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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
11/04/05
Author (S):
A. DAHL,
S. BUGAT,
R. FERNANDES
Key
:
R7.02.10-B
Page
:
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R7.02 booklet: Breaking process
HT-66/05/002/A
Organization (S):
EDF-R & D/MMC, AMA















Manual of Reference
R7.02 booklet: Breaking process
Document: R7.02.10



Analyze simplified harmfulness of defect by the method
K-beta





Summary:

The method of analysis presented (method K-beta) is applied to the analysis of harmfulness of a defect located under
the coating of the tanks ITEM. The purpose of it is codified in the RSE-M and is to evaluate the factors of intensity
stresses corrected plastically for the coating (in first point of the defect) and for the metal of
base or welded gasket (in second point of the defect).
With this intention, one calculates the stress intensity factors elastic to the two points of the defect, with aid
stresses with the nodes resulting from the mechanical resolution and residual stresses given by
the user. The reports/ratios of critical tenacities on the stress intensity factors obtained determine
factors of margins.
The theoretical aspects of the method K-beta and its implementation data-processing make the objects of
following paragraphs.
This method corresponds to the Rupt1D approach in the nomenclature of project EDF Epicure.
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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S. BUGAT,
R. FERNANDES
Key
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Count
matters
1
Theoretical aspects of the method K-beta ............................................................................................. 3
1.1
Validity of the method K
................................................................................................................ 3
1.2
Stage n°1: Calculation of the stress intensity factors elastic of a defect bandages in one
plate of infinite size ......................................................................................................... 4
1.2.1
Change of reference mark .......................................................................................................... 5
1.2.2
Method of calculation .................................................................................................................. 8
1.3
Stage n°2: Geometrical corrections for a Defect Under elliptic Coating ...................... 9
1.3.1
Correction by the factors of edge ...................................................................................... 10
1.3.2
Correction by the factors of ellipticity .................................................................................. 10
1.3.3
Stress intensity factors of an elliptic DSR ....................................................... 11
1.4
Stage n°3: Plastic correction known as “correction
“...................................................................... 11
1.4.1
Formulation of the correction
.............................................................................................. 11
1.4.2
Plastic correction progressively with the history of the loading ................................... 11
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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Author (S):
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S. BUGAT,
R. FERNANDES
Key
:
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1
Theoretical aspects of the method K-beta
1.1
Validity of the method K
The method suggested applies to a defect under coating located partly current of one
ferritic steel tank covered by subjected austenitic stainless steel either:
·
with a thermal transient applied on the surface possibly interns combined with one
loading of pressure limited
·
with a loading of direct compression.
Method
is valid only for defects under coating of which the point, side coating,
penetrate slightly in the coating. This is why for calculation, with the initial size of the defect
considered prof_def, one adds the penetration in the coating
|decaf|. [Figure 1.1-a the] precise one
the difference between the initial defect (dimensions re-entered in POST_K_BETA) and the defect
considered in calculation (defect taking account of the penetration in the coating) by
method
.

Center tank
Initial defect
Defect considered
in calculation
prof_def
- decaf
Coating
Base metal
2b
ray_int
ep_rev
2a
ep_mdb
Appear 1.1-a: Diagram of the defect under coating considered
Conditions of validity of the method:
· penetrating defect in the coating,
·
2
,
0
_
rev
ep
decaf
and
3
_
2
rev
ep
has
and
10
1
)
_
_
(
2
+
mdb
ep
rev
ep
has
.
By convention in control POST_K_BETA one selected decaf
0. The default value selected
is decaf = - 2. 10
- 4
.
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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S. BUGAT,
R. FERNANDES
Key
:
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1.2 Stage n°1: Calculation of the stress intensity factors elastic
of a defect bandages in a plate of infinite size
The stress intensity factors elastic of a defect bandages in a plate of dimensions
infinite are given by the following relations:
K
01:




-
+
=
+
-
=
+
-
+
-
has
has
IB
has
has
IA
dx
X
has
X
has
has
X
K
dx
X
has
X
has
has
X
K
)
(
)
(
where 2a is the bandwidth (depth of the defect), A and B is the two ends
(respectively in ­ has and +a).
The stress
(X) is the normal stress useful for the plan of the fissure (forced elastic added
the residual stress).
The configurations “defect circumferential” and “longitudinal defect” are defined by the two sketches
hereafter.
X
y
B
With
- has
+a
Z
Defect bandecirconférentiel
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Analyze simplified harmfulness of defect by the method K-beta
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Key
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X
y
B
With
- has
+a
Defect bandages longitudinal
Z

