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Law of behavior (in 2D) for the steel-concrete connection: JOINT_BA
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Document: R7.01.21
Law of behavior (in 2D) for the connection steel
concrete: JOINT_BA
Summary:
The law of behavior JOINT_BA describes the phenomenon of degradation and rupture of the connection between
bars of steel and concrete, in the reinforced concrete structures. This documentation presents the theoretical writing
in the thermodynamic framework and the numerical integration of the law, as well as the parameters which manage it
model.
For his use, one will be pressed on the finite elements of joint type (see Doc. [R3.06.09]) already existing in
the code.
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Law of behavior (in 2D) for the steel-concrete connection: JOINT_BA
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Short description of the steel-concrete connection ............................................................................................ 3
3 theoretical Writing ................................................................................................................................... 5
3.1 Presentation of the model ................................................................................................................... 5
3.2 Analyze damage in the tangential direction ......................................................... 7
3.3 Analyze damage in the normal direction ............................................................... 9
3.4 Analyze contribution of the friction of fissures by slip .......................................... 10
3.5 Summary of the equations .................................................................................................................. 11
3.6 Form of the tangent matrix ................................................................................................ 12
4 numerical Integration .......................................................................................................................... 13
4.1 Calculation of the part “friction of the fissures” with a method of integration implicit ............ 13
4.2 The algorithm of resolution ............................................................................................................. 14
4.3 Variables intern model ........................................................................................................ 15
5 Parameters of the law ............................................................................................................................. 15
5.1 The initial parameters .................................................................................................................. 15
5.1.1 The parameter “hpen” ......................................................................................................... 15
5.1.2 The parameter G or module of rigidity of the connection .............................................................. 16
5.2 Parameters of damage .............................................................................................. 17
5.2.1 Limit of elastic strain 1T or threshold of perfect adherence ................................... 17
5.2.2 The parameter of damage A1DT for the passage of the small deformations with
great slips .............................................................................................................. 17
5.2.3 The parameter of damage B1DT ................................................................................. 18
5.2.4 Limit of deformation 2T or threshold of the great slips ............................................. 19
5.2.5 The parameter of damage A2DT ................................................................................. 20
5.2.6 The parameter of damage B2DT ................................................................................. 20
5.3 Parameters of damage on the normal direction ....................................................... 21
5.3.1 Limit of deformation 1N or threshold of great displacements .......................................... 21
5.3.2 The parameter of damage ADN .................................................................................. 21
5.3.3 The parameter of damage BDN .................................................................................. 21
5.4 Parameters of friction ....................................................................................................... 22
5.4.1 The parameter material of friction of the fissures ............................................................ 22
5.4.2 The parameter kinematic material of work hardening .......................................................... 22
5.4.3 The parameter of influence of containment C ......................................................................... 23
5.5 Summary of the parameters ............................................................................................................... 23
6 Bibliography ........................................................................................................................................ 25
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1 Introduction
The law of behavior JOINT_BA describes the phenomenon of degradation and rupture of the connection
existing enters the steel bars (smooth or ribbed) and the concrete surrounding it. Key point for
the purpose of structural design out of reinforced concrete, the modeling of the steel-concrete connection is
representation as well as simplification of this phenomenon complexes interaction between the two
materials which develops in the interface and which undergoes an increasing degradation when one
exceed certain thresholds of resistance, specific for each material. The structural models which
do not take into account the linkage effects, are generally unable to predict
localization of the fissures as well as the networks created. In addition, the degradation of the rigidity of
connection increases the period of vibration, reduced the capacity of dissipation of energy and conduit to one
significant redistribution of the internal forces (according to Bertero, 1979, cf [bib2]).
The law of behavior JOINT_BA is described within the framework of the thermodynamics of the processes
irreversible: the writing and L `use of a “traditional” material model coupling cracking and
friction makes it possible to integrate in a robust way of the fine mechanisms nonlinear concomitant into
particular description of the kinematics of slip. This last point allows us not
to resort to traditional modelings of the type “contact” very often used in this context in spite of
many sources of numerical instabilities. Thus, in monotonous loading the taking into account
coupling normal effort shearing makes it possible to treat cases of strong multiaxial pressures; in
cyclic, the behavior hysteretic and corresponding dissipations are expressed thanks to
coupling between the state of damage and kinematic work hardening. Use of an implicit scheme
allows to obtain a robust implementation.
The paragraph [§2] described in short form the phenomenon of the connection steel concrete. The paragraph [§3]
present the thermodynamic writing of the law of behavior, while the paragraph [§4] precise
the numerical stage of integration of the law. The parameters which manage the model and which could be
obtained starting from the properties of implied materials, are described in the paragraph [§5].
2
Short description of the steel-concrete connection
Conceptually, the phenomenon of connection corresponds to the physical interaction of two materials
different, which occurs on a zone of interface by allowing the transfer and the continuity of the efforts
and of the constraints between the two bodies in contact. In the case of reinforced concrete structures, it
phenomenon is also known as the “rigidity of tension” which develops around an element of
reinforcement, partially or completely embedded in a volume of concrete. Forces of traction which
appear inside the reinforcement are transformed into shear stresses on
surface, and are transmitted directly to the concrete in contact which will balance them finally, and vice
poured. The response of the unit will depend on the capacity of the concrete to become deformed as much as steel,
since steel will tend to slip inside the concrete surrounding it. The phenomenon of connection
corresponds to this capacity of the concrete to become deformed and to degrade themselves locally by creating a species
of layer, or wraps, around the reinforcement, of which the properties kinematics and material
differ from those of the remainder from the concrete or reinforcement employed.
