Download - Batura A . S . , Orynyak I.V
Batura A.S., Orynyak I.V.Batura A.S., Orynyak I.V.
IPS NASUIPS NASU
Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine
Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine
ENGINEERING METHODS FOR STRESS INTENSITY FACTOR CALCULATION FOR 2-D AND 3-D BODIES WITH CRACKS
ENGINEERING METHODS FOR STRESS INTENSITY FACTOR CALCULATION FOR 2-D AND 3-D BODIES WITH CRACKS
IPS NASUIPS NASUWeight Function Method for plane bodiesWeight Function Method for plane bodies
x a
a
I dxxxGhSIFK0
)(),(
),( xGh - weight function,
x - the law of stress distribution,
G – geometry parameters.
xhGAxhxGh CA 00,
axxhA -1
1~)( - asymptotical (singular) part of WF,
axxhC -1~)( - correction (regular) part of WF.
Then for any specified stress law x (for example i
i a
xx
) obtain
,00
i
IC
i
IA
i
I IGAIK where .,00
a
C
i
i
IC
a
A
i
i
IA dxxhax
Idxxhax
I
i
IAI and i
ICI doesn’t depend upon geometry.
IPS NASUIPS NASUWeight Function Method for plane bodiesWeight Function Method for plane bodies
In particular, for a plane body with an edge crack
The main idea of Weight Function Methods:The main idea of Weight Function Methods:If we have the SIF solution for one particular loading we can obtain the SIF If we have the SIF solution for one particular loading we can obtain the SIF solution for any other law of loading.solution for any other law of loading.
;
1
2)(
cA
ax
a
cxh
;1151
)1(32
1)(
)(2
00
ax
ax
a
xaAxhC
.4
22
2
c
IPS NASUIPS NASUApplication of WFM for a pipesApplication of WFM for a pipes
In the circular pipe additional force N and moment M appear. Angle and displacement discontinuity can be expressed in the next form:
Crack compliance methodCrack compliance method(modification of (modification of Cheng & Finnie approachCheng & Finnie approach))
),( ,' NNMMqqEtu
),(6
,' NNMMqqE
,/6 2tMM ,/ tNN ,)()()(0
dYY iNi ,)()()(0
dYY iMi
where YN, YM – are the dimensionless SIF, induced by M and N as in the plane body,
.1
,/ 2/
E
Eta .1
,/ 2/
E
Eta
IPS NASUIPS NASUApplication of WFM for a pipesApplication of WFM for a pipes
Crack compliance methodCrack compliance method(modification of (modification of Cheng & Finnie approachCheng & Finnie approach))
qIq qYaK NNIN YaK
MMIM YaK
- caused by loading, - caused by force,
- caused by moment.
IqIMINI KKKK Obtain result SIF :
SIF is smaller than in the case of straight plane !Using equilibrium equations for a ring and initial parameter method, get the expression for a dimensionless SIF decrease from the case of straight plane (Y0):
)),(1(6
)(9
6
1 0
/0
/
00
Y
Ep
tR
EYY
YY
M
pM where - dimensionless
pressure.p
IPS NASUIPS NASUApplication of WFM for a pipesApplication of WFM for a pipes
Crack compliance methodCrack compliance method(modification of (modification of Cheng & Finnie approachCheng & Finnie approach))
Result plotsResult plots
Conclusion:Conclusion:Advanced SIF formula for pipes was obtained. The feature of the SIF Advanced SIF formula for pipes was obtained. The feature of the SIF decreasing at rising of the pressure was found. decreasing at rising of the pressure was found.
IPS NASUIPS NASUWeight Function Method for 3-D bodiesWeight Function Method for 3-D bodies
x
y
Q/ Q
r
R
Qx Qxy
Qy
a
b
)(
/ ,)( /
SQQ
dsyxWQSIF (1)
CQQ
AQQQQ
WGDWW /// , (2)
EAQQ
AQQ
WW ,// - elliptical crack,
EAQQ
EAQQ
AQQ X
WWW ,,/// - for semi-
elliptical crack,
EAQQ
EAQQ
EAQQ
EAQQ
AQQ XYYX
WWWWW ,,,,///// - for quarter-elliptical crack
ba
l
dl
R
ra
W
QQQQ
AQQ
/,cossin
cossin)(,
)(
)(1
)(2222
242
22
21
2
2
41
/
/
/
RrWW AQQ
CQQ
/1// -correction part
- asymptotical part for elliptical crack.