For the defect bandages circumferential, one takes
()
()
X
X
yy
=
For the defect bandages longitudinal, one takes
()
()
X
X
zz
=

1.2.1 Change of reference mark
1) Basic change

·
Case 1: passage of the local Cartesian base (in the plan of cut of the model
axisymmetric) at the cylindrical base

One a:
E
E
R
X
=
E
E
Z
y
=
E
E
-
=
Z
Z
X
y
Z
R
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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Author (S):
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S. BUGAT,
R. FERNANDES
Key
:
R7.02.10-B
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:
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The basic change for the tensor of the stresses is written:




-








-
=




0
1
0
1
0
0
0
0
1
0
1
0
1
0
0
0
0
1
zz
yz
xz
yz
yy
xy
xz
xy
xx
ZZ
Z
rZ
Z
R
rZ
R
rr
One obtains finally:
and




-
=
=
-
=
=
=
=
yz
Z
xy
rZ
xz
R
yy
ZZ
zz
xx
rr

·
Case 2: passage of the total Cartesian base (model 3D) at the cylindrical base


One a:


=
+
=
-
=


=
+
-
=
+
=
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
Z
Z
R
Y
R
X
Z
Z
Y
X
Y
X
R
where
of
cos
sin
sin
cos
cos
sin
sin
cos

The basic change for the tensor of the stresses is written:




-








-
=




1
0
0
0
cos
sin
0
sin
cos
1
0
0
0
cos
sin
0
sin
cos
ZZ
YZ
XZ
YZ
YY
XY
XZ
XY
XX
ZZ
Z
rZ
Z
R
rZ
R
rr
Z
X
y
Z
R
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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Author (S):
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S. BUGAT,
R. FERNANDES
Key
:
R7.02.10-B
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:
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One obtains finally:
(
)






=
+
-
=
+
-
=
+
=
+
-
+
-
=
+
+
=
ZZ
ZZ
YZ
XZ
Z
YY
XY
XX
YZ
XZ
rZ
YY
XY
XX
R
YY
XY
XX
rr
cos
sin
cos
cos
sin
2
sin
sin
cos
cos
sin
sin
cos
cos
sin
sin
cos
sin
2
cos
2
2
2
2
2
2

·
Synthesis: components used for the calculation of the stress intensity factors
Circumferential defect:
zz
in the cylindrical base is
yy
with an axisymmetric model
zz
with a model 3D
Longitudinal defect:
in the cylindrical base is
zz
with an axisymmetric model
yy
xy
xx
2
2
cos
cos
sin
2
sin
+
-
with a model 3D

2) Translation of the origin
The origin of the reference mark must be relocated radially to coincide with the point medium of
bandage:
R
R ­ R
0
with R
0
= (ray_int + ep_rev + decaf) + has
With:
ray_int: radius interns tank
ep-rev: thickness of the coating
|decaf| : penetration of the defect in the coating
half a: length of the defect considered for calculation
All these sizes are schematized [Figure 1.1-a].
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Analyze simplified harmfulness of defect by the method K-beta
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1.2.2 Method of calculation
Integrals giving
IA
K
and
IB
K
are calculated per pieces: the decomposition comes from one
subdivision of the interval [­ A/2; +a/2] in NR elementary subintervals on which the stress
useful
(X) is linearized:
(X) =
I
X +
I
for X
I
I
= [has
I
; has
i+1
]
[­ A/2; +a/2]
The meeting of the NR subintervals I
I
for 1
I NR reconstitutes the tape [­ has; +a].
Contributions of subinterval I
I
= [has
I
; has
i+1
] to the calculation of the FIC are given by:
K
02:




-
+
+
=
+
-
+
=
+
+
1
1
I
I
I
I
has
has
I
I
I
IB
has
has
I
I
I
IA
dx
X
has
X
has
has
X
K
dx
X
has
X
has
has
X
K
These integrals can be calculated analytically. One obtains finally the relations K
03-a:
















-
-


+






-


+
-


×
=










-
+


+






-


-
+


-
×
=
=
+
=
+
NR
I
I
I
I
I
IB
NR
I
I
I
I
I
IA
has
X
has
X
Arc
has
X
has
X
has
X
Arc
has
has
has
has
K
has
X
has
X
Arc
has
X
has
X
has
X
Arc
has
has
has
has
K
1
2
2
1
1
2
2
1
1
sin
1
2
sin
2
1
sin
1
2
sin
2

N.B.
There are formulas equivalent to the relations, established above after the changes of
variables.






