shape factor for an elliptical crack inner and outer radii of eyl_nder cylinder wall ... is shown in figure. I. The elastlc cylinder of wall thickness t, internal radius ..... for Internal. Longitudinal. Semi-Elliptical. Surface. Flaws in a Cylinder under.
iiiil,.o
NASA
Technical
Memorandum
80073
INASA-TH-80073) STRESS-I NTENSITT FACTORS FOR INTERNAL _UEF-_CE CRACKS IN CTLINDBICAL PRESSURE VESSELS (NASA) 11 p HC A02/HF A01 CSCL 20K
N79-27532
Unclas G3/39
2930_
STRESS-INTENSITY FACTORS FOR INTERNAL SURFACE CRACKS IN CYLINDRICAL PRESSIIREVESSELS
J. C. Newman, Jr., and I. S. Roju
JULY
1979
:-'-,..
.¥:[ ., _.. "1.ii.;".5 _,.
,-_ National
Aeronautics
and
Space Administration LangleyResearchCenter Hamplon, Virgl :_a23665
•:. ,
-_"
(_-
STRESS-INTENSITY CRACKS
FACTORS
IN
FOR
CYLINDRICAL
J. NASA
C.
SURFACE VESSELS
Newman,
Langley
Hampton,
INTE_AL
PRESSURE
Jr.
Research
Virginia
Center
23665
and I. Joint
Institute
George
for
Washington
S.
Raju
Advancement
University Hampton,
of
at
Flight
Langley
Virginia
Sciences
Research
Center
23665
SUMMARY Failures analyses
of many
of these
crack-growth
A few cylinders
been
traced
strengths.
are
needed
engineering
Because
estimates
of
to surface for the
cracks.
reliable
predietlon
complexities
or approximate
Accurate
analytical
of
stress their
of such
problems,
all
methods
to
the
obtain
factors.
been
surface
to-vessel-radius The
used
have
components
fracture
three-dimensional have
internal
and
have
stress-intenslty
vessels
surface-cracked
rates
investigators
pressure
purpose
reported
crack
with
ratio
of
of this
seml-elllptlcal
stress
surface
analyses
recently.
of seml-elllptlcal
However,
these
surface
investigators
a craek-depth-to-crack-l_ngth
ratio
cracks
in pressurized
considered
of
I/3
and
only
an
a wall-thickness-
0.I. paper
is to present
stress-intensity
on
the
inside
of pressurized
from
0.2
to ]; the
cracks
factors
for a wide
cylinders.
The
range
ratio
of
of
crack
q.
depth
to crack
from
0.2
length
to 0.8;
stress-lntenslty finlte-element elements
and
the
factors models
were
crack
The
evaluated
plane
and
ratio
of wall
were
from
ahead
by
singularity
models
had
about
a nodal-force
of
the
crack
ratio
thickness
calculated
employed
elsewhere.
factors the
ranged
of
crack
to vessel
depth
radius
a three-dimenslonal elements
6500
along
degrees
method.
front
used
crack
0.]
or 0.25.
front
freedom.
In this
were
was
thickness
finlte-element
the
of
to wall
The
method,
to obtain
the the
ranged
The
method.
and
The
linear-strain
stress-int6nslty nodal
forces
normal
to
stress-intenslty
factors. An empirical present
analysis
radius. about
The
equation
5 percent The
equation
present
for
as a function
of
applies
the
were
stress-intensity
of crack over
present
results
the
depth,
a wide
range
factors
crack
was
length,
fitted
wall
of configuration
to the
thickness, parameters
results and
and
of
the
vesse! was
within
results. compared
to other
analyses
of internal
surface
cracks
in cylin-
Xm
ders.
The
surface-crack
configuration
wall-thlckness-to-vessel-radius integral method
equation were
The latlng
in fair
were
agreement
stress-intensity
fatigue-crack-growth
in a pressurized
mm
method
ratio
cylinder.
in good
had
n crack-depth-to-crack-length
of 0. I. agreement
(i8 percent)
with
Results
from
(±2 percent) the
factors
and
eq_ations
rates
and
in calculating
present
presented
the and
ratio
literature those
from
of
using
I/3
and
a
a boundary-
a finite-element
results. herein
fracture
should
toughness
be useful for
the
_n cerresurface
crack
I.