IPS NASUIPS NASUWeight Function Method for 3-D bodiesWeight Function Method for 3-D bodies
)1(0 const
CI
AII IGDIK ,0
geometry dependentgeometry dependentloading dependentloading dependent
If 0IK is known we obtain GD , and can calculate IK
for any law of loading.
,)(~
)(~)(~,
C
AI
I
IKGD
where IK~ - is a known SIF for any law of loading.
Similarly to the 2-D case,
So
.,)()(
// S
CQQ
CI
S
AQQ
AI dsWIdsWI
SIF along crack front (angle), homogeneous loadingSIF along crack front (angle), homogeneous loading
IPS NASUIPS NASUCheck of the PWFM accuracy for
semi-elliptic cracks
Check of the PWFM accuracy for semi-elliptic cracks
a/l=0.2 (a/t=0.8)
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
0 20 40 60 80 100
Angle, degree
Tension by the PWFM Tension by Raju-Newman
Bending by the PWFM Banding by Raju-Newman
a/l=0.4 (a/t=0.8)
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 20 40 60 80 100
Angle, degree
Tension by the PWFM Tension by Raju-Newman
Bending by the PWFM Bending by Raju-Newman
0
90
IPS NASUIPS NASU
a/l=0.6 (a/t=0.8)
0
0,2
0,4
0,6
0,8
1
1,2
1,4
0 20 40 60 80 100
Angle, degree
Tension by the PWFM Tension by Raju-Newman
Bending by the PWFM Bending by Raju-Newman
a/l=1.0 (a/t=0.8)
-0,2
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 20 40 60 80 100
Angle, degree
Tension by the PWFM Tension by Raju-Newmen
Bending by the PWFM Bending by Raju-Newman
a/l=2.0 (a/t=0.8)
-0,2
0
0,2
0,4
0,6
0,8
1
0 20 40 60 80 100
Angle, degree
Tension by the PWFM Tension by Raju-Newman
Bending by the PWFM Bending by Raju-Newman
IPS NASUIPS NASU
IPS NASUIPS NASU
Homogeneous loading
1
1,2
1,4
1,6
1,8
2
0 0,2 0,4 0,6 0,8 1 1,2
a/l
90 degree by the PWFM 90 degree by Murakami
0 degree by the PWFM 0 degree by Murakami
Linear loading
0,2
0,4
0,6
0,8
1
1,2
1,4
0 0,2 0,4 0,6 0,8 1 1,2
a/l
90 degree by the PWFM 90 degree by Murakami
0 degree by the PWFM 0 degree by Murakami
Dependence SIF from ratio a/lDependence SIF from ratio a/l
IPS NASUIPS NASU
Quadratic loading
0
0,2
0,4
0,6
0,8
1
1,2
0 0,2 0,4 0,6 0,8 1 1,2
a/l
90 degree by the PWFM 90 degree by Murakami
0 degree by the PWFM 0 degree by Murakami
Cubic loading
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,2 0,4 0,6 0,8 1 1,2
a/l
90 degree by the PWFM 90 degree by Murakami
0 degree by the PWFM 0 degree by Murakami
Dependence SIF from ratio a/lDependence SIF from ratio a/l
IPS NASUIPS NASUWeight Function Method for 3-D bodies.
Simplified (speed up) approach.Weight Function Method for 3-D bodies.
Simplified (speed up) approach.
The problem: The problem: triple integraltriple integral (square and contour) with (square and contour) with singularity at the edge high computation singularity at the edge high computation costcost (especially for repeating – fatigue, stress-corrosion,… – (especially for repeating – fatigue, stress-corrosion,… – calculations) !!!calculations) !!!
The solution: approximation of the stress law with The solution: approximation of the stress law with
function of the next type: , function of the next type: ,
calculation of the SIF array for each stress functioncalculation of the SIF array for each stress function
. Approximate SIF function can be . Approximate SIF function can be
build as linear combination of precalculated .build as linear combination of precalculated .
ji
ijijapprox yxfMyx,
,,
1,, ijijij Myxfyx
ijY
ijY
IPS NASUIPS NASUWeight Function Method for 3-D bodies.