=


=
+
+
has
has
Arc
has
has
Arc
I
I
I
I
1
1
sin
sin
The FIC are then given by the new expressions K
03-b:
(
) (
) (
)
(
)
(
) (
) (
)
(
)






-
-
-
+
-
-


+
×
=


-
+
-
-
+
-


-
×
=
=
+
+
+
=
+
+
+
NR
I
I
I
I
I
I
I
I
I
I
I
I
IB
NR
I
I
I
I
I
I
I
I
I
I
I
I
IA
has
has
has
has
K
has
has
has
has
K
1
1
1
1
1
1
1
1
2
sin
2
sin
4
cos
cos
2
2
sin
2
sin
4
cos
cos
2
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Titrate:
Analyze simplified harmfulness of defect by the method K-beta
Date:
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S. BUGAT,
R. FERNANDES
Key
:
R7.02.10-B
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Note:
In practice, the calculation of
IA
K
and
IB
K
be carried out on the segment of support of the postulated defect. On
this segment, points A (side coating) and B (side base metal or welded gasket) of the fissure
necessarily do not coincide with nodes of the mesh.
A first stage thus consists in positioning A and B on the path of radial support on the basis of
external skin and finishing in external skin. This positioning takes account of the shift of
defect compared to the localization of reference of a DSR, and also depth of the defect.
A translation of the origin is then carried out, the new origin being located in the middle of
segment [A, B] (cf preceding paragraph concerning the change of reference mark).
The NR subintervals on which the calculation of the FIC is broken up are defined by the succession
[A, NO
1
], [NO
1
, NO
2
],…, [NO
N2
, NO
N-1
], [NO
N-1
, B]. The nodes of the mesh determine them
terminals. Linear interpolations of the useful stress
(X) are thus realized on these
subintervals; for the first and the last, one respectively uses the interpolations on
[NO
0
, NO
1
] and [NO
N-1
, NO
NR
], which will thus be used for calculation of the FIC only on part of their
field of definition (NO
0
is the immediate predecessor of A on the radial path, NO
NR
is it
immediate successor of B).
The formulas K
03-a or K03-b is then applied for the calculation of
IA
K
and
IB
K
.
It is important to note that this calculation uses the stresses with the nodes of the mesh, to leave
whose the linear interpolations per pieces are given.
1.3 Stage n°2
: Geometrical corrections for a Defect Under
Elliptic coating
Stress intensity factors
IA
K
and
IB
K
determined at the end of the stage n°1 concern
a defect bandages in a plate of infinite size.
The postulated defect is a Defect Under elliptic Coating of profile. Factors of intensity of
stresses determined for this type of geometry are obtained by application of corrections
geometrical on
IA
K
and
IB
K
.
[Figure1.1-a] allows to define the geometry of the DSR considered for calculation.
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Analyze simplified harmfulness of defect by the method K-beta
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R. FERNANDES
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Certain conventions are fixed:
·
The depth 2a of a longitudinal or circumferential DSR corresponds to its radial dimension,
i.e. according to the direction carried by
E
R
.
·
The length 2b of a longitudinal DSR corresponds to its axial dimension, i.e according to
direction carried by
E
Z
.
The presence of DSR of longitudinal orientation is postulated in the base metal.
[Figure 1.1-a] thus precisely this configuration of defect represents.
·
The length 2b of a circumferential DSR corresponds to its dimension orthoradiale, i.e according to
direction carried by
E
.
The presence of DSR of circumferential orientation is postulated in the welded gasket. By
report/ratio with [Figure 1.1-a], this configuration of defect would be obtained by carrying out one
rotation of 90° of the face of fissure around the small axis of the ellipse.
1.3.1 Correction by the factors of edge
This first correction holds account owing to the fact that the defect is located in a noninfinite medium.
localization of the DSR defined by [Figure 1.1-a] implies corrections in points of fissure side
coating and side base metal.
One defines beforehand the reduced variable of space
(
)
(
)
decaf
_
+
+
=
rev
ep
has
has
Z
, where ep_rev is
the thickness of the coating and decaf is the penetration of the DSR in the coating (see [Figure 1.1-a]).
Point A side coating: formulate K
04
5
4
3
2
765252
5
981896
8
491256
5
133379
1
142801
0
998742
0
Z
,
Z
,
Z
,
Z
,
Z
,
,
F
Ba
+
-
+
-
+
=
Point B side base metal (or welded gasket): formulas K
05



