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Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
1/10

Organization (S): EDF/AMA, AUSY France

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
V6.01.100 document

SSNA100 - Tube de Bree: Method Zarka-Rack
(ZAC)

Summary:

This test of nonlinear quasi-static mechanics consists in modelling an infinite cylinder subjected to a pressure
intern and a variation in temperature in the thickness depending recurringly on time (tube of BREE).
behavior is elastoplastic with a criterion of Von Mises and a linear kinematic work hardening. One
validate thus the simplified method Zarka-Casier (ZAC) into axisymmetric for a radial loading in a case
of adaptation (T=30°C) then in a case of accommodation (T=120°C). This method is a postprocessing
realized by command POST_ZAC [U4.83.21]. For more details see document [R7.06.01].
Two modelings make it possible to test the method in 2D and 3D:

· in modeling a: the cylinder is modelled by axisymmetric elements 2D (QUAD8),
· in modeling b: the cylinder is modelled in 3D (meshs HEXA20).

The results obtained by Code_Aster are very close to the reference solution.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
2/10

1
Problem of reference

1.1 Geometry

Z
D
+ E
E
a=0.2
p
B
With
- E
C
F
R
R=10.
d=0.12


1.2
Material properties

E = 200.000 MPa

= 0.3

= 1.e-4 °C1

Tréf = 0.

Elastoplastic relation between behavior and criterion of Von Mises and kinematic work hardening
linear:

T
E =
.
18181 MPa


y =
.
200 MPa

1.3
Boundary conditions and loadings

On AB: uz = 0

Loading:

· constant uniform pressure p = 2. MPa on CD.
· linear temperature in the thickness and varying according to time:

(
2 R - R)
T

(R) = -
(T) with (T) = (
) F (T)
D
E
E
2


E
T/2
T
0
1
3
5
7
9
11
13…
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
3/10

2
Reference solution

2.1
Method of calculation used for the reference solution

Elastoplastic calculation carried out with Code Aster on twenty cycles in the case of adaptation
(T = 30°C).
Analytical solution in the case of accommodation with T = 120°C ([bib1]).

2.2
Results of reference

p
moy
p


p


p



and
in the case of adaptation (
,
,
,
,
,
in the case
zz
zz
zz
zz
zz
zz
lim
zzlim
moy
moy
inf
inf
zz
sup
sup
of accommodation) statements at the points C (skin interns), E (average surface) and F (external skin).

2.3 References
bibliographical

[1]
Contract EDF SEPTEN NC 4158. Study relating to the behavior of the components of
nuclear engines. Laboratory of mechanics of the solids. Polytechnic school,
Palaiseau, June 1985.
[2]
S. TAHERI: “Method ZAC”. Note EDF/DER HI-71/6139 of 14 Juin 1989.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
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SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
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Author (S):
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:
V6.01.100-A Page:
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3 Modeling
With

3.1
Characteristics of modeling

It is about an axisymmetric modeling.

Z
N17
N33
N2
M1
M2
M3
M4
M5
M6
N26
N3
N18
N32
N4
N19
R
6 elements


The loading and the boundary conditions are modelled by:

DDL_IMPO: (NOEUD: (N26 N3), DY: 0.)
TEMP_CALCULEE: temple

where temple is the higher definite positive function for the moments t=1 and 3 [§1.3]

PRES_REP: (GROUP_MA: L4, PRES: 2.)

Two preliminary calculations are carried out:

· an elastic design for the loadings min. and max. on a cycle (tmin = 1 and T
= 3)

max
· an optional elastoplastic calculation until the maximum loading before discharge (your = 3)

3.2
Characteristics of the grid

A number of nodes:
33
A number of meshs and type:
6 elements QUAD8

3.3 Functionalities
tested

Commands



AFFE_MODELE
AFFE
MODELISATION
AXIS

DEFI_MATERIAU
ECRO_LINE
D_SIGM_EPSI

SY
AFFE_CHAR_MECA
PRES_REP
GROUP_MA

TEMP_CALCULEE
STAT_NON_LINE
COMP_INCR
RELATION
VMIS_CINE_LINE

MECA_STATIQUE
OPTION
SIEF_ELGA_DEPL

POST_ZAC
EVOL_ELAS

EVOL_NOLI

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
5/10

4
Results of modeling A

4.1 Values
tested

· T = 30°C, case of adaptation.
The results of reference come from an elastoplastic calculation Aster on 20 cycles.

