Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
1/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
R7.01.01 document
Relation of behavior of Granger
for the clean creep of the concrete
Summary:
This document presents the clean model of creep of “Granger”, which is a way of modelizing creep
clean of the concrete.
One also details there the writing and the digital processing of the model.
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
2/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
3/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
1 Introduction
Within the framework of the studies of the long-term behavior of structures out of concrete, a share
dominating of the deformations measured on structure relates to the differed deformations which
appear in the concrete during its life. They comprise the withdrawals at the youth, the withdrawal of
desiccation, clean creep and the creep of desiccation.
The model presented here is dedicated to the modeling of the differed deformation associated creep
clean. Clean creep is, in complement of the creep of desiccation, the share of creep of the concrete
that one would observe during a test without exchange of water with outside. In experiments concrete in
clean creep presents a growing old viscous behavior. The deformation of creep observed is
proportional to the stress of loading, depends on the temperature and the hygroscopy.
longitudinal deflection is accompanied as in elasticity by a transverse deformation by sign
opposed.
The selected model is that proposed by L. Granger [bib1]. It is model of a viscoelastic type which
takes into account the effect of ageing as well as the history of stress, temperature and of
the hygroscopy. It thus allows this fact of modelizing the experimental facts quoted above.
One initially carries out a short recall on the linear viscoelastic models and one
present then the model itself like its numerical integration in Code_Aster.
In Code_Aster, 3 versions are available: GRANGER_FP_V the complete model,
GRANGER_FP
who does not take into account the effect of ageing and GRANGER_FP_INDT, which in more does not depend
temperature.
2
Recall on behavior in creep of a material
viscoelastic linear [bib3]
The conventional curve of creep represents the evolution according to the time of the deformation of one
material subjected to a constant unidimensional stress
. Deformation of creep
fl
is,
in opposition to the instantaneous strain, the share of deformation which evolves/moves with time.
If a material has a linear viscoelastic behavior, then whatever
the constant load
applied as from the time of loading
C
T
, the deformation of creep (1D) can be written:
-
=
)
(
)
(
C
fl
T
T
F
T
éq 2-1
where
)
(
)
,
(
C
C
T
T
F
T
T
J
-
=
is related to creep, increasing function of
(
)
C
T
T
-
and null for
(
)
C
T
T
-
negative.
2.1
Principle of superposition of Boltzmann
The relation [éq 2-1] is valid only for one constant loading. For a history of loading
nonconstant the principle of superposition of Boltzmann is applied; history of loading
)
(T
is broken up into increments of load:
Heavyside.
of
function
is
where
H
)
H (
)
(
0
=
-
=
N
I
I
I
T
T
T
One can then write:
=
-
=
N
I
I
I
fl
T
T
F
T
0
)
(
)
(
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
4/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
what uninterrupted gives:
)
(
)
(
)
(
0
=
=
-
=
=
F
T
F
D
T
F
T
T
fl
éq
2.1-1
where
represent the product of convolution.
2.2
Model of Kelvin in series
One can show that any linear viscoelastic body can be modelized by a series connection
models of Kelvin and that the function of creep can then be put in the form
))
exp (
1
.(
)
(
1
=
-
-
=
R
S
S
S
T
J
T
F
S
S
J
and
are plus coefficients identified on the experimental curves of creep.
3
Presentation of the clean model of creep of Granger [bib1]
3.1
Experimental properties of the clean creep of the concrete in loading
uniaxial
The clean creep tests on test-tube reveal the following properties:
·
in a range of stress lower than 50% of the breaking strength, clean creep
is proportional to the stress,
·
the clean creep of a test-tube with hygroscopy
ext.
H
is almost proportional to
ext.
H
.
clean creep of a no-slump concrete is almost null and it is maximum for a concrete saturated with water,
·
when the temperature T increases one has an acceleration of creep,
·
clean creep is a strongly growing old phenomenon,
·
a longitudinal deflection of creep is accompanied by a transverse deformation by
sign opposite (effect Poisson).