Introd_ctlon Failures
of
of
these
analyses crack-growth
many
and
have
stress-£ntenslty
cyllnders
Include
the [3]
have
effects
have of
made
wall
thickness.
been
reported
the
[6]
considered
and
a wall-thlckness-to-vessel-radius
front
purpose
in
0. I or
of
crack
length 0.25. An
presented
a
surface
cracks.
rellable
the
Accurate
prediction
complexities
of
analytical
and,
analyses
and
of
such
methods
for
Kobayashi
however, for
nf
surface
[2].
and
internal
surface
all
to
the
obtain
Polvanich,
cracks
that
crack
paper
is
to
depth
The
to
wall
from
0.2
to
stress-intenslty for
the
parameters.
were
the
factors
of surface
b
half-length
of pressurized
c
half-length
of surface
and
and
pressurized Raymund
ratio
varlatio;is
References
[5]
from
on 0.2
was
factor
the
to
of wall
of
boundary-correction
inside
0.8;
the
thickness by
also
using
from
the
ratio
of
crack
the
factors
(cylinder
to flat
stress-lntensity
P
internal
q
shape
factor
inner
and
t
cylinder
x,y,z
local
factor
pressure for
outer wall
Cartesian
on an
for an for
(Mode
internal .th j
surface
stress
for
a wide
the
crack
literature.
plate)
I)
the cylinder elliptical
radii
crack
of eyl_nder
thic_less coordinates
centered
at
crack
crack
distribution
mouth
on
depth
radius
a nodal-force
along
by
cylinders.
to vessel
developed
variations
solutions
of
crack
factors
_3
surface
cylinder
factor
of
calculated
seml-elliptlcal
crack
boundary-correctlon
and
"atlo
factors,
range
located
other
of boundary-correction
[5]
Pelllssler-Tanon factor
stress-lntenslty
factors
with
HeGowan
parameters.
calculated
stress-intenslty compared
and
stress-intenslty
a wide
ratio
were
stress-intensity The
I
for
ranged
l; and
in
a crack-depth-to-crack-length
Hode
cracks
depth
R,R O
include
0.1.
[7-9],
thickness
obtain
with of
present
method
s
KI
not and
did
Ss,_bols
cj ,Hi
presdid
Emery,
cracks
Labbens,
configuration
ratio
possible,
F
in
estimates
C
°
their
problems,
surface [4]
Hellot,
to
of
surface
g_thiresan
while
method,
where
f
•
stress
cracks
Their
Kobayashi,
seml-e111ptlcal
Atlurl
range
cylinders.
ranged
The
factors
[I]
factors
llmlted
in.ernal
this
equation
configuration
2.
approximate
methods,
finite-element
of
[7-9].
an
pressurized
ratio
to crack
or
Recently,
equation
for
only
three-dlmensional
The
for
of
Underwood
flnlte-element
along
cracks
needed
stress-lntensity
recently.
boundary-integral
crack
by
stress
three-dlmenslonal
The
to
thickness.
three-dlmensional
the
the
stress-intensity
wall
have
used
for
been
estimated of
cylinders
[6]
traced
Because
estimates
estimates
effects
A few
used
are
strengths.
engineering
engineering
surized
the
been
components
fracture
used
have
factors.
Some
Love
vessels
surface-cracked
rates
investigators
pressure
crack
was
method range front
of are
a
3.
X,Y,Z
global
8
angular
measurement
_j
applied
stress
¢
parametric
Three-Dimensional A surface
cylinder
crack
intensity
of
dimensional 3.1
types
surface
larity
elements
A typical element
in
conform
to the
ranged
from
0.25.
Further
apart).
and
the
surfaces
were
applied
to the
were
degrees
and
used
in the
along of
cracks
b/c
the
I.
The
elastlc
a seml-elliptical
cylinder.
by using
and
plane
not
were
The
stress-
a three-
and
Anp1ied were
except
model
for
the
were
and
The
the
crack
2(b).
used
the models
of
the
the
two
front. finite-
in refercurved
ratio,
t/R
ratio
of
elements
types
singu-
The
were
half-length-to-radius
i0 to 50, And
where
[8].
at
to those
one-
front,
used
in figure
that
an internal
(llnear-straln,
crack
strain
to model
idealized
isoparametrlc
is shown
vessel
from
in combination
freedom,
near
identical
plates,
repeated
simulated
was
to
b/R, 0.1
used
or are
here.