The phenomenon can be broken up into three well defined mechanisms:
·
a chemical adherence of origin,
·
a mechanism of friction between two rough surfaces (steel-concrete or concrete-concrete),
·
a mechanical action created by the presence of the veins of the steel bar on the concrete
bordering.
According to this decomposition, one can clearly deduce that for a smooth bar, the mechanism
dominating is friction between two materials, while for a bar ribbed (in
French usually called “braces ha: High Adhérence”), the mechanism dominating is
the mechanical interaction between surfaces. When the reinforcement is consisted the strands with
steel wire ropes, it is possible to control or combine the various mechanisms since they are
function directly of the surface of the cables.
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The connection will undergo a degradation different according to the type of loading applied, that is to say monotonous,
that is to say cyclic. In addition, among the most important parameters which influence the behavior of
the connection, one can quote:
· characteristics of the loading,
· geometrical characteristics of the steel bar,
· spacing between active bars,
· characteristics of the concrete,
· containment by passive reinforcement,
· side pressure.
At the time of the study of a cylindrical bar embedded in an infinite medium, one can identify the surface of
discontinuity where one will place the linkage effects, which develops in a certain concrete zone
fissured and crushed around the steel bar. At a given moment, this surface will correspond to
cylindrical fissure created during the coalescence of the fissures of shearing. By looking at the network
fissures, one can suppose that, in ideal conditions, the plan of cracking is always
perpendicular (normal direction) on the surface of the bar and parallel (tangential direction) with sound
longitudinal axis (see [Figure 2-a]). That enables us to project the components of displacement on
normal and tangential direction of the plan of cracking, and consequently to obtain the deformations
and corresponding constraints.
BODY A
Sd
A1
N
F
A2
B1
T
crushed concrete
B2
slipped
BODY B
Appear 2-a: real description of the phenomenon of connection and simplification finite elements:
co-ordinates in the local reference mark of the element of interface used like support of law JOINT_BA
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3 Writing
theoretical
The formulation presented here was developed within the framework of the thermodynamics of the processes
irreversible; it gives the constitutive relation between the normal effort, the shear stress and it
slip by considering the influence of the cracking of the concrete, friction and the various couplings
in the phenomenon. For that, the constitutive relations which connect the tensor of the constraints and it
tensor of the deformations must include:
·
the cracking of material of interface by shearing
·
inelastic deformations because of the slip
·
the behavior hysteretic due to friction
·
coupling between the tangential answer and the normal constraints
3.1
Presentation of the model
One places oneself within the framework of a plane formulation in 2D, in the definite local reference mark [Figure 2-a].
tensors of the constraints and the deformations are written:
NR
NR
=
and =
éq 3.1-1
0
0
where NR is the normal constraint and is the tangential constraint of the element of interface;
NR
corresponds to the normal deformation and the tangential deformation. Normal deformation
in the tangential direction with the interface is regarded as null. This mode of deformation for
an element of adherence is with null deformation energy.
Normal and tangential behaviors being regarded as uncoupled on the level from the state, it
thermodynamic potential obtained starting from the free energy of Helmholtz is expressed way
following:
= 1
[E + E 1 - D
NR -
NR -
NR +
(
NR)
NR +
2
+ G 1 - D + -
D -
2
+ +
éq
3.1-2
T
(
T) T
(
F
T
T) G
T (
F
T
T)
] (Z)
where is the density, is the D NR, Young modulus is the internal variable of normal damage
and DT the variable interns tangential damage, both being related to cracking and
ranging between 0 and 1. G is the module of rigidity or shearing, F
is the deformation
T
irreversible induced by slip with friction of the fissures, is the internal variable
of kinematic work hardening, is a parameter material and Z, the variable of pseudo “work hardening
isotropic “by damage, with its function of consolidation (Z).
and
define
-
+
respectively positive and negative parts of the tensor considered.
One can notice in the equation [éq 3.1-2] that in the normal direction, the damage will be
activated during the appearance of the positive deformations produced by forces of traction, while if
the deformations are negative because of the effects of compression, the behavior will remain
rubber band. With regard to the tangential part of the behavior, one can recognize a coupling
traditional élasticitéendommagement as well as a new term allowing to associate the state
elasticity-endommageable, a state of slip with friction. Coupling between slip and
cracking is possible thanks to the presence of the variable of damage like multiplier
in the second element of the right part of the equation [éq 3.1-2].
