Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
1/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
Organization (S):
EDF/AMA, IPSN
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
V6.04.150 document
SSNV150 - Triaxial traction with the law
of behavior BETON_DOUBLE_DP
Summary
This case of validation is intended to check the model of behavior 3D BETON_DOUBLE_DP formulated in
tally of the thermo plasticity, for the description of the nonlinear behavior of the concrete, in traction, and in
compression, with the taking into account of the irreversible variations of the thermal characteristics and
mechanics of the concrete, particularly sensitive at high temperature.
The description of cracking is treated within the framework of plasticity, using an energy equivalence,
by identifying the density of energy of cracking in mode I, with the plastic work of a homogeneous medium
equivalent, where the plastic deformation is uniformly distributed, in an “elementary” area. This approach
preserve the continuity of the formulation of the model, on the whole of its behavior, and contributes to avoid
possible numerical difficulties during the change of state of material.
Pathological sensitivity of the numerical solution to the space discretization (mesh), generated by
the introduction of a softening behavior of the concrete in traction and compression, is partially solved
by introducing an energy of cracking or rupture, dependant a characteristic length L
C
, related to
cut elements.
The resolution of the equations constitutive of the model is carried out by an implicit scheme.
It is about a cube with 8 nodes subjected to a triaxial traction, in imposed displacement. This led loading
with the particular case of a hydrostatic state of stress, solved by projection at the top of the cone of traction,
when one places oneself in a hydrostatic diagram forced equivalent/forced. It is about a case test
with analytical solution.
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
2/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
It is about a cube with 8 nodes, whose three faces have a normal displacement no one and the three faces
opposed an imposed and identical normal displacement.
The cube makes 1 mm on side. In modeling A, the cube is directed according to the Oxyz reference mark.
Modeling A
U
2
= U
1
= U
3
1.2
Material properties
To test the irreversible evolution of the mechanical characteristics with the temperature, one applies
a field of temperature decreasing. Certain variables depend on the temperature, others of
drying. Lastly, one applies a coefficient of withdrawal of desiccation not no one, equal to the coefficient of
thermal dilation, to test “data-processing” operation. The thermal deformations will be
thus equal and opposed to the deformations of withdrawal of desiccation. These dependences
intervene only for purely data-processing checks, the mechanical characteristics
can be regarded as constants.
X
y
Z
Uz = 0
Ux = 0
U
1
U
2
Face1xy
Facexy
Faceyz
Face1yz
N1
N2
Face1xz
Facexz
U
3
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
3/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
For the usual linear mechanical characteristics:
Young modulus:
E = 32.000 MPa
of
0°C with 20°C
E = 15.000 MPa
with
400°C (linear decrease)
E = 5.000 MPa
with
800°C (linear decrease)
Poisson's ratio:
N = 0.18
Thermal expansion factor:
has = 10
5
/°C
Coefficient of withdrawal of desiccation: K = 10
5
For the nonlinear mechanical characteristics of model BETON_DOUBLE_DP:
Resistance in uniaxial pressing
:
f' C = 40 NR/mm ²
of
0°C with 400°C
f' C = 15 NR/mm ²
with
800°C (linear decrease)
Resistance in uniaxial traction:
f' T = 4 NR/mm ²
of
0°C with 400°C
f' T = 1.5 NR/mm ²
with
800°C (linear decrease)
Report/ratio of resistances in compression
biaxial/uniaxial pressing:
B = 1.16
Energy of rupture in compression:
Gc =10 Nm/mm ²
Energy of rupture in traction:
WP =0.1 Nm/mm ²
Report/ratio of the limit elastic to resistance
in uniaxial pressing:
30%
1.3
Boundary conditions and loadings mechanical
Field of temperature decreasing of 20°C with 0°C.
Lower face of the cube (facexy):
locked according to OZ.
Higher face of the cube (face1xy):
displacement U
Z
= 0,15 mm
Left face of the cube (faceyz):
locked according to OX.
Right face of the cube (face1yz):
displacement U
X
= 0,15 mm
Front face of the cube (facexz):
locked according to OY.
Face postpones cube (face1xz):
displacement U
y
= 0,15 mm
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
4/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is calculated in an analytical way, knowing that in traction, only the criterion of
traction is activated, and that in the case of a hydrostatic loading, one is projected at the top of
cone of traction. It is thus necessary to solve a linear system of an equation to an unknown factor, which allows
to obtain the plastic deformation cumulated in traction. The aforementioned makes it possible to calculate then
strains and stresses.