Loadln_
appl_ed a vessel
on
the
with
X = 0
two
plane;
syn_metric
Y = 0
surface
plane,
cracks
and
(180
degrees
free.
factor
solutions
crack
surfaces.
constant,
surfaces
plane
formulation
are
was
the
and
The
except
in stress
crack
The
ranged
of
of a pentabedro_,
the crack
shape.
degrees
0 _ X _ b).
shape
used
finlte-element
everywhere
in flat
ratio
conditions
analyzed:
given
of
obtained
[7]) were
6500
the
the model
crack
in figure
2b, contains
surface were
a typical
nearly
on model_ng
on
inner
singular
pattern
Conditions
stresses
shows
were
eacL
_'
singularity
stress-intensity
of applied
on the
is shown
length
configurations
a square-root
[7-9]
X = b
cylinder R, and
employed
cylindrical
boundary
The
The
linear,
as shown
were
obtained
Four
applied
quadratic,
in figure
by solving stress
and
2(c),
complementary
distributiens
cubic.
were
the
These
symmetric
on the
stresses,
about
problem crack
which
the
y = 0
were plane
by
S
mD P
= (_)
oj
for
where
z
is measured
stress
distrlbutlons
cylinder. other
mm
desired
Boundary
a
2(a)
which
surface
details
plane,
depth
the neighborhood
1 to 5,
Symmetry
and
Figure
elements
for
radius
(isoparametric
(0 _ 0 _ 90
in references 3.2
crack
pressurized
internal
and
elements,
[9]
elliptical
Idealization
produced
models
surfaces
analysis.
finite-element
[8] and
crack
surface-crack
model,
vessel
singularity
Z = 0
2c
the
hexahedron)
eight
cylinder
internally
elements
The
of the
elght-noded
given
for
vessels.
crack.
ences
length
of
on
of
t,
finlte-element
cylindrical
eighth
in an
Finite-Element
Two
on
angle
thickness
factors
coordinates
Analysis
crack
of wall
surface
the
Cartesian
stress
(These
J = 0 to 3 from
were four
the crack
mouth
superimposed
solutions
dlstributions,
(I)
such
can as
toward
to obtain be
the
caused
front.
stress-intenslty
superimposed
those
crack
by
to obtain thermal
Solutions factors
for
stress-intenslty
shock.)
for
these
the
pressurized
factor_
four
for
4.
$tress-lntenslt_ The
to
Mode
I
stress-lntenslty
factor,
KI,
at
any
point
along
the
surface
crack
was
taken
be
9"
for
GlaaR'\ j _,_-,_,q,)
_
J
-
0 to
distribution square
of
always
3.
Oj
Is
from
eq.
(I).
the
complete
chosen
(b/c
>
were
by
enough
where
shape
_ntegral that
appropriate
the
factor
factor
for
of
the
second
would
have
length
r_-
pR/t
the
corresponding
an
to
e11iptlcal kind.
The
the
jth
crack,
is
vessel
length
a negllglble
effect
stress
g__ven
on
by
the
(2b)
was
stress
intensity
the
of from
for
the
stress
where
each
factor
F
faces.
Values
0.1
surface
crack
of
written
the
in
results
an
Internally
given
pressurized
by
eq.
(2).
cylinder
For
convenience,
as
Gj
(3) stress
was
four
an
also
The
the
boundary-correctlon
cylinder. of
the
The
a pover-serles
pressurized
from the
F
is
terms
Internally
obtained
for
F
pressurized
first in
0.25.
and
internally
the
includes
and
was
hoop
an
obtained hoop
a
aaR
is
inside
for
superpositlon
factor
was
and
the
factors
stress-lntenslty pR
of
boundary-correctlon Q,
stress-intensity
"t on
the
elliptic
large
obtained
the
(2)
10).
The
of
the
were
calculated
as
a
a/c
ratios
were
expression of
The
function
0.2.
O.h,
of and
and
crack
terms
of
solutlon
The
acting
a/t,
Gj,
[I0]
_
a/t
correction
on
and
the
In
surface
is
p,
a/c, 1;
¥,
solution.
pressure,
a
Lame's
result
finite-element internal
for
for
expansion
cyllnder.
appropriate
influence
factor
the
for
crack t/R
sur-
values
ratios
were
0.2,
obtained
by
using
0.5,
0.8. The
stress-intensity
nodal-force nodal
the
stress-intensity
forces
Results In
face
the
of
wide
range
single
of
sections,
the
inside
front An
which
are
crack
plane
for
finite-element given
in
aud
models
references
ahead
of
from
the
also
were
[7] the
and
crack
[9].
front
In are
this
used
a
method, to
evaluate
Surface
3 shows
the a
(a/c
=
0.25.