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The laws of state are obtained classically by derivation of the thermodynamic potential, and thus
allow to define the associated thermodynamic variables. The normal constraint is expressed
like:
E
if
0
NR
NR
=
=
NR
éq
3.1-3
NR
(
1 - DNN) E
if
> 0
NR
NR
and the total tangential constraint like:
T =
= G 1
(- DT) T + G DT (
F
T - T) éq
3.1-4
T
One can also define the tangential constraint due to the slip with friction (deformation
S
):
F
= -
= D -
T
G
T (
F
T
éq
3.1-5
F
T)
T
Note:
Such a formulation moves away amply from a traditional formulation of coupling plasticity
damage. The assumption bringing to the introduction of the damage into the constraint
by slip bases itself on an experimental observation which is that all the phenomena
inelastic in a fragile material come from the growth of the fissures.
The rate of energy restored by damage-friction can be written like:
1
1
- = -
= T G T - (
F
T - - = -
+
T) G (
F
T
T)
(DT fT) éq
3.1-6
DT
2
2
In this last expression, DT corresponds to the rate of energy restored by damage and
fT at the rate of energy restored by friction of the fissures.
The law of state of kinematic work hardening brings to the definition of the constraint of recall:
=
= éq
3.1-7
Concerning the law of work hardening of the isotropic damage, it is expressed by:
=
= '(Z)
éq 3.1-8
Z
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It is necessary for us now to clarify in a more detailed way the evolution of the mechanism
of damage in the connection, in other words to specify the expression of (Z). For weak
value of damage, the mechanism which prevails is the interaction of the concrete with the veins of
the steel bar, while for a value much larger, it is friction between the concrete and
the steel which prevails. During the evolution of the damage, 2 principal phases could be
identified:
· the first phase corresponds to a stable growth of transverse fissures related to
presence of veins on steel (positive apparent work hardening of the law of evolution),
· the second does not utilize any more but the coalescence of these transverse fissures bringing to
not to more consider but the mechanisms of friction (negative work hardening towards a constraint
of friction residual).
3.2
Analyze damage in the tangential direction
The law of evolution of the damage is divided into three stages:
1) area of perfect adherence,
2) area of passage of small deformations to the great slips,
3) area of maximum resistance of the connection and degradation until residual resistance
ultimate.
To identify these areas, two thresholds are established:
· the threshold of perfect adherence 1
,
T
· the threshold of continuity before coalescence of fissures 2
.
T
Thus, by taking again the expressions related to the damage with knowing that of the rate of refund
of energy [éq 3.1-6] and that of the variable interns associated with isotropic work hardening [éq 3.1-8], one can
to note:
· a true separation between L `damage and the friction of the fissures (what allows
to amend only the law of evolution of the damage without affecting the part
“friction”),
· the partition in two parts of isotropic work hardening since one has two different stages in
the damage.
From now on we will write for work hardening related to the variable of damage:
,
if 1 < 2
T1
T
T
T
=
=
T
'(Z)
=
éq
3.2-1
Z
2
, if
<
T1
T 2
T
T
Components and
express themselves in the following way:
T1
T 2
2
1
G
=
T
+
+ Z
T1
T1
ln (1
T)
éq
3.2-2
With
1
1
2
DT
T
1 - Z
=
T 2
+
T
T 2
éq
3.2-3
With
1 Z
2DT + T
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The function threshold DT is also defined which depends on DT and which S `writes like:
=
DT
D -
T
(+
T1
T) 0 éq
3.2-4
The thresholds which manage the law of evolution of the damage are also expressed in terms of T
D
(see [Figure 3.2-a]). The first expression corresponds to the threshold of perfect adherence and S `written:
1 1
1
T1 = elas = G éq
3.2-5
T
2 T
T
Where T1 is the initial threshold of damage defined according to the limiting deformation of adherence
perfect 1
T, which will correspond to the limiting deformation of shearing or traction of the front concrete
the initialization of the damage. In addition, T 2 is the threshold of initiation of coalescence of
microscopic cracks which is defined according to the initial tangential deformation of the great slips
2
T:
1 2
2
T 2 = G
éq
3.2-6
2 T
T
2
1
0
0
2
T
2t
Appear 3.2-a: construction of the functions thresholds in terms of energy
The laws of evolution of the internal variables within the framework of the standard associated laws allow
to obtain the derivative of the multiplier of damage D:
·
·
·
·
·
·
D
D
D = D
= D and Z = D
= - D
éq 3.2-7
D
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By using the condition of consistency in addition, one obtains the expression of the damage:
B
1 T
T
2
D
D = 1
1
-
exp
T
1
With
-
DT
(
DT
T1)
*
DT
G
1
éq 3.2-8
1 + A
-
B 2D
2
DT
DT
T1 +
T
In this expression, one can identify the part which corresponds to the area of the passage of small
deformations with the great slips with two parameters: A1
B
DT and 1DT, as well as the part
of damage finale in mode 2, with the A2 parameters
B
DT and 2 DT. It should be noted that the relation
-
DT
T1 is managed by a function of Macaulay, i.e. this difference in energy must
to be always positive or null.
The functions which manage isotropic work hardening in the tangential direction are expressed like:
Z
= Y
- Y;
T1
DT
T1
éq 3.2-9
,
0
if
<
=
1
D
2
T 2
T
T
T
éq
3.2-10
-, if
<
DT
T 2
T 2
DT
According to these expressions, one can notice that
is not taken into account in the area of
T 2
transition from the small deformations to great slips.