Simplified (speed up) approach.Weight Function Method for 3-D bodies.
Simplified (speed up) approach.
Polynomial examplePolynomial example
22 1 k
ji
jiij b
x
a
yMyx
0,
,
),,(),(),()(
)(),(),(
41
DII
aM
kEKG C
ijAij
ij
ijij
The expression for dimensionless SIF functions :ijG
dSWbx
ay
a
kEI A
ji
S
Aij /
)(41
)(
)(),(
dSW
bx
ay
a
kEI C
ji
S
Cij /
)(41
)(
)(),(
IPS NASUIPS NASU
Weight Function Method for 3-D bodies.Simplified (speed up) approach.
Weight Function Method for 3-D bodies.Simplified (speed up) approach.
Polynomial examplePolynomial example For 2,0 simple expressions for Iij
A,C(α) were obtained :
0761.00515.00108.0;0
0018.01576.01369.00439.0;0
0285.03858.04071.01572.0;0
96.019.04747.02448.0;0
2303
2302
2301
2300
A
A
A
A
I
I
I
I 0327.00297.00357.00127.0;0
0703.00578.00717.00267.0;0
2.01049.01262.00452.0;0
1;0
2303
2302
2301
00
C
C
C
C
I
I
I
I
4216.00196.01513.00621.0;2
496.00236.01475.00655.0;2
6297.00420.01005.00524.0;2
9808.01753.02455.00901.0;2
2303
2302
2301
2300
A
A
A
A
I
I
I
I 1335.00227.00484.00165.0;2
205.004.00343.00081.0;2
374.00481.00278.00057.0;2
1;2
2303
2302
2301
00
C
C
C
C
I
I
I
I
Depth of crack Loading (0, j) I0j(0) I0j(π/2)
=a/t j Exact Approx. Exact Approx.
0.2 0123
1.1400.1970.0740.038
1.1400.1960.0780.040
1.0150.7150.5880.512
1.0150.7260.6050.533
0.5 0123
1.2190.2210.0850.044
1.2190.2140.0850.044
1.0500.7290.5960.515
1.0500.7420.6100.540
Semi-elliptical crack on the inner surface of the cylinder.
IPS NASUIPS NASU
Application of the peveloped methods:Software “ReactorA”Application of the peveloped methods:Software “ReactorA”
• Residual life is calculated Residual life is calculated deterministically and deterministically and probabilistically (MASTER probabilistically (MASTER CURVE approach) for CURVE approach) for various points of crack frontvarious points of crack front
• Residual life is calculated Residual life is calculated deterministically and deterministically and probabilistically (MASTER probabilistically (MASTER CURVE approach) for CURVE approach) for various points of crack frontvarious points of crack front
• This program is intended This program is intended for calculation of reactor for calculation of reactor pressure vessel residual life pressure vessel residual life and safety margin with and safety margin with respect to brittle fracturerespect to brittle fracture.
• This program is intended This program is intended for calculation of reactor for calculation of reactor pressure vessel residual life pressure vessel residual life and safety margin with and safety margin with respect to brittle fracturerespect to brittle fracture.
• User sets loading and User sets loading and temperature fields in the temperature fields in the different moments of time. different moments of time. Then material fracture Then material fracture toughness, embrittlement toughness, embrittlement parameters are also set by parameters are also set by useruser.
• User sets loading and User sets loading and temperature fields in the temperature fields in the different moments of time. different moments of time. Then material fracture Then material fracture toughness, embrittlement toughness, embrittlement parameters are also set by parameters are also set by useruser.
IPS NASUIPS NASUReactorA advantagesReactorA advantages
• The sizes of stress and temperature fields' aren't bounded• Number of time moments is bounded only by the
memory size • Cladding is taken into account • Welding seam and heat-affected area are taken into
account • Deterioration is taken into account not only as shift of
the material fracture toughness function but also as its inclination
• Original feature of the software is using of the author variant of the weight function method. It allows to set loading on the crack surface in the form of table.