One chooses to modelize the clean creep of the concrete with a linear viscoelastic model which will have in
to more take into account the dependence of creep with respect to the temperature and the hygroscopy.
3.2
Modeling by a series connection of models of Kelvin
One uses a series connection of models of Kelvin whose coefficients are identified from
experimental curves of creep. It is shown in practice that one reproduces in a satisfactory way them
curves of concrete creep with
R
= 8 models in series.
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
5/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The following function of creep is thus used:
=
-
-
-
=
8
1
exp
1
.
)
,
(
S
S
C
S
C
T
T
J
T
T
J
éq
3.2-1
In practice it is very difficult to determine at the same time them
S
J
and
S
as soon as the number of series of
Kelvin exceeds 2. One thus makes generally a choice a priori on
S
,
1
1
10
-
=
S
S
and one
then determine by linear regression them
S
J
.
The expression [éq 3.2-1] is related to basic creep of the model. One shows below how
taken into account of the effect of the temperature, the hygroscopy and ageing is integrated in
final model.
3.3
Effect of the temperature
To take account of the effect of the temperature on the kinetics of creep, one defines a “time
equivalent “
)
(T
T
eq
who will replace time
T
in the model.
ds
T
S
T
R
U
T
T
T
T
S
ref.
C
eq
C
=
-
-
=
1
)
(
1
exp
)
(
éq
3.3-1
Note:
·
The temperature and the term of activation of the law of Arrhenius
R
U
C
are expressed in
degrees
K
.
·
To modelize the effect of the temperature thus
T
exploit only the kinetics of creep. For
to really utilize
T
on the amplitude of the phenomenon of creep, in particular on
the level of the value ad infinitum of the function of creep,
T
is also introduced into
the expression of J like a multiplicative function of the coefficients of creep such as:
=
-
-
-
-
-
=
R
S
S
C
eq
S
ref.
C
T
T
J
T
T
T
T
T
J
1
exp
1
.
45
)
45
(
)
,
,
(
éq
3.3-2
·
ref.
T
is the temperature of reference. It is chosen by the user. It is
generally taken equalizes with 20°C. In the continuation of the document
ref.
T
will be taken equalizes with
20°C.
·
For the version independent of the temperature, one has simply
T
T
T
eq
=
)
(
and
=
-
-
-
=
R
S
S
C
eq
S
C
T
T
J
T
T
T
J
1
exp
1
.
)
,
,
(
.
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
6/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.4
Effect of the hygroscopy
In the model,
H
is also introduced like a multiplicative parameter of the coefficients of creep
so that:
=
-
-
-
-
=
R
S
S
C
eq
S
C
T
T
J
T
H
H
T
T
T
J
1
exp
1
.
45
248
)
,
,
,
(
éq
3.4-1
Note:
It is the noted variable drying
C
that one has at the end of Code_Aster calculation of drying and it is
isothermal curve of sorption-desorption which makes it possible to pass from the variable
C
with the hygroscopy of
ambient conditions
H
. That is to say C the isothermal curve of desorption:
C
= C (
H
) and
H
= C
- 1
(
C
). The curve
H
= C
- 1
(
C
) must be informed by the user.
3.5
Effect of ageing
For a growing old viscoelastic material, the function of creep varies for two times of
loading different. Ageing is associated the hydration at the youth and others
phenomena like polymerization for the old concrete. The effect of ageing is modelized in
multiplying the coefficients of creep by a function of ageing
()
C
T
K
depending on time on
loading. Modeling chosen to take into account ageing associated with the hydration
is that of the CEB [bib2]:
day.
in
expressed
is
C
C
C
T
T
T
K
1
1
.
0
28
)
(
2
.
0
2
.