3.3
Analyze damage in the normal direction
The two most important mechanisms which can appear on the normal direction are it
detachment between the concrete and the bars of steel, and the penetration of the reinforcement in the body of
concrete. These two conditions can be interpreted respectively like an opening or one
closing of fissure, and can be described by a particular law of behavior in
normal direction uncoupled from the tangential behavior.
In order to simplify the resolution for compression between surfaces, one decided to allow small
penetration between those, which implies that 0, and by adopting a law of behavior
NR
rubber band, one will have:
= - if 0
NR
E
NR
NR
éq
3.3-1
The case of the decoherence of the interface can be described by a behavior endommageable in
normal direction, is:
=
NR
(1 - DNN) E + if > 0
NR
NR
éq
3.3-2
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with D NR variable scalar of the damage in the normal direction, calculated with the expression
following:
0
if
1
NR
NR
D =
NR
éq
3.3-3
1
if 1 <
NR
B
NR
1+
- D NR
ADN
D NR
N1 +
In this expression, two parameters material, AD NR and data base NR, control decoherence by
the damage in traction of the concrete. In addition, is the threshold of damage defined in
NR 1
term of energy, are equivalent to the elastic threshold in the normal direction
and which is expressed
elas NR
like:
1 1
1
N1 = elas
= E
éq
3.3-4
NR
2 NR
NR
1
being limiting deformation of perfect adherence, which corresponds to the limiting deformation of the concrete in
NR
traction before the initialization of the damage. It should be mentioned that when the detachment or
the opening of fissure reaches the maximum value of resistance to traction, no force of
shearing will not have to be transmitted between two materials: it east is the single condition in
which the scalar variable of damage in the tangential direction becomes 1 because of
the damage in the normal direction
3.4 Analyze contribution of the friction of fissures by
slip
With regard to the part “slip” of the formulation, one supposes that it has a behavior
pseudo-plastic, with nonlinear kinematic work hardening. Initially introduced by Armstrong &
Frederick, 1966, cf [bib1], nonlinear kinematic work hardening makes it possible the formulation to surmount
the principal disadvantage of the kinematic law of work hardening of Prager, namely, the linearity of the law
of state which connects the forces associated with kinematic work hardening. Here, the nonlinear terms are
additions in the potential of dissipation. The criterion of slip takes the traditional form of
function threshold of Drucker-Prager which takes into account the effect of radial containment on the slip:
= F
- + C I
0
F
T
1
éq
3.4-1
Here is the constraint of recall, C is a parameter related to material, translating the influence of
containment, while I corresponds to the first invariant of the tensor of the constraints, which for our
1
case is expressed like:
1
1
I =
1
Tr [] = NR
éq
3.4-2
3
3
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In addition, the initial threshold for the slip is 0. Moreover, by considering the principle of dissipation
figure maximum, the laws of evolution can be derived from the expression of the plastic potential which
is:
p
F
3
2
= - + C I
F
T
1 +
has
éq
3.4-3
4
Where A is a parameter material. It should be mentioned that the quadratic term allows some
to introduce the non-linearity of kinematic work hardening. Laws of evolution for the deformation of
slip as for kinematic work hardening take the following forms:
·
·
p
·
·
p
F
=
F
F
and = F
F
T
éq 3.4-4
F
T
·
The multiplier of slip F is calculated numerically by imposition of the condition of
consistency.
3.5
Summary of the equations
We show here, a summary of the equations which constitute the law of behavior of the connection
steel-concrete:
1
. = [
-
E
-
+
+
E
+
NR
NR
NR
. (1 - DNN)
Free energy
NR
2
of Helmholtz
+ G
2
T
(1 - T
D) T + (
F
T -
- + +
T) G.
T
D (
F
T
T)
] H (Z)
= Y
DT
D -
T
(Y + Z
T1
T) 0;
Function threshold
= F
- X + C. I 0
F
T
1
E. NR
if NR 0
NR = (
;
1 - DNN). E. NR if NR > 0
Laws of state
T = G (1 - T
D) T + G. T
D (
F
T - T);
F
=. D -
G
T (
F
T
T)
- Y = -
= - (D
Y + Yf);
D
Dissipation
X =
=
;
'
Z =
= H (Z)
Z
D & = &.
D = &; Z & = &. D = - &;
D
Y
D
D
Z
D
Laws
D
of evolution
p
F
p
& = &. F; & = &. F
T
F
F
F
X
T
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3.6
Form of the tangent matrix
In order to ensure the robustness and the effectiveness of the model in the numerical establishment and for the analysis
total of the massive structures, it is necessary to calculate the tangent matrix, which can be given with
to leave the following expression:
·
·
=
T
T
éq 3.6-1
After some analytical calculations, one can deduce the expression from the tangent module while using
condition of consistency and respective laws of evolution:
G (1 - (G ()
T
T) F
T)
H =
éq
3.6-2
(
F
T)
p
(
F
T)
1+ G DT (
F
) 2
2
p
(
) (
F
)
With
G
(T)
DT DT F G H -
'F “G H - F G” H
=
=
G
éq
3.6-3
2
T
T
DT
T
H
Where F, G and H are the following functions, obtained thanks to [éq 3.2-8]:
T
F
1
=
éq 3.6-4
DT
B
2
1DT
G = expA
1
éq
3.6-5
DT
(
-
DT
T1)
G
B2DT
H = 1 + A2
-
DT
DT
T1 +
éq
3.6-6
Note:
In practice in Aster, the tangent matrix was not established, only the secant matrix is
E 1
(- D)
0
NR
used either H =
.
