Code_Aster ®
Version
6.4
Titrate:
SSNV163 - Calcul of clean creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
1/16
Organization (S): EDF-R & D/MMC
Handbook of Validation
V6.04 booklet: Non-linear statics of the voluminal structures
Document: V6.04.163
SSNV163 - Clean Calcul of creep
with model UMLV
Summary:
This test makes it possible to validate the model of clean creep UMLV. The results of this test are compared with
analytical solution for three modelings: 3D, axisymmetric and plane constraints.
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
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2/16
1
Problem of reference
1.1 Geometry
Z
E
H
y
X
L
height:
H = 1,00 [m]
width:
L = 1,00 [m]
thickness:
E = 1,00 [m]
1.2
Properties of material
E = 31 GPa
=,
0 2
Here one informs also the curve sorption-desorption which connects the water content C to the hygroscopy h.
In this case one supposed that the numerical values of C and H are the same ones.
Parameters specific to clean creep:
K Sr =,
2 0E + 5 [MPa]
spherical part: rigidity connects associated with the formed skeleton
by blocks of hydrates on a mesoscopic scale
K if = 0
,
5th + 4 [MPa]
spherical part: rigidity connects intrinsically associated
with the hydrates on a microscopic scale
K Dr. =,
5 0E + 4 [MPa]
deviatoric part: rigidity associated with the capacity with water
adsorbed to transmit loads (load bearing toilets)
S
R =,
4 0E +10 [MPa.s]
spherical part:viscosity connects associated with the mechanism
of diffusion within capillary porosity
S
I = 0
,
1 E +11 [MPa.s]
spherical part: viscosity connects associated with the mechanism
of diffusion interlamellaire
D
R =,
1 0E +10 [MPa.s]
deviatoric part:viscosity associated with the water adsorbed by
layers of hydrates
D
I = 0
,
1 E +11 [MPa.s]
deviatoric part: viscosity of interstitial water.
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
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1.3
Boundary conditions and loadings
In this test, one creates a homogeneous field of drying invariant in the structure, moisture is worth
100% (condition of a sealed test-tube). The mechanical loading corresponds to a compression
one-way according to the vertical direction (Z in 3D or there of 2D); its intensity is 1 [MPa].
load is applied in 1s and is maintained constant for 100 days.
1.4 Conditions
initial
The beginning of calculation is supposed the moment 1. At this moment there is neither field of drying, nor forced
mechanics.
At moment 0, one applies a field of drying corresponding to 100% of hygroscopy.
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SSNV163 - Clean Calcul of creep with model UMLV
Date:
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Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
4/16
2
Reference solution
2.1
Method of calculation
This section presents the analytical resolution supplements problem of a body of test subjected to
a homogeneous and one-way stress field applied instantaneously to the initial moment
and maintained constant thereafter (case of a creep test in simple compression):
= E
0 Z E Z
éq 2.1-1
Whose partly spherical and deviatoric decomposition is written:
= 1
2
1
01 + E
0
Z ez -
0 (ex ex + E y E y)
éq
2.1-2
3 3
2
1
1
3
4
4
4
4
4
4
4
3 2
4
4
4
4
4
4
4
3
part
die
part
viatoric
spherical
By operating a spherical/deviatoric decomposition identical to that of the constraints,
axial deformation is written in the form:
= fs
zz
(
fd
0
) 3+ (20) 3
éq
2.1-3
It is thus necessary successively to solve the response to a level of spherical constraint and one
level of deviatoric constraints.
2.2
Resolution of the equations constitutive of spherical creep [bib2]
The process of deformation spherical of creep is controlled by the system of coupled equations
according to (equations [éq 2.2-1] and [éq 2.2-2], cf [R7.01.06]):
fs
& = 1 H - K -
éq
2.2-1
S [
S
S
fs
R
R]
fs
I
&
R
where S
Kr indicates rigidity connect associated with the skeleton formed by blocks with hydrates on the scale
mesoscopic;
and S
R viscosity connect associated with the mechanism with diffusion within capillary porosity.
fs
1
I
& =
K
K
K
H
K
éq
2.2-1
S [Sr
fs - (Sr + if) fsi] - [S - S fs
R
R] +
I
where
S
ki indicates rigidity connect intrinsically associated with the hydrates on the scale
microscopic;
and S
I viscosity connects associated with the interfoliaceous mechanism of diffusion.