0
+
+
=
To reveal a sensitivity of the phenomenon of ageing compared to the temperature one
also a time of equivalent loading defines
)
(
C
eq
T
Tc
who replaces
C
T
in the function of
ageing.
ds
T
S
T
R
U
T
Tc
C
T
T
S
ref.
v
C
eq
=
-
-
=
0
1
)
(
1
exp
)
(
0
T
: corresponds to the age of the concrete at the youth, it
is generally taken equal to 28 days
C
T
: the time or age of loading expressed in
days
Note:
·
T
and
R
U
v
are in degrees
K
,
·
for the old concrete it would be necessary to use another equivalent time and another function of
ageing,
·
if one does not take into account ageing, one has simply
1
)
(
=
C
T
K
.
The function of creep, which will be related final to creep of the model, is written then:
=
-
-
-
-
=
N
S
S
C
eq
S
eq
C
T
T
J
Tc
K
T
H
H
T
T
T
J
1
exp
1
.
)
(
45
248
)
,
,
,
(
éq
3.5-1
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
7/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.6 Modeling
3D
The conventional assumption consists in supposing the existence of a Poisson's ratio of constant creep
and equal to the elastic Poisson's ratio, that is to say
2
.
0
=
F
. From where for
H
T
,
,
constant:
]
)
(
)
1
[(
)
,
,
,
(
)
(
I
tr
v
v
H
T
T
T
J
T
F
F
C
fl
-
+
=
and thus:
-
=
+
=
tr
H
T
T
T
J
T
tr
H
T
T
T
J
T
F
C
fl
F
C
fl
)
2
1
(
)
,
,
,
(
))
(
(
~
)
1
(
)
,
,
,
(
)
(
~
3.7
Superposition on the stress, the temperature and the hygroscopy (1D)
To simplify the demonstration, one takes in this part like function of creep one of
components of the series of Kelvin, without taking into account of the effect of ageing, nor of time
equivalent parameterizing the temperature, is:
-
-
-
-
=
S
C
S
C
T
T
J
T
H
H
T
T
T
J
exp
1
.
45
248
)
,
,
,
(
deformation of then being written creep:
-
-
-
-
=
S
C
S
fl
T
T
J
T
H
exp
1
.
45
248
.
It is reminded the meeting that this writing of the deformation of creep is valid for
,
T
and
H
constant
(in this case the model is equivalent in fact to take a Young modulus decreasing according to
time).
For a history of loading, temperature and hygroscopy nonconstant one applies it
principle of superposition of Boltzmann.
Let us suppose that for an element of volume given, one knows at time
N
T
sizes
)
,
,
,
(
N
N
N
N
fl
H
T
. At time
1
+
N
T
the sizes will be
)
,
,
,
(
1
1
1
1
+
+
+
+
N
N
N
N
fl
H
T
.
For
1
+
<
<
N
N
T
T
T
one proposes to calculate the deformation of creep in the following way:
)
,
,
,
(
)
,
,
,
(
)
(
)
(
1
1
1
1
+
+
+
+
+
-
=
N
N
N
N
N
N
N
fl
N
fl
H
T
T
T
J
H
T
T
T
J
T
T
N
N
i.e.:
-
-
-
-
-
-
-
-
+
=
+
+
+
+
S
N
S
N
N
N
S
N
S
N
N
N
N
fl
N
fl
T
T
J
H
T
T
T
J
H
T
T
T
exp
1
45
248
exp
1
45
248
)
(
)
(
1
1
1
1
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
8/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The superposition is thus considered not only on the stress but also on the temperature and
the hygroscopy which is treated mathematically in the same way. From where:
I
N
I
S
I
S
fl
H
T
T
T
J
T
-
-
-
-
=
=
45
248
exp
1
.
)
(
0
One has then in integral writing, the deformation of creep of a component S of the series of Kelvin:
-
-
-
-
=
=
H
T
D
T
J
T
T
T
S
S
fl
S
45
248
exp
1
.