-
without elastoplastic semi-cycle given (0=0)

Identification Reference
ZAC - Aster %
difference
p
with the N32 node (skin interns)
zz
1.7898 10­3 2.7142
10­3
+51.65%
lim
moy
with the N32 node (skin interns)
86.55 72.27 ­ 16.50%
zzlim

p
with the node N4 (external skin)
zz
3.1031 10­3 3.8715
10­3
+24.76%
lim
moy with the node N4 (external skin)
114.03 128.35 +12.56%
zzlim


-
with elastoplastic semi-cycle given (0_0)

Identification Reference
ZAC - Aster %
difference
p
with the N32 node (skin interns)
zz
1.7898 10­3 1.7872
10­3
­ 0.14%
lim
moy
with the N32 node (skin interns)
86.55 89.09 +2.93%
zzlim

p with the node N4 (external skin)
zz
3.1031 10­3 3.1053
10­3
+0.07%
lim
moy with the node N4 (external skin)
114.03 114.31 +0.24
%
zzlim


· T = 120°C, case of accommodation.
The results of reference come from an analytical solution given in [bib1]

-
without elastoplastic semi-cycle given (0=0)

Identification Reference
ZAC - Aster %
difference
p with the N32 node (skin interns)
zz
4.141 10­3
4.424 10­3
+6.83%
moy

with the N32 node (skin interns)
zzmoy

91.82 87.52 ­ 4.68%
p with the node N4 (external skin)
zz
9.59 10­3 9.75
10­3
+1.64%
moy

with the node N4 (external skin)
zzmoy

191.90 195.17 +1.70%
p with the node N18 (surf.moyenne)
zz
5.96 10­3 6.20
10­3
+4.06%
moy

with the node N18 (surf.moyenne)
zzmoy

319.18 320.45 +0.39%
p with the N32 node (skin interns)
zz
3.6363 10­3 3.6574
10­3
+0.58%
inf
with the N32 node (skin interns)
zzinf

472.72 474.66 +0.41%
p with the node N4 (external skin)
zz
3.6328 10­3 3.6098
10­3
­ 0.63%
inf
with the node N4 (external skin)
zzinf

472.72 473.97 +0.26%
p with the N32 node (skin interns)
zz
3.6363 10­3
3.6604 10­3
+0.66%
sup

with the N32 node (skin interns)
zzsup

472.72 473.31 +0.12%
p with the node N4 (external skin)
zz
3.6328 10­3 3.6097
10­3
­ 0.63%
sup

with the node N4 (external skin)
zzsup

472.72 473.18 +0.09%
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
6/10

-
with elastoplastic semi-cycle given (0_0)

Identification Reference
ZAC - Aster %
difference
p with the N32 node (skin interns)
zz
4.141 10­3 4.177
10­3
+0.89%
moy

with the N32 node (skin interns)
zzmoy

91.82 83.43 ­ 9.13%
p with the node N4 (external skin)
zz
9.59 10­3 9.51
10­3
­ 0.87%
moy

with the node N4 (external skin)
zzmoy

191.90 190.39 ­ 0.79%
p with the node N18 (surf.moyenne)
zz
5.959 10­3 5.963
10­3
+0.08%
moy

with the node N18 (surf.moyenne)
zzmoy

319.18 313.5
­ 1.78%
p with the N32 node (skin interns)
zz
3.6363 10­3
3.6574 10­3
+0.58%
inf
with the N32 node (skin interns)
zzinf

472.72 474.66 +0.41%
p
with the node N4 (external skin)
zz
3.6328 10­3 3.6098
10­3
­ 0.63%
inf
with the node N4 (external skin)
zzinf

472.72 473.97 +0.26%
p with the N32 node (skin interns)
zz
3.6363 10­3
3.6604 10­3
+0.66%
sup

with the N32 node (skin interns)
zzsup

472.72 473.31 +0.12%
p with the node N4 (external skin)
zz
3.6328 10­3 3.6097
10­3
­ 0.63%
sup

with the node N4 (external skin)
zzsup

472.72 473.18 +0.09%

4.2 Remarks

In the case of accommodation (T=120°C), the M1 meshs, m2, M5 and M6 form a zone
adapted whereas the central zone (meshs m2 and m3) remains adapted.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
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5 Modeling
B

5.1
Characteristics of modeling

Modeling 3D

Angle nets: 5°

Z
y
X


The loading and the boundary conditions are modelled by:

DDL_IMPO: (GROUP_NO: facemoy,
DZ: 0.)
face0
DY: 0)
FACE_IMPO: (GROUP_MA: face10
DNOR: 0)
TEMP_CALCULEE: temple

where temple is the higher definite positive function for the moments t=1 and 3 [§1.3]

PRES_REP: (GROUP_MA: facepres, PRES: 2.)