+
+
1
In [éq 2.2-2], hooks
appoint the operator of Mac Cauley: X
= (X + X)
2
The resolution of the preceding system of coupled equations requires to distinguish two cases according to
sign quantity ranging between the hooks of Mac Cauley. In the continuation, one presents
analytical resolution of the response to a level of constraint
S
. The relative humidity is supposed
invariant; the medium is saturated with water.
Handbook of Validation
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HT-26/03/023/A
Code_Aster ®
Version
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
5/16
2.2.1 Case of short-term creep
At the initial moment, T = 0, one applies a spherical constraint S
positive. Deformations of creep
reversible and irreversible are equal to zero (initial conditions). The equation of the system [éq 2.2-2]
is thus written:
fs
&
I (
+
+
T =)
1
0 =
K
K
éq
2.2.1-1
S [2 Sr 0 - S
I -
S]
1
0
=
[- S
S
] =0
I
I
The speed of irreversible deformation of creep is thus equal to zero. One deduces from it that
irreversible deformation of creep is also equal to zero. The speed of deformation unrecoverable remains
equalize to zero until the moment T = t0, defined by the relation [éq 2.2.1-2]:
S
S
fs
2 Kr R (t0)
S
fs
- = 0 R (t0) =
éq
2.2.1-2
S
2 Kr
Until the moment T = t0, the reversible deformation of creep is defined by the following relation:
S
S
1
T
&r =
K
T
1 exp
éq
2.2.1-3
S [S - Sr Sr]
Sr () =
-
-
S
K
S
R
R
R
S
S
R
R =
is the characteristic time associated the reversible deformation of creep. The moment T is
S
0
Kr
thus defined by the relation [éq 2.2.1-4]:
fs
T
0
R (t0)
S
S
=
=
1 - exp -
t0 = ln (2) S
S
R 0.69 R éq
2.2.1-4
S
S
S
2 K
R
Kr
R
The reversible and irreversible deformations of creep are thus determined by:
S
fs (
T)
T
=
R
S
1 - exp - S
K
éq
2.2.1-5
R
R
fs
(T)
= 0
I
During the calculation of the deformations of creep for T > t0, the new initial conditions are thus:
S
fs (
t0)
=
R
S
2 Kr éq
2.2.1-6
fs
(t0)
= 0
I
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HT-26/03/023/A
Code_Aster ®
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
6/16
2.2.2 Case of long-term creep
By expressing speeds of deformations of reversible and irreversible creep according to
deformations of creep, the relation then is obtained:
S
S
K
K
S
K
fs
R
R
fs
I
fs
1
2
& = -
- 4
R + 2
I
+
+
S
R
S
S
S
S
S
R
I
I
R I
éq 2.2.2-1
S
K
K
fs
R
fs
I
fs
1
&
2
I
=
R + -
I + -
S
S
S
S
I
I
I
In order to simplify calculations, the following intermediate variables are defined:
Kr
1
ki
1
Kr
urr =
:
=
, uii =
:
=
and uri =
:
éq
2.2.2-2
R R
I I
I
The system of equations [éq 2.2.2-1] can be put then in the following matric form:
-
- 4
2
fs
& fs
fs
U
U
U
1
U
+ 2u
&
R
rr
laughed
II
R
S
rr
laughed
= =
+
éq
2.2.2-3
& fs
fs
S
2 U
- U
K
- U
I
1
4
4
4
ri2
4
4
4
3
II
I
3
2
1
1r 4
4 2 4
4ri 3
With
S
B
I.e.:
fs
& = A fs
S
+ B éq
2.2.2-4
Let us suppose that matrix A is diagonalisable (this property will be checked thereafter):
1
-
With = P D P where D indicates the diagonal matrix of the eigenvalues of matrix A, P
the matrix of the clean vectors of matrix A and
1
-
P the matrix reverses P. matrix In
carrying out term in the long term the product by the quantity
1
-
P, [éq 2.2.2-4] can be put under
form:
fs, *
&
= D
fs, *
S
+ B *
fs, *
with
= P-1
fs
and B * = P-1 B éq
2.2.2-5
That is to say 1
and the 2 eigenvalues of matrix A. the quantities are defined:
*
*
fs
B
, *
=
: 1
*
and B =
: 1
*
2
* 2
B
[éq 2.2.2-5] is written then:
*
S
1 & (T) =
1 *
1 (T) +
*
B
1
éq
2.2.2-6
*
S
&2 (T) = 2 *
2 (T) + *
b2
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Code_Aster ®