0
G 1
(- D)
T
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4 Integration
numerical
Separation in two parts in the formulation: damage slip, allows us
to treat each one of it separately. Thus, the integration of the damage part is carried out of
explicit way by the definition of two surfaces threshold. On the other hand, the part “slip” is solved
in an implicit way by a traditional method with knowing the algorithm of the type “return-mapping”
proposed by Ortiz & Simo, cf [bib4], which will ensure the effective convergence of way.
4.1 Calculation of the part “friction of the fissures” with a method
of implicit integration
The effects on the connection associated with the phenomenon of friction with the fissures can be calculated in
the framework of a behavior pseudo-plastic with a nonlinear kinematic work hardening. For
the establishment with the method of integration suggested, we will carry out a linearization of
function threshold around the current values of the variables intern associated. With the iteration (i+1),
surface threshold is written:
(I)
(I)
(i+)
1
(I)
(I)
F
F
F
= +
:
éq
4.1-1
F
F
-
F
T
T
+
: ((i+) 1 - (I)) 0
F
T
According to the equations [éq 3.1-7], [éq 3.1-8], and [éq 3.6-5], one a:
·
·
· p
= = -
F
F
éq
4.1-2
·
·
p
·
F
F
= - D = - D
F
T
G
T
T
G
T
F
éq
4.1-3
F
T
That one can discretize in the following way:
p
= (i+) 1 - (I) = = -
F
F
éq
4.1-4
p
(i+)
1
(I)
F
F
F
F
=
-
= - D
= - D
F
T
T
T
G
T
T
G
T
F
éq
4.1-5
F
T
By combining these expressions with the expression of surface threshold and by writing that is equal to
F
zero, one can deduce the increment from multiplier
with each iteration I:
F
(I)
=
F
F
éq
4.1-6
(I)
(I)
(I)
p
(I)
p
F
G D
F
F
F
T
+
F
F
T
T
After obtaining the value of
, one can substitute it in the equations [éq 4.1-4] and [éq 4.1-5]
F
in order to bring up to date the thermodynamic forces
F
T and. The iterations will have to continue until
moment when the condition of consistency is checked.
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4.2
The algorithm of resolution
In a general way, one seeks to check the balance of the structure at every moment, in a form
incremental. As clarified previously, for the damage a simple scalar equation
allows to obtain the corresponding value, which makes it possible to avoid a recourse to the iterative methods.
On the other hand, an iterative method is applied for integration of the friction part of the fissures.
Then, the algorithm is as follows:
(I) Réactualisation
geometrical:
(
=
+ U
T) +
(
N 1
T)
S
N
T
(II) Prédiction
rubber band:
(F () =
T) 0
n+1
(fT) N;
(E () =
-
T) 0
n+1
(T) n+1 (fT) n+1;
(0
) =
n+1
N;
(0)
=
,
n+1
((and) (0) (0)
n+1
n+1)
(III) Evaluation of
threshold:
(F) (0) 0?
n+1
if so, end of the cycle; if NON, beginning of the iterations
OUI:
(F =
(E =
(0)
=
(0)
=
T) n+1
(and) (0)
T) n+1
(fT) (0) n+1;
n+1; n+1
n+1; n+1
n+1
NON:
I = 0
(iv) Correction
plastic:
(F) (I)
= (
N 1
+
F
F
F
/
+
T) (I) G. D
/
/X
.
/X
N 1
+
T (
p
F
F
T) (I)
(
1
+
F
) (I)
N
N 1
+
(PF) (I) n1+
(i+) 1
(I)
=
-
-
N 1
+
N
G.
1
+
T
D.
.
F
(p F
/
F
T) (I)
.
.
F
(p/X
F
) (I)
(i+1)
(I)
=
+
n+1
n+1
F
(PF) (I)
(v) Vérification of
convergence:
(F) (i+1) TOL
+
(F) (0)?
N 1
n+1
if so, end of the cycle; if NON, to continue the iterations in (iv)
OUI:
(i+1)
=
n+1
n+1;
(i+1)
=
n+1
n+1;
(E =,
T)
E
n+1
T (n+1
n+1);
(F =
-
T) +
(
N 1
T) n+1
(and) n+1
NON:
I = I + 1
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4.3
Variables intern model
We show here the internal variables stored in each point of Gauss in the implementation of
model:
Number of
Feel physical
variable
intern
1
D NR: Scalar variable of the damage in the normal direction
2
DT: Scalar variable of the damage in the tangential direction
3
zT1: Scalar variable of isotropic work hardening for the damage in mode 1
4
zT2: Scalar variable of isotropic work hardening for the damage in mode 2
5
F
: Deformation of slip cumulated by friction of the fissures
T
6
: Value of kinematic work hardening by friction of the fissures
5
Parameters of the law
The law of behavior presented here is controlled by 14 parameters, of which 3 manage the answer in
the normal direction and the others affect the response in the tangential direction. In addition, it
Young modulus is recovered starting from the elastic data provided by the operator ELAS, who must
to always appear in the command file.