For
t/R
1)
for
the
The (a/c
= 0.2
for
symmetric
semi-elliptical
stress-intensity and
stress-intensity
An estimate The
t_o
1)
factor are
factor the
stress-intensity
also
as
factor are
a
func-
developed
stress-lntensity factors
variations
presented is
sur-
compared
for for with
a a
other
literature.
Semi-Circular
cracks
cracks
for
presented.
presented
cylinders.
surface
equation parameters.
is
are
pressurized
various
empirical
crack
results
of
configuration
surface
Figure
the
the
Discussion
crack
5.1
to
following
a/t.
soluttov.s
of
from
factors.
on
the
tion
details normal
and
cracks
along
factors
method,
the
5.
t
Factor
as = 0
Cracks
boundary-correction function
(flat
plate
of
the [8,9]),
factors parametric the
for angle,
pR/t
stress
two _,
symmetric and
in
eq.
semi-circular
a/t (3)
for is
surface
t/R
= 0,
replaced
by
0.1, St,
and a
remote
uniform
for
larger
for
smaller
occurred
applied
a/t
ratios.
t/R
at
5.2
stress.
the
Also,
ratios.
The
intersection
Sc_i-Elllptical
Figure
4 shows
For
the
of
as a function
fixed
t/R
ratio,
the
a semi-clrcular occurred
at
5.3
t/R
angle,
so
t/R
=
0.1
and
function bars) or
of give
1
and
results
the
crack,
were
ratio,
the
correction
factors
are
higher
correction
factor
(or stress-intenslty
Factor
for
the Figure F0
the
(and of
to
be
are
the
inner
surface
(_ = 0).
for for
hlghez
two t/R
for
correction
correction
symmetric
seml-elllptlcal
= 0, 0. I, and
larger
a/t
factors. factor
0.25.
Again,
ratios.
For
In contrast
(or
surface for
a _Iven
to results
stress-lntenslty
a
for
factor)
(@ = w/2). Equation 4
3 and
for
range)
(t/R
= 0
can
fc
(the
ratio
factor fc
clarity,
closely
that
t/R
of
(For
suggest
plate
correction
_.
a/t,
higher
5 shows
is
and
a flat
curve
with
factors
_
point
those
that
front
factor)
Cracks
gave
and
found
a/t
factors
figures
value
for a fixed
the maximum
depth
average
any
higher
of
in
t/R.
are
the crack
shown
0.25.
factors
ratios
surface
for a given metric
t/R
the maximum
results
correction
correction
Stress-lntensity
The
the
boundary-correction
= 0.2)
smaller
ratio,
Surface
(ale
ratio,
t/R
maximum
cracks
a/t
a fixed
be
of a
approximated
to
F
t/R
a/t
= 0.5
The
not
para-
results to
with
are
factors
of the
the
[8,9].
of
a/t
independent
a given
plate
value
correction
approximate
for
flat
for
of the
nearly
scaled
a given
results
ratio
= 0) are
for
for
the
for
FO) abels
a/c
as
a
(and
= 0.2,
shown.)
0.4,
These
by
"1
R2 - 0 5,_I
In figure
5,
The
curve
upper
solutlon The
for
other The
surlzed been
the upper was
curve
show
fitted
to
is given the
present
the
from
exact
Lame's
limiting
stresses
in a semi-infinite
results
stress-lnte1_sity cylinder
shows
obtained
an edge-crack
curves
t
from
eq.
factor
for
by
(3) where
eq.
results
two
and
[I0] on
plate,
(5) for
and
various
symmetric the
those
solution the
for inside
is-given a/t
surface
following
of references
by
a/c
= 0
of
the
and
alt
cylinder
eq.
(5) with
the
inside
= O. and
a/t
the
= 0.
ratios. cracks
on
approximate [$] and
expression
of a presfor
F
has
[9]:
(6)
M1 =
1.13
-
0.09
_C
(7)
(8)
0.89 H 2 = 0.54
+ -0.2+--
0.65
g : 1 +
0.i
a
c
+ --
+ 0.35
(10)
(l - sin
5
"_
and
f_ =
for
0 -