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
7/16
System whose solution is written:
S
*
*
B
1 (T) = -
1 + 1
µ exp (1
T)
1
1
0
éq
2.2.2-7
S
*
B
2 (T)
*
0
= -
2 + µ2
2
exp (2 T)
2
One can then return to initial space, by the means of the matrix of passage; deformations of
creep reversible and irreversible are linear combinations of *
1 and *
2. Eigenvalues
matrix A, 1
and 2 are obtained while solving:
det (A - I)
1 = 0
- urr - 4 uri -
2
I
U
éq 2.2.2-8
II
= 0
2
I + (urr + 4uri + uii) I + urr uii = 0
2 uri
- uii - I
By noticing that urr, uri and uii are strictly positive, the discriminant is thus always
strictly positive. The eigenvalues are thus real and distinct, matrix A is thus
diagonalisable. In addition, none of the two eigenvalues is equal to zero
(= urr uii 0
1
2
). The two eigenvalues are defined by:
- (urr + 4uri + uii) -
1
=
2
éq
2.2.2-9
- (urr + 4uri + uii) +
2 =
2
One can show that the two eigenvalues are indeed negative. Let us show that
second eigenvalue is negative. The spherical deformation of creep is thus asymptotic,
assumption put forth in the model of clean creep spherical [bib1]. Let us determine one now
base clean vectors (X 1, X 2) associated the eigenvalues 1
and 2. It is determined in
solving equation (A - I)
1 X I = 0.
A particular base of clean vectors is written:
x1
1
1
+ uii
2 uri
X 1 =
and X
2 =
with X
1 =
and x2 =
éq
2.2.2-10
1
X
2
2 uri
2 + uii
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SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
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:
V6.04.163-A Page:
8/16
After having checked that P can be reversed indeed, one deduces the solution in space from it
physics:
B *
B *
fs
1
2
R (T) =
S
X
-
1
+
+ x1 1
µ exp (1
T) + µ2 exp (2 T)
1
2
B *
B *
fs
1
2
I (T) =
S
X
-
+ 2
+ 1
µ exp (1
T) + x2 µ2 exp (2 T)
1
2
with
éq 2.2.2-11
1
B * 1 =
[x2 (urr + 2uri) + uri]
x1 x2 - 1
1
B * 2 =
[- (urr + 2uri) - x1 uri]
x1 x2 - 1
Lastly, 1
µ and µ2 are defined by the relations:
1
1
1
µ1 = -
x2 exp 2 t0 - exp 2 t0
(1x x2 -) 1exp ([1
+ 2) t0]
(
)
(
)
2 K
K
R
I
éq
2.2.2-12
1
1
1
µ2 = -
-
exp 1
t0 + 1
X exp 1
t0
(1x x2 -) 1exp ([1
+ 2) t0]
(
)
(
)
2 K
K
R
I
2.3
Resolution of the equations constitutive of creep deviatoric
The deviatoric constraints comprise a reversible part and an irreversible part
(cf [R7.01.06]):
fd
fd
fd
=
+
éq
2.3-1
{
{
R
{
I
contribution
deformation
contribution
water
ue
déviatoriq
water
absobée
total
free
The principal component jème of the total deviatoric deformation is governed by the equations
[éq 2.3-2] and [éq 2.3-3]:
D D, J
D D, J
D J
R R & + K
,
R R
= H
éq
2.3-2
where D
Kr indicates rigidity associated with the capacity with water adsorbed to transmit loads (load
bearing toilets);
and D
R viscosity associated with the water adsorbed by the layers with hydrates.
D D, J
D J
I I & = H
,
éq 2.3-3
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SSNV163 - Clean Calcul of creep with model UMLV
Date:
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Author (S):
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:
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where
D
I indicates the viscosity of interstitial water. The system of equations [éq 2.3-2] and [éq 2.3-3] is more
simple to solve that that governing the spherical behavior owing to the fact that it is uncoupled. One
always suppose that moisture remains equal to 1 during all the loading. The equation [éq 2.3-2]
corresponds to the viscoelastic model of Kelvin whose response to a level of constraint is of
exponential type. As for the equation [éq 2.3-3], the response in deformation is linear with time.