2.2
Calculation of the reference solution of reference
For more detail on the notations and the setting in equation, one will refer to the reference document
[R7.01.03]. Only, the main equations are pointed out here.
One notes “has”, imposed displacement following directions X, y and Z. The tensor of deformation is
form (has, has, has, 0., 0., 0.) by taking the usual notations of Code_Aster (three components
main, three components of shearing).
The tensor of stress is form (
, 0., 0., 0.), in modeling A.
General equations of the model:
The equations constitutive of the model are written by distinguishing the isotropic part of the part
deviatoric of the tensors of stresses and deformations.
()
H
tr
= 13
()
S
tr
I
= -
1
3
()
H
tr
= 13
()
~
= - 13tr I
= +
S
I
H
and
= +
~
H
I
The equivalent stress is written then:
()
eq
tr S
= 32
2
In the case of an incremental formulation, and of a variable law of behavior, while noting with one
exponent “E” the elastic components of the stress and the deformation, one obtains:
S
S
E
=
+
+
-
-
+
µ
µ
µ
2
~
and
H E
H
H
K
K
K
=
+
+
-
-
+
3
Criteria in compression
()
F
comp
and in traction
(
)
F
trac
express themselves in the following way:
()
()
F
has
B
F
B
has
B
F
comp
Oct.
Oct.
C
C
eq
H
C
C
=
+
-
=
+
-
.
2
3
()
()
F
C
D
F
D
C
D
F
trac
Oct.
Oct.
T
T
eq
H
T
T
=
+
-
=
+
-
.
2
3
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
5/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
()
()
with
Oct.
Oct.
tr S
tr
=
=
2
3
3
C
: plastic multiplier in compression
T
: plastic multiplier in traction
and
B C D has
,
coefficients of the model
The plastic deformations in traction and compression are expressed:
~
PC
C
eq
B
S
= 2
p Hc
C
has
B
=
3
~
p T
T
eq
D
S
= 2
p HT
T
C
D
=
3
One obtains for the stress:
(
)
S S
E
PC
p T
=
-
+
+
2
µ
~
~
(
)
H
H E
H PC
H p T
K
=
-
+
+
3
S
B
D
S
C
T
E eq
E
=
-
+
+
1 2
2
2
1
µ
H
H E
C
T
K
has
B
C
D
=
-
+
+
3
3
3
for the equivalent stress:
µ
eq
E eq
C
T
B
D
=
-
+
+
2
2
2
The two criteria lead then to a system of two equations to two unknown factors
C
and
T
with
to solve:
(
)
(
)
2
3
2
3
2
3
0
2
3
2
3
2
3
0
2
2
2
2
2
2
B
has
B
B
K has
B
data base
K ac
data base
F
D
C
D
data base
K ac
data base
D
K C
D
F
E eq
H E
C
T
C
C
C
E eq
H E
C
T
T
T
T
µ
µ
µ
µ
+
-
+
-
+
-
+
=
+
-
+
-
+
-
+
=
+
+
+
+
-
+
+
+
+
-
In a similar way, in the case of the only criterion of traction activated, configuration of the case test, one
a system of an equation to an unknown factor obtains
T
to solve:
(
)
2
3
2
3
0
2
2
2
D
C
D
D
K C
D
F
E eq
H E
T
T
T
T
µ
+
-
+
-
+
=
+
+
-
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
6/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
Resolution with projection at the top of the cone of traction:
One thus seeks to solve this system, by using the particular shape of the tensors of stresses
and of deformations, uniforms on the structure.
On the basis of
= (has, has, has, 0., 0., 0.) and of =
(
, 0., 0., 0.). one obtains:
The elastic tensor of stress
(
)
(
)
(
)
µ
µ
µ
X
y
Z
has
has
has
=
+
=
+
=
+
3
2
3
2
3
2
The elastic diverter of stress
S
S
S
X
y
Z
=
=
=
0
0
0
The elastic hydrostatic stress
(
)
()
µ
E H
has
aK
=
+
=
3
2
3
The elastic equivalent stress
E eq
= 0
In the case of a curve of linear work hardening post-peak in traction, the expression of the parameter
of work hardening is as follows:
(
)
()
()
()
F
F
T
tp
T
tp
U
,
,
=
=
-
1
with
()
()
()
U
F
C
T
G
L F
=
2.
.
where
indicate the maximum of temperature during the history of loading,
()
F
T
resistance in traction.
