Fig. 21} A
common
slope of -3.0
guided the S TO Speci f icat ions 2
although
round off of s tress range
values l e f t
the slope sl ight ly
off
of
the
mark.
Regardless,
equating n
to
3.0
is
reasonable in the l i fe integral
procedure
Eq. 52 .
Since
both of the above regression
equations
are ba.sed
upon
the
s pe ci fi c s er ie s under study here, close agreement between predicted and
actual cycle
l i fe is
expected and,
indeed,
found Table
6 and
Fig.
20 .
However, is also f rui t ful
to
compare
the
value a t
the
approximate
upper 95
confidence
l imi t .
Again assuming
log-normal dis tr ibution and
-45-
-
8/10/2019 [HAY] 386_10_79_
51/99
incorporat ing the s tandard devia t ion 5 0.1024 from
the
yp regres-
sion
analysis ,
Eq. 57
i s
adjusted to the
upper
95 percent
confidence
l imi t
lo g .96N
10.5772 -
3.505
log S
r
58
=
10.7820
-
3.505 lo g
Sr
The large
separation
between the upper
confidence l imi t
and the
mean value emphasizes the
wide v r i b i l i ty of experimental
data.
This
separation t he reby g ives
a
measure of
accuracy
of the unified estimate
t
the
average in i t i l
crack s ize.
4.2 Cover Plates With Transverse
End
Welds
Reference 9 is
a source o f con sid er ab le d ata on
cover
plates .
Combined ser ies
eWE eWe
was
selected
for
invest igat ion
and
the
important
geometrical parameters are summarized in Table 6. Series
CW
had a
sl ight ly
different flange
thickness,
Table D-2
of
Ref. 9, and was there-
fore
omitted
from the combination.
The s t ress
range assumed is
16
ksi.
Thus,
using
the
logarithmic average of the twelve available data poin ts
Tables F-2 and F-3 Ref. 9 ,
the 96 percent
l i fe is se t t 0.356
million cyclese
The l i fe estimates
for
the
average
and maximum
in i t i l
-crack
sizes
and crack growth rates are given in Table
6 and
Fig. 22 . The
estimates
-46-
-
8/10/2019 [HAY] 386_10_79_
52/99
given
in
Table
6 bound
the average l i fe
N t
for
both crack growth rates.
ac
The resul ts plot ted in Fig. 22
are
compared
with the
resul ts of
regres-
sian
analyses
of
a l l of the
t es t data.
Reference
9 gives
the
following.
regression
equation
for a ll cover
plates with transverse end welds:
log
N
=
9.2920 - 3.095 log Sr
59
Incorporat ion
of
the 96 percent
l i fe for crack growth through
the flan ge
modifieds Eq. 59
s
follows:
log .96N
9.2743
-
3.095 log S
r
60
For the specific s t ress
range
in
Table
6,
can
be seen that th e
es t i
mated
l i fe
from
Eq.
60- by
chance
precisely
equals
th e
actual
l i fe
for
the pa rt icu la r s e ri es
being
studied. By
making
use of the
standard
error for Eq. 59
8
0.101 ,
is again possible to define the
equation
for
the upper 95 percent confidence l imit .
log
~ 9 6 N = 9.2743
3.095
log
S
r
61
=
9.4763
-
3.095
log
S
r
Equations 60 and
61.are
plotted in Fig. 22 . Reasonable agreement is
seen
to exis t
between
the
predicted
fatigue
l i fe
and the experimental resul ts
-47-
-
8/10/2019 [HAY] 386_10_79_
53/99
he minimum
and
m ximum l i fe
estimates corre la te well with the
lower nd
upper confidence
l imits
of the
t es t
data.
t
is apparent
from this analysis that the
crack
shape relationship
and
the
crack growth
r te both have
s ig n if ic a nt e ff e ct
on the f t i g ~
l i fe
estimate. t
seems
reasonable to
expect
the wide v ar ia ti on i n
fa-
tigue
l i fe that is experienced in laboratory studies
cons idering the
r ndom nature of in i t i l
crack
s ize s tress concentrat ion condi tions and
crack growth rates that are
known
to exist .
48
-
8/10/2019 [HAY] 386_10_79_
54/99
SUMM RY
ND ON LUS IONS
In th is
report
the
stress
concentration
effects on fatigue
cracks
that
grow a t weld terminations
has been investigated.
Fin i te
element
studies were used to develop the s t ress
concentration
factor decay curves
for
s t i f f ener and
cover plate deta i l s .
These
resul ts
were then
used
to
express the s t ress
concentration
conditions in . terms of jo in t geometry
and weld s ize .
To
permit a fa t igue l i fe a na ly sis of typical welded deta i l s the
to ta l s t ress in tensi ty a t
the weld
toe was developed
including
the stress
gradient
correction factor ,
the crack
sha