Code_Aster ®
Version
7.4
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
: 1/12
Organization (S): EDF-R & D/MMC, AMA
Handbook of Référence
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 tanks REP. The purpose of it is codified in the RSE-M and is to evaluate the factors of intensity
constraints corrected plastically for the coating (in first point of the defect) and for the metal of
base or welded joint (in second point of the defect).
With this intention, one calculates the stress intensity factors elastic to the two points of the defect, with the assistance
constraints 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.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 2/12
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 Défaut Sous Revêtement elliptic ...................... 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
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 3/12
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.
The 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.
Initial defect
Defect considered
in calculation
2b
2a
- decaf
prof_def
Coating
Base metal
ray_int
ep_rev
ep_mdb
Center tank
Appear 1.1-a: Schéma of the defect under coating considered
Conditions of validity of the method:
· penetrating defect in the coating,
decaf
2a
2a
1
·
,
0 2 and
3 and
.
ep _ rev
ep _ rev
(ep _ rev + ep _ mdb)
10
By convention in command POST_K_BETA one selected decaf 0. The default value selected
is decaf = - 2. 10-4.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 4/12
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:
+a (X) has - X
K
=
dx
IA
has has +
X
- has
K01:
+a (X) has + X
K
=
dx
IB
has
- X has
- has
where 2a is the bandwidth (depth of the defect), A and B is the two ends
(respectively in has and +a).
The constraint (X) is the normal constraint 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.
y
Defect bandecirconférentiel
X
With
B
- has
+a
Z
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 5/12
y
Defect bandages longitudinal
X
With
B
- has
+a
Z
For the defect bandages circumferential, one takes (X) =
yy (X)
For the defect bandages longitudinal, one takes (X) =
zz (X)
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
y
Z
X
R
Z
One a: E =
=
Z = -
X
er ey eZ E
E
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 6/12
The basic change for the tensor of the constraints is written:
rr R rZ 1
0
0 xx xy xz 1
0
0
R Z =
0
0 -
1 xy yy yz 0 0
1
rZ Z
ZZ
0
1
0 xz yz
zz
0 - 1
0
=
rr
xx
= -
R
xz
One obtains finally: =
and
zz
=
rZ
xy
=
ZZ
yy
= -
Z
yz
·
Case 2: passage of the total Cartesian base (model 3D) at the cylindrical base
Z
Z
y
R
X
er = cos E
X + sin E
Y
eX = cos E
R - sin E
One a: E = - sin E
cos of
X +
eY
where eY = sin E
R + cos E
eZ = eZ
eZ = eZ
The basic change for the tensor of the constraints is written:
rr R rZ cos
sin
0 XX XY XZ cos
- sin
0
R Z =
- sin
cos
0 XY YY YZ sin
cos
0
rZ Z
ZZ
0
0
1
XZ YZ
ZZ
0
0
1
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 7/12
One obtains finally:
2
2
rr = cos XX + 2 sin cos XY + sin
YY
2
2
R = - sin cos XX + (cos - sin)
XY + sin cos YY
rZ = cos XZ + sin
YZ
2
2
= sin XX - 2 sin cos XY + cos YY
Z = - sin XZ + cos
YZ
ZZ = ZZ
·
Synthesis: components used for the calculation of the stress intensity factors
Circumferential defect: in the cylindrical base is
zz
with an axisymmetric model
yy
with a model 3D
zz
Longitudinal defect: in the cylindrical base is
with an axisymmetric model
zz
2
sin - 2 sin cos
2
+ cos with a model 3D
xx
xy
yy
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 r0 with r0 = (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].
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 8/12
1.2.2 Method of calculation
Integrals giving K
K
and
are calculated per pieces: the decomposition comes from one
IA
IB
subdivision of the interval [ A/2; +a/2] in NR elementary subintervals on which the constraint
useful (X) is linearized:
(X) = I X + I for X II = [have; ai+1] [ A/2; +a/2]
The meeting of the NR subintervals II per 1 I NR reconstitutes the tape [ has; +a].
The contributions of the subinterval II = [have; ai+1] with the calculation of the FIC is given by:
ai+1 X
X has
I
+
I
I
-
K
=
dx
IA
has
+ X has
I
has
K02:
ai+1 X
X has
I
+
I
I
+
K
=
dx
IB
has
- X has
I
has
These integrals can be calculated analytically. One obtains finally relations K03-a:
ai+1
NR
2
2
has
has
X X
X
X
X
K
=
×
Arc sin
1 has
Arc sin
1
IA
I -
+ - - + I
+
-
i=1
2
has 2
has
has
has
have
have 1
NR
2
2 +
= has
K
×
has
X X
X
X
X
Arc sin
1 has
Arc sin
1
IB
I
- + - + I
-
-
i=1
2
has 2
has
has
has
have
N.B.