~
p T
= 0
p HT
T
C
D
=
3
H
H E
T
K
C
D
=
-
+
3
3
The equation characterizing projection at the top of the cone of traction is as follows:
C
D
K C
D
F
L F
G
H E
T
T
T C
T
T
-
-
-
=
+
2
2
1
2
0
.
.
éq
2.2-1
G
T
being the energy of rupture in traction (characteristic of material).
What makes it possible to obtain the plastic multiplier:
()
(
)
()
T
H E
T
C
T
T
T
C
T
T
C
D
F
K C
D
L
F
G
C
D aK
F
K C
D
L
F
G
=
-
-
=
-
-
+
+
2
2
2
2
2
2
2
3
2
.
.
.
.
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
7/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
Then the stress:
H
H E
T
T
K
C
D
K
has
C
D
=
-
=
-
+
3
3
3
.
Knowing has, imposed displacement, one obtains all the unknown factors of the problem.
2.3
Uncertainty on the solution
The solution being analytical, uncertainty is negligible, about the precision of the machine.
2.4 References
bibliographical
The model was defined starting from the following theses and is described in the report/ratio of specification:
[1]
G. Heinfling, at the time of its thesis “Contribution to the numerical modeling of the behavior of
concrete and of the concrete structures reinforced under mechanical thermo stresses with high
temperature ",
[2]
J.F. Georgin, at the time of its thesis “Contribution to the modeling of the concrete under stress of
fast dynamics. The taking into account of the effect speed by viscoplasticity ".
[3]
SCSA/128IQ1/RAP/00.034 Version 1.2, Development of a model of behavior 3D
concrete with double criterion of plasticity in Code_Aster - Specifications “.
Code_Aster
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Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
8/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
3D (HEXA8)
1 element, stress field and uniform deformation.
3.2
Characteristics of the mesh
A number of nodes: 8
A number of meshs and type: 1 HEXA8
3.3
Functionalities tested
Controls Options
“MECHANICAL” AFFE_MODELE
“3D”
DEFI_MATERIAU “BETON_DOUBLE_DP”
DEFI_MATERIAU “ELAS_FO”
“K_DESSIC”
AFFE_CHAR_MECA “SECH_CALCULEE”
STAT_NON_LINE “BETON_DOUBLE_DP”
X
y
Z
Uz = 0
Ux = 0
U
1
U
2
Face1xy
Facexy
Faceyz
Face1yz
N1
N2
Face1xz
Facexz
U
3
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
9/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Components xx and zz of the stress field were tested
SIEF_ELNO_ELGA
, and
plastic deformation cumulated in traction (second variable internal, second component of
field
VARI_ELNO_ELGA)
. Displacement being imposed, the field
EPSI_ELNO_DEPL
is not
tested.
The three moments correspond to a displacement of 0.005, 0.01 and 0.015 Misters.
Field
SIEF_ELNO_ELGA
component
SIXX
Identification Reference
Aster %
difference
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.005
1.9182065 1.9182066 7.10
6
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.010
1.161
6770
1.1616769 3.10
6
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.015
0.4051470 0.4051473 7.10
5
Field
SIEF_ELNO_ELGA
component
SIZZ
Identification Reference
Aster %
difference
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.005
1.9182065 1.9182066 7.10
6
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.010
1.1616770 1.1616769 3.10
6
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.015
0.4051470 0.4051473 7.10
5
Field
VARI_ELNO_ELGA
component
V2
(plastic deformation cumulated in traction)
Identification Reference
Aster %
difference
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.005
0.0099232717 0.0099232717 4.10
7
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.010
0.0199535329 0.0199535329 1.10
7
For an imposed displacement
in load U
1
= U
2
= U
3
= 0.015
0.0299837941 0.0299837941 3.10
8
4.2 Parameters
of execution
Version: 5.04.22
Machine: Claster
Overall dimension memory: 32 Mo
Time CPU To use: 17.66 seconds
Code_Aster
®
Version
5.0
Titrate:
SSNV150 - Tractiontriaxiale with law BETON_DOUBLE_DP
Date:
22/01/02
Author (S):
C. CHAVANT, B. CIREE
Key
:
V6.04.150-A
Page:
10/10
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
5
Summary of the results
This case test offers very satisfactory results compared to the analytical solution, lower than
7.10
- 5
% with a low iteration count (1 or 2 iterations). The solution is obtained from one
linear equation in the case of a linear curve of work hardening in traction, but the resolution uses
an algorithm of Newton within a more general framework.
One can note the work hardening of the criterion of traction which takes place during the loading, involving one
reduction in the stress (component xx, yy and zz) in addition equalizes with the hydrostatic stress.