These parameters, or the analytical expressions which make it possible to obtain them, were obtained or
determined starting from the digital simulation of the experimental tests carried out by Eligehausen and
Al, 1983, cf [bib3]. The realization of multiple simulations made it possible to determine a relation enters
geometrical and material characteristics of materials in question (steel and concrete) and them
parameters which manage the model of the interface.
5.1
Initial parameters
5.1.1 The parameter “hpen”
The element joint functioning on the concept of jump of displacement, it is necessary to introduce one
dimension characteristic of the zone of degraded interface allowing to define the concept of
deformation in the interface. With this intention it was introduced the principle of penetration between surfaces:
parameter “hpen” makes it possible to define this zone surrounding the bar of steel. This parameter corresponds
with the possible maximum penetration which depends on the thickness of the compressed concrete - crushed. Into same
time, “hpen” manages the dissipation of energy in the element as well as the kinematics of the slip.
In order to give a reference to the user for the choice of this parameter, one proposes to calculate it with
to leave the diameter of the bar D
B and the relative surface of the veins
defined by:
Sr
K FR
sin
Sr =
éq
5.1.1-1
D C
B
where K is the number of veins on the perimeter; R
F the transverse surface of a vein; is the angle
between the vein and the axis longitudinal of the steel bar; and C is the measured distance between veins
center in center. Finally, “hpen” will be calculated with the expression:
hpen = dB Sr
éq
5.1.1-2
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According to Eligehausen and Al, the reinforcements usually used in the United States have values of
Sr
between 0.05 and 0.08. For the smooth bars, since one needs a small value for
“hpen”, one proposes values of between 0.005 and 0.02.
Sr
The following table gives the values of “hpen” according to the diameter of the bar:
Diameter (mm)
Relative surface
Hpen (mm)
Description
8 0.01
(0.08) 0.1
Smooth commercial bar
8
0.08
0.64
Ribbed commercial bar
20
0.08
1.50
Ribbed commercial bar
25
0.08
2.00
Ribbed commercial bar
32
0.08
2.54
Ribbed commercial bar
The unit of “hpen” must of course correspond to the unit used for the grid.
5.1.2 The parameter G or module of rigidity of the connection
Generally, because of difficulty in measuring the deformations by shearing, the module of
rigidity of a material is calculated starting from Young and the Poisson's ratio modulus, parameters
currents obtained in experiments. However, for our case, the interface is a pseudo-material
whose characteristics must depend on the properties corresponding to materials in contact,
steel and concrete. Since the material which one expects to damage is the concrete, one proposes
to initially use for the connection the same value of G that for the studied concrete but it can be
higher up to a value similar to the value of the Young modulus E, when one increases
value of “hpen”. In the case of reinforcements with rigidities higher than those of the bars
commercial current (because of a provision or special geometry of the veins), one can make
a correction of the value chosen, by multiplying the module of rigidity by a coefficient of correction
calculated starting from the relative surfaces of the commercial bars, with the expression:
(SR) bars
Carm = (
éq
5.1.2-1
SR) barrecomm
Then, the module of rigidity of the connection G will be:
Gliai = Carm Gbeton
éq
5.1.2-2
In the last expressions, Glia is the module of rigidity of the connection; Gbeton is the module of
rigidity of the concrete; Carm is the coefficient of correction per reinforcement; () bars
SR
, relative surface of
veins of the bar concerned; and (SR) barrecom, relative surface of the veins of the bar
commercial of the same diameter (preferably, 0.08).
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5.2
Parameters of damage
5.2.1 Limit of elastic strain 1 or threshold of perfect adherence
T
To define the threshold of perfect adherence, it is considered that the damage by shearing must
to initiate itself at the time of the going beyond of a certain threshold of deformation. So one proposes to adopt them
limiting deformations of the concrete in traction, i.e., between 1x10-4 and 0.5x10-3, which corresponds to
shear stresses between 0.5 and 4 MPa in perfect adherence.
5.2.2 The parameter of damage A for the passage of the small deformations
1DT
with the great slips
In this area, the law of evolution of the damage is expressed in term of deformations and its
construction depends on the definite elastic slope for the linear behavior (constraint on
shearing vs. deformation) in the area of perfect adherence: this parameter controls the value of
the constraint compared to the slip in the passage of small deformations to large
slips.
The determination of the value of this parameter is a key and delicate point model, since the evolution
damage must be carried out with certain conditions noticed by several researchers;
for example:
·
the resistance of the connection is directly proportional to the compressive strength of
concrete. However, as the resistance of the concrete is increased, it
behavior becomes more rigid, bringing to the brittle fracture of the connection,
·
the particular rigidity of the reinforcement, which is related on the diameter and the quantity of the veins on
surface, must increase the resistance of the connection,
·
the relation between the moduli of elasticity of two materials concerned must manage directly
the kinematics of the connection.