Two preliminary calculations are carried out:

· an elastic design for the loadings min. and max. on a cycle (tmin = 1 and T
= 3)

max
· an optional elastoplastic calculation until the maximum loading before discharge (your = 3)

5.2
Characteristics of the grid

A number of nodes:
793
A number of meshs and type:
120 elements HEXA20

5.3 Functionalities
tested

Commands



AFFE_MODELE
AFFE
MODELISATION
AXIS

DEFI_MATERIAU
ECRO_LINE
D_SIGM_EPSI

SY
AFFE_CHAR_MECA
PRES_REP
GROUP_MA

TEMP_CALCULEE
STAT_NON_LINE
COMP_INCR
RELATION
VMIS_CINE_LINE

MECA_STATIQUE
OPTION
SIEF_ELGA_DEPL

POST_ZAC
EVOL_ELAS

EVOL_NOLI

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
8/10

6
Results of modeling B

6.1 Values
tested

· T = 30°C, case of adaptation.
The results of reference come from an elastoplastic calculation Aster on 20 cycles.

-
without elastoplastic semi-cycle given (0=0)

Identification Reference
ZAC - Aster %
difference
p
at the point C (skin interns)
zz
1.7898 10­3 2.7151
10­3
51.700
lim
p
at the point F (external skin)
zz
3.1031 10­3 3.8794
10­3
25.014
lim

-
with elastoplastic semi-cycle given (0_0)

Identification Reference
ZAC - Aster %
difference
p
at the point C (skin interns)
zz
1.7898 10­3 1.7911
10­3
0.075
lim
p at the point F (external skin)
zz
3.1031 10­3 3.1133
10­3
0.327
lim

· T = 120°C, case of accommodation.
The results of reference come from an analytical solution given in [bib1]

-
without elastoplastic semi-cycle given (0=0)

Identification Reference
ZAC - Aster %
difference
p at the point C (skin interns)
zz
4.141 10­3 4.4393
10­3
7.203
moy

at the point C (skin interns)
zzmoy

91.82 8.782E+01
­ 4.345
p at the point F (external skin)
zz
9.59 10­3 9.766
10­3
1.803
moy

at the point F (external skin)
zzmoy

191.90 1.9537E+02
1.803
p at the point C (skin interns)
zz
3.6363 10­3 3.6299
10­3
­ 0.176
inf
at the point C (skin interns)
zzinf

472.72 4.7328E+02
0.117
p at the point F (external skin)
zz
3.6328 10­3 3.6401
10­3
0.202
inf
at the point F (external skin)
zzinf

472.72 4.7367E+02
0.200
p at the point C (skin interns)
zz
3.6363 10­3 3.632
10­3
­ 0.119
sup

at the point C (skin interns)
zzsup

472.72 4.7205E+02
­ 0.142
p at the point F (external skin)
zz
3.6328 10­3 3.6405
10­3
0.212
sup

at the point F (external skin)
zzsup

472.72 4.7323E+02
0.106

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
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Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
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S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
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-
with elastoplastic semi-cycle given (0_0)

Identification Reference
ZAC - Aster %
difference
p at the point C (skin interns)
zz
4.141 10­3 4.1905
10­3
1.195
moy

at the point C (skin interns)
zzmoy

9.182 10+1 8.305
10+1
­ 9.552
p at the point F (external skin)
zz
9.59 10­3 9.5225
10­3
­ 0.741
moy

at the point F (external skin)
zzmoy

1.9190 10+2 1.9031
10+2
­ 0.832
p
at the point C (skin interns)
zz
3.6363 10­3 3.6299
10­3
­ 0.176
inf
at the point C (skin interns)
zzinf

4.7272 10+2 4.732810+2
0.117
p
at the point F (external skin)
zz
3.6328 10­3 3.64041
10­3
0.202
inf
at the point F (external skin)
zzinf

4.7272 10+2 4.7367
10+2
0.200
p at the point C (skin interns)
zz
3.6363 10­3 3.632
10­3
­ 0.119
sup

at the point C (skin interns)
zzsup

4.7272 10+2 4.720510+2
­ 0.142
p at the point F (external skin)
zz
3.6328 10­3 3.6405
10­3
0.212
sup

at the point F (external skin)
zzsup

4.7272 10+2 4.7323
10+2
0.106

6.2 Remarks

In the case of accommodation (T=120°C), the M1 meshs, m2, M5 and M6 form a zone
adapted whereas the central zone (meshs m2 and m3) remains adapted.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

Code_Aster ®
Version
5.4
Titrate:
SSNA100 - Tube de Bree: Zarka method - Rack (ZAC)

Date:
13/09/02
Author (S):
S. TAHERI, J. Mr. PROIX, Key Mr. BONNAMY
:
V6.01.100-A Page:
10/10

7
Summary of the results

It appears at the end of this test that the results obtained are in concord with the solutions of
reference. However, in the case of the adaptation (T=30°C) projection starting from an initial state
corresponding to the preliminary calculation of an elastoplastic semi-cycle is necessary to approach
accurately solutions. Contrary, the case of accommodation (T=120°C) has already results
correctly approximate starting from an initial state no one.

One will find in [bib1] and [bib2] of other results relating to this test like incremental calculation
realized with the numerical COCAINE or solutions obtained with programmed method ZAC
in the INCA code.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-66/02/001/A

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