• The sizes of stress and temperature fields' aren't bounded• Number of time moments is bounded only by the
memory size • Cladding is taken into account • Welding seam and heat-affected area are taken into
account • Deterioration is taken into account not only as shift of
the material fracture toughness function but also as its inclination
• Original feature of the software is using of the author variant of the weight function method. It allows to set loading on the crack surface in the form of table.
Input Data
1) Stress field for time1) Stress field for time it
Table arbitrary sizeTable arbitrary size
IPS NASUIPS NASU3. Residual Life calculation of the NPP
pressure vessel using fracture mechanics methods
3. Residual Life calculation of the NPP pressure vessel using fracture mechanics
methods
IPS NASUIPS NASU
2) Temperature field for time2) Temperature field for time0t it
Input Data
Table arbitrary sizeTable arbitrary size
a) Axial with weld seama) Axial with weld seam
IPS NASUIPS NASU
Input Data
weld seamheat-affected zonebase materialcladdingcrack
weld seamheat-affected zonebase materialcladdingcrack
base materialcladdingcrack
base materialcladdingcrack
b) circumferentialb) circumferential
3) Crack types3) Crack types
)f(TAKcI
IPS NASUIPS NASU
4) The basic material characteristics4) The basic material characteristics
1. Arctangents 1. Arctangents 0arctan2 TTBAK
cI
2. Exponent2. Exponent
0exp TTBAKcI
Common shape of the crack growth resistance function is
for user function A takes from coordinates of first point
Common shape of the crack growth resistance function is
for user function A takes from coordinates of first point
3. User (pointed) function3. User (pointed) function
IPS NASUIPS NASU
1. Shift1. Shift
TTAKcI f
2. Inclination2. Inclination
TT
TTTAK
cI
1
1f
A
ICK
T
T
A
ICK
T
T
5) Shift and inclination conceptions 5) Shift and inclination conceptions
nn
FF YTF
ffAAT
exp
0
00
IPS NASUIPS NASU
a)Analytical forma)Analytical form
b)Table formb)Table form
6) Dependence of shift on radiation6) Dependence of shift on radiation
IPS NASUIPS NASU Results
Scenario – Break of the Steam Generator Collector WWER-1000 operated at full powerScenario – Break of the Steam Generator Collector WWER-1000 operated at full power
It is given : - stress field, - temperature field,
= 1000, 2000, 2800, 3000, 3160, 3600, 4000 sec - time points
It is given : - stress field, - temperature field,
= 1000, 2000, 2800, 3000, 3160, 3600, 4000 sec - time points
Axial crack. Half-length l - 40 мм., depth a - 50 мм.
Axial crack. Half-length l - 40 мм., depth a - 50 мм.
ii tT
it
ii t
IPS NASUIPS NASU
a) Dependences of the calculated and critical SIF from temperature for time = 3000 sec
a) Dependences of the calculated and critical SIF from temperature for time = 3000 sec
SIF for base material --//-- for weld seam
Critical SIF for base material --//-- for weld seam
--//-- for heat-affected area
SIF for base material --//-- for weld seam
Critical SIF for base material --//-- for weld seam
--//-- for heat-affected area
it
IPS NASUIPS NASU
history for basic material --//-- for weld seam critical SIF for basic material --//-- for weld seam
--//-- for heat-affected area
history for basic material --//-- for weld seam critical SIF for basic material --//-- for weld seam
--//-- for heat-affected area
b) History of the dependences calculated SIF from temperature for some points and all times intervals and
critical SIF
b) History of the dependences calculated SIF from temperature for some points and all times intervals and
critical SIF
T
IPS NASUIPS NASU
fields for chosen history pointsminimal marginmargin for time points
fields for chosen history pointsminimal marginmargin for time points
c) Table of the calculated temperature margin
for all points of crack front and time points
c) Table of the calculated temperature margin
for all points of crack front and time points
T
T
calculated temperature marginshift of the temperature by user table
shift of the temperature by analytical model
calculated temperature marginshift of the temperature by user table
shift of the temperature by analytical model
IPS NASUIPS NASUd) Figure of the calculated margind) Figure of the calculated margin
IPS NASUIPS NASU
New geometry for axial crackNew geometry for axial crack