The total deformation of creep is thus written as the sum of the contribution of a chain of
Kelvin and of the contribution of a damping device and series:
K D
R
-
T
D, J
T
1
D
(T) =
+
1
D,
- E
J
R
H (T)
éq
2.3-4
D
K D
I
R
2.4
Summary of the analytical solution
For a uniaxial loading the analytical solutions of the two components of deformation are
known. The contribution of the deviatoric part is written:
D
2
T
1
K T
fd
(T)
R
= 0
+
1 - exp -
éq
2.4-1
3
D
D
K
D
I
R
R
As for the contribution of the spherical part, the solution is defined on two intervals:
S
K T
S
0
R
1 - exp -
T R
S
S
ln 2
S
fs
(T) 3k
R
R
K
=
R
éq 2.4-2
S
0 1
1
R
+
+ µ1 (1+ 1
X) exp (1
T) + µ2 (1+ x2) exp (2t) T >
ln 2
3 S
S
S
K
K
R
I
Kr
The axial deformation is a linear function of the two preceding contributions:
= fs
zz
(
fd
0
) 3+ (20) 3
éq
2.4-3
2.5
Sizes and results of reference
The test is homogeneous. One tests the deformation in an unspecified node.
2.6
Uncertainties on the solution
Exact analytical result.
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Code_Aster ®
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
10/16
2.7 References
bibliographical
[1]
BENBOUDJEMA, F.: Modeling of the deformations differed from the concrete under stresses
biaxial. Application to the buildings engines of nuclear thermal power stations, Mémoire of D.E.A.
Materials Avancés Ingénierie of Structures and Enveloppes, 38 p. (+ appendices) (1999).
[2]
BENBOUDJEMA, F., MEFTAH, F., HEINFLING, G., the POPE, Y.: Numerical study and
analytical of the spherical part of the clean model of creep UMLV for the concrete, notes
technique HT-25/02/040/A, 56 p (2002).
[3]
The POPE, Y.: Relation of behavior UMLV for the clean creep of the concrete,
Reference material de Code_Aster [R7.01.06], 16 p (2002).
Handbook of Validation
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Code_Aster ®
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
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:
V6.04.163-A Page:
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3 Modeling
With
3.1
Characteristics of modeling
Modeling 3D
Z
NO7
NO6
NO8
NO5
NO2
NO3
y
NO4
NO1
X
3.2
Characteristics of the grid
A number of nodes: 8
A number of meshs: 1 of type HEXA 8
6 of type QUAD 4
The following meshs are defined:
S_ARR
NO3 NO7 NO8 NO4
S_AVT
NO1 NO2 NO6 NO5
S_DRT
NO1 NO5 NO8 NO4
S_GCH
NO3 NO2 NO6 NO7
S_INF
NO1 NO2 NO3 NO4
S_SUP
NO5 NO6 NO7 NO8
The boundary conditions in displacement imposed are:
On nodes NO1, NO2, NO3 and NO4: DZ = 0
On nodes NO3, NO7, NO8 and NO4: DY = 0
On nodes NO2, NO6, NO7 and NO8: DX = 0
The loading is consisted of the same field of drying and the same nodal force 1/4 applied
on the four nodes of S_SUP.
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
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:
V6.04.163-A Page:
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3.3 Functionalities
tested
Commands
Key word
DEFI_MATERIAU ELAS_FO
FONC_DESORP
UMLV_FP
K_RS
K_IS
K_RD
V_RS
V_IS
V_RD
V_ID
CREA_CHAMP “AFFE”
NOM_CMP=' TEMP'
TYPE_CHAM=' NOEU_TEMP_R'
AFFE_CHAR_MECA DDL_IMPO
LIAISON_UNIF
FORCE_NODALE
SECH_CALCULEE
STAT_NON_LINE COMP_INCR
RELATION= `UMLV_FP'
3.4
Sizes tested and results
The component with node NO6 was tested.
xx
Moment Reference
Aster %
difference
0. 0. 0.