)
(
0
éq
3.7-1
4
Relations of Code_Aster behavior
One introduces into Code_Aster three relations of behavior associated with clean creep:
·
GRANGER_FP_V
·
GRANGER_FP
·
GRANGER_FP_INDT
The first takes account of the whole of the effects (forced, temperature, hygroscopy and
ageing), the second does not take account of the phenomenon of ageing and the last does not hold
count neither ageing nor of the effect of the temperature. They are available in modeling 2D,
3D and plane stresses.
The various parameters of the model are indicated in
DEFI_MATERIAU
. Are well informed under
the key word
GRANGER_FP
, the use is common to the relations of behavior
GRANGER_FP
and
GRANGER_FP_V
, the following characteristics materials:
GRANGER_FP:
·
(2x8) constant characteristics of
function of creep
I
J
,
I
,
J1:
1
J
TAUX_1:
1
.
.
.
J8:
8
J
TAUX_8:
8
·
the curve of sorption-desorption giving
H
according to the variable drying
C
FONC_DESORP:
C
- 1
()
C
·
the constant of energy of activation for
time-temperature equivalence.
QSR_K
:
R
U
C
If one uses the growing old relation of behavior then one informs in more the key word
V_GRANGER_FP
under which the characteristics associated with ageing are indicated, with
to know energy of activation for the calculation of the time of equivalent loading and the function of
ageing
)
(
eq
Tc
K
.
V_GRANGER_FP:
QSR_VEIL:
R
U
v
FONC_V:
)
(
eq
Tc
K
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
9/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
For the law
GRANGER_FP_INDT
, parameters to be informed under the key word
GRANGER_FP_INDT
of
DEFI_MATERIAU
are:
GRANGER_FP_INDT:
·
(2x8) constant characteristics of
function of creep
I
J
,
I
,
J1:
1
J
TAUX_1:
1
.
.
.
J8:
8
J
TAUX_8:
8
·
the curve of sorption-desorption giving
H
according to the variable drying
C
FONC_DESORP:
C
- 1
()
C
5
Numerical integration of the model
5.1 Discretization
(1D)
Let us pose
45
248
-
=
=
T
T
H
T
S
The expression [éq 3.7-1] is thus written:
D
S
T
J
T
T
T
S
S
fl
S
.
.)
exp
1
.(
)
(
0
=
-
-
-
=
The discretization in time is such as for
]
[
,
1 N
N
T
T
T
-
one considers a linear evolution of
S
(decomposition of
()
T
S
in linear functions per piece). One has then:
D
T
J
T
S
T
I
I
T
T
S
N
S
N
I
I
I
N
fl
S
-
=
=
-
-
-
=
1
exp
1
.
.
)
(
1
-
-
-
-
-
-
=
-
=
=
S
I
N
S
I
N
S
S
N
I
I
I
I
S
N
I
I
I
N
fl
S
T
T
T
T
J
T
S
T
J
T
S
T
1
1
1
exp
exp
)
(
-
-
-
-
-
=
=
=
S
I
S
I
N
I
S
S
N
I
I
N
I
I
S
N
fl
S
T
T
T
T
J
S
S
J
T
exp
1
exp
)
(
1
1
éq
5.1-1
Note:
Notation
1
-
-
=
I
I
I
X
X
X
.