<
+
-
+
-
+
-
+
-
=
1
92
0
14949
515
11970
1436
75998
1336
20286
414
92
0
0
432714
0
527964
0
395205
0
012328
0
1
3
2
4
3
2
Z
,
Z
,
Z
,
Z
,
,
,
Z
Z
,
Z
,
Z
,
Z
,
F
bB
if
if
1.3.2 Correction by the factors of ellipticity
This second correction holds account owing to the fact that the defect found an elliptic profile. It must be
applied to the estimates determined for a defect bandages.
Two cases are distinguished, according to the preponderance of one or the other of two dimensions of the profile
elliptic.
First case: has
B Depth of the defect Length
K
06:


+
=
=
B
has
F
F
B
With
65
,
1
464
,
1
1
1
Second case: B
length of the defect Depth has
K
07:


+
×
=
=
has
B
has
B
F
F
B
With
65
,
1
464
,
1
1
1
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Analyze simplified harmfulness of defect by the method K-beta
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S. BUGAT,
R. FERNANDES
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:
R7.02.10-B
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1.3.3 Stress intensity factors of an elliptic DSR
Stress intensity factors of a Defect Under elliptic Coating, obtained by correction
FIC of a defect bandage in a plate of infinite size, are given by the relations
Point A side coating:
K
08-a:
×
×
=
IA
Ba
With
IA
K
F
F
K
Point B side base metal (or welded gasket): K
08-b:
×
×
=
IB
bB
B
IB
K
F
F
K
1.4
Stage n°3: Plastic correction known as “correction
1.4.1 Formulation of the correction
Stress intensity factors determined by the relations K
08-a and K08-b are those of one
Elliptic DSR, under the assumption of an elastic behavior of materials.
Correction
, specific to the DSR stuck to the interface, allows to take account of plasticization
with the two points of the fissure side coating (point A) and side base metal or gasket welded (point
B).
The corrective factors are defined by the following relations:
K
09:




=








×
+
=




×
+
=
teststemyà
IA
teststemyà
teststemyà
B
teststemyà
With
K
R
rev
ep
R
rev
ep
R
2
6
1
_
36
tanh
5
,
0
1
_
36
tanh
3
,
0
1
where
ep_rev is the thickness of the coating,
teststemyà
is the yield stress of the coating at the temperature of
point A.
From where FIC corrected with the two points of the fissure:
K
10:


×
=
×
=
IB
B
B
IA
With
With
K
K
K
K
1.4.2 Plastic correction progressively with the history of the loading
The plastic correction is calculated according to the formulas K
09 and K10 above for a phase of
charge considered separately in the history of the loading.
To evaluate the plastic correction progressively history of the loading, one must retain with one
moment given the maximum correction obtained on all the preceding phases of load.
Principle
With each new phase of load, one revalues a plastic correction
K
11: K = K
­ K
I
= (
­ 1) × K
I
(even calculation with two points A and B of the fissure, from where the omission of the indices). If this news
plastic correction is higher than the maximum correction
K
max
obtained hitherto, one updates
K
max
. The correction finally applied is written
K
12: K
CP
= K
I
+
K
max
In phase of discharge, the plastic correction applied is the addition of
K
max
obtained on all them
preceding phases of load:
·
no plasticization in phase of discharge,
·
the correction corresponds to the plasticized residue of the preceding phases of load.
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Code_Aster
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Analyze simplified harmfulness of defect by the method K-beta
Date:
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Author (S):
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S. BUGAT,
R. FERNANDES
Key
:
R7.02.10-B
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:
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R7.02 booklet: Breaking process
HT-66/05/002/A
Algorithmic
One initializes
K
max
= 0
K is initialized
I_ast
with a high arbitrary value
·
at the first moment one will be in phase of discharge per comparison to K
I_ast
·
no plasticization at the first moment
Loop over the moments of the history of the loading
If K
I
(T
N
)
K
I_ast
then (phase of discharge)
K
CP
(T
N
) = K
I
(T
N
) +
K
max
If not (phase of load)
If
(T
N
)
× K
I
(T
N
) > K
I
(T
N
) +
K
max
then
K
CP
(T
N
) =
(T
N
)
× K
I
(T
N
)
K
max
= K
CP
(T
N
) ­ K
I
(T
N
)
If not
K
CP
(T
N
) = K
I
(T
N
) +
K
max
End If
End If
K
I_ast
= K
I
(T
N
)
Fine Loops
The same algorithmic one described above is implemented for the plastic corrections of the FIC
with two points A and B of the fissure progressively with the history of the loading.