From the digital simulations that one carried out, one observed that this value is located
between a minimum of 1 and one maximum of 5, and which it will have to be adjusted according to the test of
reference selected. Optionally, one proposes an expression which makes it possible to adopt an initial value
and which depends on the particular characteristics of materials:
1
F 'C
Ea
A1
=
DT
(
éq
5.2.2-1
1 + SR)
30
Eb
In the last expression, Eb will be calculated with the expression provided in the section A.2.1, 2 of
BAEL'91:
E = 11000
B
× (F 'c)1 3 éq
5.2.2-2
In the two last expressions, one a:
·
F 'C, compressive strength of the concrete in MPa;
·
Ea, modulus of elasticity of steel, in MPa;
·
Eb, modulus of elasticity of the concrete, in MPa;
· Sr, relative surface of the veins of the bar concerned.
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[Figure 5.2.2-a] gives some a graphic comparison.
COMPARISON OF A1DT
- Local Law of the connection -
40
20
ad1=1.0
35
ad1=2.19
ad1=3.8
-
O
30
15
MT
ent
25
)
has
)
Pa
L
E
m
P
20
M
10
Y (M
C
I
sai
Y (
X
S
SX
T
E
of
15
has
I
N
10
-
contr
5
5
0
0
0
0.5
1
1.5
2
0
0.05
0.1
0.15
defo EXY
defo EXY
Appear 5.2.2-a: Comparaison of A1DT: growth of the resistance of the connection
5.2.3 The parameter of damage B
1DT
The purpose of this parameter is to soften the shape of the curve of behavior, like facilitating
transition from the elastic slope towards the nonlinear area. It can have a value included/understood enters
0.1 and 0.5 (never higher than 0.5 since it is the equivalent of the square root of the formula) .On can
to advise to adopt the value of 0.3 for ordinary calculations. (See [Figure 5.2.3-a]).
VARIATION DE Bd1
- Local Law of the connection -
25
20
Bd1=0.1
Bd1=0.2
Bd1=0.25
20
Bd1=0.3
Bd1=0.5
-
15
O
NT MT
E
15
)
Pa)
I
L
E
m
Pa
M
its
10
E
Ci
XY (M
SXY (
D
S
T
E
10
I
N
T
ruffle
N
-
Co
5
5
0
0
0
0.5
1
1.5
2
0
0.05
0.1
0.15
defo EXY
defo EXY
Appear 5.2.3-a: Comparaison of B1DT: Modification of the curvature
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5.2.4 Limit of deformation 2 or threshold of the great slips
T
According to several authors, the great slips are overall higher 1 mm from displacement,
but that is an indicator which depends on the form and dimensions of the specimens tested; therefore, one
propose that this deformation never exceeds 1.00 (adimensional value). In way more
specify, one proposes to apply the following expression:
N
With
With
2
1
(
=
1DT)
1
(
-
=
1DT) 4
T
(
éq 5.2.4-1
hpen)
1
-
2 C + (
N
1
WITH T) (hpen)
1
2
D
+ (1
WITH T)
1.0
9
4
D
In this expression, one applied a sigmoid function whose coefficients C and N allow
to adjust the kinematic effect of A1DT on the slip, i.e., when the connection becomes more
resistant because of an increase in rigidity, the slip is reduced gradually. One adopted
values 9.0 and 4.0 respectively, but they are always optional.
The choice of the value of the limit of deformation 2
is very important because it introduces one more or less
T
great brittleness of the response by translation of the threshold of passage of the small deformations to large
slips. This brittleness is related to the stiffness of the concrete via parameter A1DT. It is necessary
to note that the following parameters which manage the damage must be also adjusted on the level
room to ensure the correct continuity of behavior in shearing of the connection and to thus be able
to obtain the desired or awaited response of a system real steel connection concrete.
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5.2.5 The parameter of damage A
2DT
The damage, such as it was conceived in the model, obeys two laws of evolution which are
expressed using only one traditional scalar variable which will ensure the coherence of
the damage. The parameters of each of the 2 laws are independent and numerically
stable, but they are likely to generate serious errors in the continuity of the behavior
if one does not pay attention to the shape of the local curve stress-strain: to see the case of
curve shown in graphics of [Figure 5.2.5-a], with a value A
= 1x10-3 MPa-1. Us
2 DT
let us not be able to propose an analytical relation for the choice of this parameter, but
the gained experience enables us to affirm that the value of this parameter must be included/understood enters
1x10-3 and 9x10-2 MPa-1 roughly.
COMPARISON OF A2DT
- Local Law of the connection -
40
35
-
Ad2 = 1e-3
O
30
Ad2 = 3rd-3
MT
Ad2 = 6th-3
Ad2 = 9th-3
ent
25
Ad2 = 1.2e-2
has
)
P
L
E
m
M
20
(
Y
C
I
sai
SX
T
E
of
15
has
I
N
10
-
contr
5
0
0
0.5
1
1.5
2
defo EXY
Appear 5.2.5-a: Comparaison of A2DT: damage and rupture of the connection
5.2.6 The parameter of damage B
2DT
This parameter, which supplements the law of evolution of damage in great slips, controls not
only growth of the resistance of the connection or shape of the curve of behavior to the peak
and in the area post-peak, but also the kinematics of the answer, which implies the determination of
slip for the maximum shear stress as well as the amplitude of the curve to the peak of
behavior. Then, although values of the parameters of A2 damage
B
DT and 2 DT
will have to adjust itself at the same time when one builds the curve of behavior of the connection in order to
to respect the continuity of the pace, one can say that the value of B2DT is inversely proportional
the amplitude of slip at the top, i.e., a value of 0.8 allows great slips
broader in the node than a value of 1.2, for example.