Now let us consider the 8 models of Kelvin in series one a:
=
=
=
S
N
fl
S
S
N
fl
S
N
fl
T
T
T
)
(
)
(
)
(
8
1
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
10/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One can then break up the deformation of creep [éq 5.1-1] on the basis of
-
S
T
exp
;
1
and
to carry out a recursion on the coefficients of this base. According to [éq 5.1-1] one has with
N
T
:
Sn
N
S
With
S
I
S
I
N
S
I
S
N
I
I
With
N
I
I
S
N
fl
S
With
With
J
T
T
T
J
T
S
S
J
T
Sn
N
-
=
-
-
-
-
-
=
=
=
0
1
1
exp
1
exp
)
(
0
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
2
1
3
2
1
With
1
+
N
T
one can also write:
-
-
-
+
-
-
-
-
-
=
+
+
+
+
+
=
=
+
S
N
S
N
S
N
S
N
S
I
S
I
N
S
S
I
N
I
I
N
I
I
S
N
fl
S
T
J
T
S
J
S
T
T
T
J
T
S
S
J
T
1
1
1
1
1
1
1
1
exp
1
exp
1
exp
1
)
(
that is to say:
-
-
-
-
-
-
-
-
-
=
+
+
+
+
=
+
=
+
S
N
S
N
S
N
S
I
S
I
N
S
N
S
S
I
N
I
I
N
I
I
S
N
fl
S
T
J
T
S
T
T
T
T
J
T
S
S
J
T
1
1
1
1
1
1
1
1
exp
1
exp
1
exp
exp
1
)
(
One can thus write:
-
-
-
-
-
+
=
+
+
+
+
+
+
S
N
S
N
S
N
S
N
S
N
S
N
N
S
N
fl
S
T
J
T
S
T
With
J
S
With
J
T
1
1
1
1
1
0
1
exp
1
exp
)
(
Let us pose
=
=
8
1
S
S
J
J
, one has then:
=
+
+
+
=
-
=
-
=
8
1
1
0
1
1
8
1
0
)
(
)
(
S
S
N
N
N
fl
S
S
N
N
N
fl
With
With
J
T
With
With
J
T
and
with
-
-
+
-
=
+
=
+
+
+
+
+
+
+
S
N
S
N
S
N
S
N
S
N
S
N
N
N
N
T
J
T
S
T
With
With
S
With
With
1
1
1
1
1
1
0
0
1
exp
1
)
exp (
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
11/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
More precisely, if:
·
one takes into account equivalent time for the temperature and ageing,
·
during a pitch of time the parameter
T
is evaluated in the middle of the pitch of time for
calculation of equivalent times
eq
T
and
eq
Tc
, its linear evolution being supposed during
this pitch of time,
then one a:
=
+
+
+
-
=
8
1
1
0
1
1
)
(
S
S
N
N
N
fl
With
With
J
T
with
1
2
/
1
1
1
1
1
exp
)
(
)
(
)
(
+
+
+
+
-
-
=
=
-
N
ref.
N
C
N
eq
N
eq
N
eq
T
T
T
R
U
T
dt
T
T
T
T
1
2
/
1
1
1
293
1
1
exp
)
(
)
(
)
(
+
+
+
+
-
-
=
=
-
N
N
v
N
eq
N
eq
N
eq
T
T
R
U
T
dtc
T
Tc
T
Tc
-
-
+
-
=
+
=
+
+
+
+
+
+
+
+
+
S
N
eq
N
eq
S
N
S
N
S
N
eq
S
N
S
N
N
N
eq
N
N
T
T
Tc
K
J
T
S
T
With
With
S
T
Tc
K
With
With
1
2
/
1
1
1
1
1
1
2
/
1
0
0
1
exp
1
))
(
(
)
exp (
))
(
(
Note:
·
If account of ageing is not taken
K
then
1
1
1
0
1
+
+
=
+
=
=
N
N
I
I
N
S
S
With
,
·
one noted
2
1
2
/
1
N
N
N
X
X
X
+
=
+
+
,
·
one noted
()
1
1
+
=
+
N
eq
eq
T
T
T
N
.
To have
fl
at time
1
+
N
T
, one should only store
0
With
and
S
With
pitch of previous time, is 9
variables. In 3D
0
With
and
S
With
are tensors. One will associate the two relations then
clean behavior of creep (9x6) variable interns corresponding to the components of the tensors
With
. They characterize the advance of creep.
The writing in increment of deformation, nearer to the programming gives as for it:
-
-
-
+
-
-
=
-
=
+
+
+
+
+
+
+
)
exp
1
(
1
))
(
(
)
exp (
1
)
(
)
(
)
(
1
1
2
/
1
1
1
1
1
S
eq
N
S
S
N
eq
N
S
N
eq
S
N
N
fl
S
N
fl
S
N
fl
S
N
T
T
J
T
Tc
K
S
T
With
T
T
T
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
12/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.2
Integration of the relation of behavior
That is to say the increment of deformation
(
)
)
(
)
(
2
1
U
U
T
+
=
.
