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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.

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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 -