There are formulas equivalent to the relations, established above after the changes of
variables.
have
= Arc sin
I
has
ai+1
= Arc sin
i+1
has
The FIC are then given by new expressions K03-b:
NR
has
ad interim
have
K
=
×
cos
cos
sin 2
sin 2
IA
-
I
(
-
i+1
I) + (
-
I
ad interim) (
-
i+1
I) +
(
-
i+1
I)
i=1
2
4
NR
has
have
K
=
×
ad interim
cos
cos
sin 2
sin 2
IB
+
I
(
-
i+1
I) - (
+
I
ad interim) (
-
i+1
I) -
(
-
i+1
I)
i=1
2
4
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 9/12
Note:
In practice, the calculation of K
K
and
be carried out on the segment of support of the postulated defect. On
IA
IB
this segment, points A (side coating) and B (side base metal or welded joint) of the fissure
necessarily do not coincide with nodes of the grid.
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, NO1], [NO1, NO2],…, [NON-2, NON-1], [NON-1, B]. The nodes of the grid determine them
terminals. The linear interpolations of the useful constraint (X) are thus carried out on these
subintervals; for the first and the last, one respectively uses the interpolations on
[NO0, NO1] and [NON-1, NON], which will thus be used for calculation of the FIC only on part of their
field of definition (NO0 is the immediate predecessor of A on the radial path, NON is it
immediate successor of B).
The formulas K03-a or K03-b are then applied for the calculation of K
K
and
.
IA
IB
It is important to note that this calculation uses the constraints with the nodes of the grid, to leave
whose the linear interpolations per pieces are given.
1.3 Stage n°2
: Geometrical corrections for Défaut Sous
Elliptic coating
Stress intensity factors K
K
and
determined at the end of the stage n°1 concern
IA
IB
a defect bandages in a plate of infinite size.
The postulated defect is Défaut Sous Revêtement of profile elliptic. Factors of intensity of
constraints determined for this type of geometry are obtained by application of corrections
geometrical on K
K
and
.
IA
IB
[Figure1.1-a] allows to define the geometry of the DSR considered for calculation.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 10/12
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 er.
·
The length 2b of a longitudinal DSR corresponds to its axial dimension, i.e according to
direction carried by eZ.
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 joint. 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 Z = has (has + (ep _ rev+ decaf), 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 K04
2
3
4
5
F
= 0 998742
,
+ 0142801
,
Z - 1133379
,
Z + 5 491256
,
Z - 8 981896
,
Z
5 765252
,
Z
Ba
+
Point B side base metal (or welded joint): K05 formulas
1 - 0 012328
,
Z + 0 395205 2
,
Z - 0 527964 3
,
Z + 0 432714 4
,
Z
if 0 Z
F
=
0 92
,
bB
- 414 20286
,
+1336 75998
,
Z - 143611970 2
,
Z + 51514949 3
,
Z
if 0 92
,
< Z 1
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 Profondeur of the Longueur defect
1
K06: F = F =
With
B
has, 165
1 +,
1 464
B
Second case: B has Longueur of the Profondeur defect
B
1
K07: F = F =
×
With
B
has
B, 165
1 +,
1 464
has
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 11/12
1.3.3 Stress intensity factors of an elliptic DSR
Stress intensity factors of Défaut Sous Revêtement elliptic, obtained by correction
FIC of a defect bandage in a plate of infinite size, are given by the relations
Point A side coating:
K08-a: K
= F × F × K
IA
With
Ba
IA
Point B side base metal (or welded joint): K08-b: K
= F × F × K
IB
B
bB
IB
1.4
Stage n°3: Plastic correction known as “correction”
1.4.1 Formulation of the correction
The stress intensity factors determined by the relations K08-a and K08-b are those of one
Elliptic DSR, under the assumption of an elastic behavior of materials.
The correction, specific to the DSR stuck to the interface, makes it possible to take account of plasticization
with the two points of the fissure side coating (point A) and side base metal or joint welded (point
B).
The corrective factors are defined by the following relations:
R
36 teststemyàs
= 1+ 3
,
0 × tanh
2
With
ep _ rev
1 K
K09:
where R
=
IA
teststemyà
R
36 teststemyàs
6 teststemyàs
= 1+ 5
,
0 × tanh
B
ep _ rev
ep_rev is the thickness of the coating, is the yield stress of the coating at the temperature of
teststemyà
point A.
From where FIC corrected with the two points of the fissure:
K = × K
K10: With
With
IA
K = × K
B
B
IB
1.4.2 Plastic correction progressively with the history of the loading
The plastic correction is calculated according to the formulas K09 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
K11: 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 Kmax obtained hitherto, one updates
Kmax. The correction finally applied is written
K12: K CP = K I + Kmax
In phase of discharge, the plastic correction applied is the addition of Kmax 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.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Code_Aster ®
Version
7.4
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
: 12/12
Algorithmic
One initializes Kmax = 0
One initializes K 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 (tn) K I_ast then (phase of discharge)
K CP (tn) = K I (tn) + Kmax
If not (phase of load)
If (tn) × K I (tn) > K I (tn) + Kmax then
K CP (tn) = (tn) × K I (tn)
Kmax = K CP (tn) K I (tn)
If not
K CP (tn) = K I (tn) + Kmax
End If
End If
K I_ast = K I (tn)
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.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A
Outline document