If one holds account, in the partition of deformation, of the thermal deformation, the deformations
associated the endogenous withdrawal and the withdrawal of desiccation, then:
dice
ret
end
ret
HT
fl
E
+
+
+
+
=
where:
-
-
=
-
=
-
=
=
D
ref.
ret
D
end
ret
D
ref.
HT
E
C
C
T
T
H
I
I
I
)
(
)
(
: elastic strain
: thermal deformation
: endogenous deformation of withdrawal
: deformation of withdrawal of desiccation
with:
: hydration,
C
: water concentration.
ref.
T
and
ref.
C
: temperature and drying of reference
,
,
,
H
: characteristics materials
Note:
In the continuation of the document, one will note
ret
end
ret
HT
With
+
+
=
.
fl
E
~
2
~
2
~
2
2
~
2
~
-
+
=
=
-
-
µ
µ
µ
µ
µ
and
-
+
=
=
+
+
+
8
1
1
0
1
1
~
~
J
)
1
(
)
(
~
S
S
N
N
F
N
fl
T
With
With
it results that:
(
)
(
)
(
)
(
)
-
-
-
-
-
-
+
-
+
=
-
-
-
+
+
+
+
+
-
-
-
+
-
-
+
+
+
S
S
N
eq
N
S
S
S
N
eq
S
N
eq
S
N
F
S
S
N
eq
N
S
S
N
eq
F
T
T
T
J
Tc
K
T
H
T
T
T
T
T
J
Tc
K
T
H
µ
µ
µ
µ
µ
)
(
exp
1
1
)
~
(
)
(
exp
1
~
)
1
) (
2
(
~
2
~
2
2
)
(
exp
1
1
1
)
2
(
1
~
1
1
2
/
1
1
1
1
2
/
1
With
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
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Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
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:
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Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
In the same way:
)
tr (
3
)
tr (
3
)
tr (
3
)
tr (
3
3
)
tr (
3
)
tr (
With
fl
E
HT
K
K
K
K
K
K
-
-
+
=
=
-
-
and
-
-
=
=
8
1
0
J
)
2
1
(
)
tr (
S
S
N
N
F
fl
With
With
from where:
(
)
()
(
)
(
)
)
tr (
3
)
(
exp
1
1
)
(tr
exp
1
tr
2
1
3
)
(
tr
3
)
tr (
3
3
)
(
exp
1
1
)
(
) (
2
1
(
3
1
)
tr (
1
1
1
1
2
/
1
1
2
/
1
With
S
S
N
eq
N
S
S
S
eq
S
T
eq
S
N
F
E
S
S
N
eq
N
S
S
eq
F
K
T
T
T
J
Tc
K
T
H
T
K
K
K
K
T
T
T
J
Tc
K
T
H
K
N
N
N
-
-
-
-
-
-
-
-
-
+
=
-
-
-
-
+
+
+
-
-
-
-
-
+
+
+
+
+
With
One deduces some then
since
ij
ij
ij
tr
3
1
~ +
=
5.3 Variables
of state
The variables of state of the two relations of behavior are thus:
·
: tensor of the stresses,
·
: tensor of the deformations,
·
T
: temperature,
·
C
: water concentration,
·
: hydration,
·
S
With
: tensors characteristic of the advance of creep, are 6x9 variable,
·
eq
Tc
: time of equivalent loading, characteristic of the age of the concrete.
S
With
and
eq
Tc
are internal variables of the laws of behavior, which thus comprise
55 internal variables.
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
14/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5.4 Stamp
tangent
D
I
)
(tr
3
1
~ +
=
kl
ij
jl
ik
kl
ij
3
1
~
~
~
~
~
-
=
=
()
ij
ij
=
=
tr
)
(tr
)
(tr
)
(tr
)
(tr
Iteration of Newton:
I
µ
µ
2
))
)
(
exp (
1
) (
(
1
)
(
).