Calculated temperature marginCalculated temperature margin
Half length l - 60мм Depth a - 40 ммHalf length l - 60мм Depth a - 40 мм
Results for other crack geometries
New geometry for axial crackNew geometry for axial crack
Half length l - 40мм Depth a - 60 ммHalf length l - 40мм Depth a - 60 мм
IPS NASUIPS NASU
Calculated temperature marginCalculated temperature margin
Half length l - 60мм Depth a - 30 ммHalf length l - 60мм Depth a - 30 мм
New geometry for circumferential crackNew geometry for circumferential crack
IPS NASUIPS NASU
calculated temperature margincalculated temperature margin
IPS NASUIPS NASU
1. Failure probability calculation for structural element 1. Failure probability calculation for structural element
bIi
f KTK
KK
B
BP i
imin0
min
0exp1
2. Failure probability calculation for crack2. Failure probability calculation for crack
N
iiff PP
1, )1(1
3. Calculation parameters 3. Calculation parameters
))(019,0exp(7731 00 xTTTK
4. In addition4. In addition
Кmin , K0(Т), В0, b - arbitrarily
Pf = 63,2% Кmin = 20 В0 = 25 мм b = 4
Implementation MASTER CURVE Conception
Implementation MASTER CURVE Conception
For time T =0 failure probability equal 1.07*10-05For time T =0 failure probability equal 1.07*10-05
IPS NASUIPS NASU
Time point t4 = 3000 sec
Axial crack half length l - 40 мм., depth a - 50 мм.
Time point t4 = 3000 sec
Axial crack half length l - 40 мм., depth a - 50 мм.
50
60
70
80
90
100
110
0 20 40 60 80 100 120 140 160 180
Angle, degree
K1
SIF dependences on angleSIF dependences on angle
Result for main scenario
Dependences of logarithm probability on TDependences of logarithm probability on T
IPS NASUIPS NASU
ln(Pf) from deltaT
-14
-12
-10
-8
-6
-4
-2
0
0 50 100 150 200
deltaT
ln(P
f)
IPS NASUIPS NASUProbability density for T = 50Probability density for T = 50
IPS NASUIPS NASU
Application of the developed methods:Software “WFM”Application of the developed methods:Software “WFM”
• SIF, grow of the crack dimensions in time and endurance are SIF, grow of the crack dimensions in time and endurance are calculated. “Until specified depth” or “until specified count of cycles” calculated. “Until specified depth” or “until specified count of cycles” modes are presented. modes are presented.
• SIF, grow of the crack dimensions in time and endurance are SIF, grow of the crack dimensions in time and endurance are calculated. “Until specified depth” or “until specified count of cycles” calculated. “Until specified depth” or “until specified count of cycles” modes are presented. modes are presented.
• This program is intended This program is intended
for for SIFSIF calculation for calculation for different (1-D and 2-D) types different (1-D and 2-D) types of cracks and for endurance of cracks and for endurance estimation with using estimation with using different fatigue and stress-different fatigue and stress-corrosion laws.corrosion laws.
• This program is intended This program is intended
for for SIFSIF calculation for calculation for different (1-D and 2-D) types different (1-D and 2-D) types of cracks and for endurance of cracks and for endurance estimation with using estimation with using different fatigue and stress-different fatigue and stress-corrosion laws.corrosion laws.
• User sets “maximum”, User sets “maximum”, “minimum” and “corrosion” “minimum” and “corrosion” loading fields. loading fields.
• User sets “maximum”, User sets “maximum”, “minimum” and “corrosion” “minimum” and “corrosion” loading fields. loading fields.
1. Damages
2. Cracks
IPS NASUIPS NASUWFM: implemented types
of damages and cracksWFM: implemented types
of damages and cracks
IPS NASUIPS NASUWFM: example of result windowWFM: example of result window
• Input and output data can be exchanged with clipboard. . • Input and output data can be exchanged with clipboard. .
IPS NASUIPS NASU CONCLUSIONCONCLUSION
1. Efficient methods of stress intensity factor (SIF)calculation are developed.
2. The computer software which reflected all modern requirements for brittle strength analysis of Reactor Pressure Vessel is created.
1. Efficient methods of stress intensity factor (SIF)calculation are developed.
2. The computer software which reflected all modern requirements for brittle strength analysis of Reactor Pressure Vessel is created.