-
1.0000E+00 3.225814D-05
3.225810D-05
1.37E-04
9.7041E+04 3.867143D-05
3.867140D-05
8.95E-05
1.8389E+06 6.088552D-05
6.088554D-05
3.25E-05
8.6400E+06 1.100478D-04
1.100473D-04
7.27E-06
Handbook of Validation
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Code_Aster ®
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Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
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:
V6.04.163-A Page:
13/16
4 Modeling
B
4.1
Characteristics of modeling
Modeling axisymmetric 2D.
y
N3
N4
N1
N2
X
4.2
Characteristics of the grid
A number of nodes: 4
A number of meshs: 1 of type QUAD 4
4 of type SEG2
The following meshs are defined:
L_INF NO1
NO2
L_DRT NO2
NO4
L_SUP NO4
NO3
L_GCH NO3
NO1
The boundary conditions in displacement imposed are:
On L_GCH: DY = 0
On L_INF: DX = 0
The loading is consisted of the same field of drying and the same nodal force 1/2 applied
on the two nodes of L_SUP.
4.3 Functionalities
tested
Commands
Key word
DEFI_MATERIAU ELAS_FO
FONC_DESORP
UMLV_FP
K_RS
K_IS
K_RD
V_RS
V_IS
V_RD
V_ID
CREA_CHAMP “AFFE”
NOM_CMP=' TEMP'
TYPE_CHAM=' NOEU_TEMP_R'
AFFE_CHAR_MECA DDL_IMPO
LIAISON_UNIF
FORCE_NODALE
SECH_CALCULEE
STAT_NON_LINE COMP_INCR
RELATION= `UMLV_FP'
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SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
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:
V6.04.163-A Page:
14/16
4.4
Sizes tested and results
The component yy with node NO3 was tested
Moment Reference
Aster %
difference
0. 0. 0.
-
1.0000E+00 3.225814D-05
3.225810D-05
1.37E-04
9.7041E+04 3.867143D-05
3.867140D-05
8.95E-05
1.8389E+06 6.088552D-05
6.088554D-05
3.25E-05
8.6400E+06 1.100478D-04
1.100473D-04
7.27E-06
5 Modeling
C
5.1
Characteristics of modeling
Modeling in Contraintes Planes.
y
N3
N4
N1
N2
X
5.2
Characteristics of the grid
A number of nodes: 4
A number of meshs: 1 of type QUAD 4
4 of type SEG2
The following meshs are defined:
L_INF NO1
NO2
L_DRT NO2
NO4
L_SUP NO4
NO3
L_GCH NO3
NO1
The boundary conditions in displacement imposed are:
On L_GCH: DY = 0
On L_INF: DX = 0
The loading is consisted of the same field of drying and the same nodal force 1/2 applied
on the two nodes of L_SUP.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/03/023/A
Code_Aster ®
Version
6.4
Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
15/16
5.3 Functionalities
tested
Commands
Key word
DEFI_MATERIAU ELAS_FO
FONC_DESORP
UMLV_FP
K_RS
K_IS
K_RD
V_RS
V_IS
V_RD
V_ID
CREA_CHAMP “AFFE”
NOM_CMP=' TEMP'
TYPE_CHAM=' NOEU_TEMP_R'
AFFE_CHAR_MECA DDL_IMPO
LIAISON_UNIF
FORCE_NODALE
SECH_CALCULEE
STAT_NON_LINE COMP_INCR
RELATION= `UMLV_FP'
ALGO_C_PLAN=' DEBORST'
5.4
Sizes tested and results
The component yy with node NO3 was tested
Moment Reference
Aster %
difference
0. 0. 0.
-
1.0000E+00 3.225814D-05
3.225810D-05
1.40E-04
9.7041E+04 3.867143D-05
3.867140D-05
9.225E-05
1.8389E+06 6.088552D-05
6.088554D-05
3.08E-05
8.6400E+06 1.100478D-04
1.100478D-04
8.22E-06
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/03/023/A
Code_Aster ®
Version
6.4
Titrate:
SSNV163 - Clean Calcul of creep with model UMLV
Date:
01/10/03
Author (S):
Y. The Key POPE
:
V6.04.163-A Page:
16/16
6
Summary of the results
The values obtained with Code_Aster are in agreement with the values of the analytical solution of
reference. This same test was turned with Castem in Laboratoire de Mécanique in Université of
The Marne Vallée, the same results were obtained.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/03/023/A
Outline document