1
(
2
1
~
~
1
1
2
/
1
=
-
-
-
+
+
+
+
+
S
S
N
eq
N
S
S
eq
F
T
T
T
J
Tc
K
T
H
N
with
jl
ik
ijkl
I
=
I
K
T
T
T
J
Tc
K
T
H
K
S
S
N
eq
N
S
S
eq
F
N
3
))
)
(
exp (
1
) (
(
1
)
(
).
2
1
(
3
1
)
(tr
)
(tr
1
1
2
/
1
=
-
-
-
-
+
+
+
+
Phase of prediction for the pitch of time [tn, tn+1]
Note:
In 1D:
T
S
Tc
K
J
With
T
eq
S
S
S
T
fl
N
S
-
=
-
)
(
.
Writing of speed at the moment
N
T
:
(
)
-
-
+
-
=
+
+
-
-
-
-
-
-
-
S
eq
S
eq
S
S
S
F
S
eq
S
F
dt
dh
T
Tc
K
J
dt
T
D
H
Tc
K
J
T
H
T
Tc
K
J
T
N
N
N
~
)
(
~
)
(
~
1
2
~
2
)
)
(
) (
1
(
2
1
~
µ
µ
µ
With
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
15/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
linearization for the phase of prediction of the pitch of time
[
]
1
,
+
tn
tn
:
-
-
+
-
=
+
+
-
-
-
-
-
-
-
S
eq
S
eq
S
S
S
F
S
eq
S
F
H
T
Tc
K
J
T
H
Tc
K
J
T
H
T
Tc
K
J
N
N
N
~
)
(
~
)
(
~
)
1
(
2
~
2
)
)
(
) (
1
(
2
1
~
µ
µ
µ
With
Writing of speed at the moment tn:
)
3
(
3
)
3
(
3
)
3
(
3
)
(
)
(
)
(tr
)
(
)
(tr
)
2
1
(
3
)
(tr
3
)
)
(
) (
2
1
(
3
1
)
(tr
dt
cd.
K
dt
D
K
dt
dT
K
dt
dh
T
tr
Tc
K
J
dt
T
D
H
Tc
K
J
K
T
K
H
T
Tc
K
J
K
T
S
eq
S
eq
S
S
S
F
S
eq
S
F
N
N
+
+
-
-
-
-
-
=
-
+
-
-
-
-
-
-
-
With
linearization for the phase of prediction of the pitch of time
[
]
1
,
+
tn
tn
:
)
3
(
3
)
3
(
3
)
3
(
3
)
(tr
)
(
)
(tr
)
(
)
(tr
)
2
1
(
3
)
(tr
3
)
)
(
) (
2
1
(
3
1
)
(tr
C
K
K
T
K
H
T
Tc
K
J
T
H
Tc
K
J
T
K
K
H
T
Tc
K
J
K
S
eq
S
eq
S
S
S
F
S
eq
S
F
N
N
N
+
+
-
-
-
-
-
=
-
+
-
-
-
-
-
-
-
With
Creep thus introduces a specific term of second member at the time of the phase of prediction which in
fact is neglected, without consequence on the results.
Code_Aster
®
Version
7.4
Titrate:
Relation of behavior of Granger for the clean creep of the concrete
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE
Key
:
R7.01.01-C
Page
:
16/16
Manual of Reference
R7.01 booklet: Modeling for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6 Bibliography
[1]
L. GRANGER: Behavior differed from the concrete in the chambers of nuclear thermal power station:
analyze and modeling. Thesis of Doctorate of the ENPC (February 1995).
[2]
CEB FIP Model (1990) General task group n°9, Evaluation off the time behavior off concrete.
[3]
J. LEMAITRE, J-L CHABOCHE: Mechanics of solid materials. Dunod.