For practical cases, one recommends to use a value ranging between 0.8 and 1.1 to reproduce
a coherent curve of behavior (Voir [Figure 5.2.6-a]).
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VARIATION OF B2DT
- Local Law of the connection -
25
20
-
O
With
T
Bd2 = 0.7
E
NT
Bd2 = 0.8
)
15
Bd2 = 0.9
Pa
I
L
lem
Bd2 = 1.0
M
I
its
Bd2 = 1.5
Y (
C
X
E
S
D
10
T
E
has
I
N
-
contr
5
0
0
0.5
1
1.5
2
defo EXY
Appear 5.2.6-a: Comparaison of B2DT: damage and rupture of the connection
5.3
Parameters of damage on the normal direction
5.3.1 Limit of deformation 1 or threshold of great displacements
NR
In a way similar to the elastic behavior in the tangential direction, it is considered that
decoherence must be initiated at the time of the going beyond of a certain threshold of deformation. We propose
to adopt a value between 10-4 and 10-3.
5.3.2 The parameter of damage A
DNN
This parameter controls primarily the slope of degradation of the normal constraint compared to
deformation due to the opening of the interface. We propose to use a minimal value of 1x10-1
MPa-1, which corresponds to a degradation similar to that of the concrete. Nevertheless, if one wishes to have
a behavior of the connection even more fragile, it is enough to increase this value.
5.3.3 The parameter of damage B
DNN
In combination with the preceding parameter, this parameter controls the damage of the connection,
in particular shape of the curve of behavior in phase post-peak.
We propose to use a value equalizes to 1, or 1,2 for more marked curves.
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25
ADN = 1e-3
ADN = 1e-2
20
ADN = 1e-1
-
NCE
R
E
E
D
H
has
15
)
has
P
of
E
M
Y (
rmal
O
SY
E NR
10
NT
T
R
have
O
N
- C
5
0
0
0.5
1
1.5
2
Normal deformation
Be reproduced 5.3.3-a: behavior of the connection on the normal direction at the time of the opening
interface (normal traction on the connection).
5.4
Parameters of friction
5.4.1 The parameter material of friction of the fissures
One of the assets of the model suggested here is that it is able to take into account the effects of
friction of the fissures, which, in the case of monotonous loading, appears by a contribution
positive with the shear strength of the connection; in addition, in the cases of loadings
cyclic, it is obvious that the pace of the loops of hysteresis depends directly on the choice of the value
this parameter material. However, the corresponding values were not gauged, since
we did not simulate tests with cyclic loadings yet to validate them.
Temporarily, one proposes to use values lower than 10 MPa, with a maximum value of
equal to 1.0 MPa-1.
5.4.2 The parameter kinematic material of work hardening
On [Figure 5.4.2-a], one can appreciate that the reduction in the value of increases dissipation
hysteretic, but also the resistance of shearing and the residual deformation pseudo-plastic.
That is very important for the cyclic modeling of the connection since in reality, when one
exceed the peak of maximum resistance, one notices that at the time of the discharge there is no more
elastic contribution of the slip, i.e. the residual deformation pseudo-plastic
corresponds exactly to the total slip reached. In other words, once connected all them
fissures in the potential of rupture, longitudinal and tangential layer with the steel bar, the single one
resistance which will prevent the displacement of the reinforcement is the friction resistance of the connection,
produced by the contact and the tangle of the asperities between surfaces concrete concrete.
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As previously, our experiment is limited: one proposes to use a maximum value of
0.1 MPa-1 which gives correct results for applications in monotonous loading, and which
seem suitable for cyclic loadings.
EFFET OF A SUR LES BOUCLES OF HYSTERESIS
20
- Friction of the fissures -
15
-
O
With
T
E
NT
10
)
Pa
I
L
E
m
I
its
Y (M
C
X
5
S
of
T
E
has
I
N
ntr
0
100 A 4
-
Co
100 A 1
100 A .5
100 A .2
- 5
0
0.5
1
1.5
2
defo EXY
Appear 5.4.2-a: Comparaison of a: effects on the loops of hysteresis into cyclic
5.4.3 The parameter of influence of containment C
In our model, the influence of containment was taken into account thanks to the application of it
parameter which controls these effects on the connection, and which appears by an increase in
maximum shear stress like by the increase in maximum displacement to the peak
when containment increases.
For the calibration, we carried out simulations with containments of 0, 5, 10 and 15 MPa,
by always using a value of 1.0 for this parameter. It was noticed that if one wants to produce one
kinematic translation of the slip caused by containment, it is enough D `to adopt a value of 1.2
or 1.5 (adimensional). Optionally, it is advised to maintain the value of 1.0 for
ordinary calculations.
5.5
Summary of the parameters
To facilitate the use of the law, the suiant table presents a synthesis of the whole of
parameters of the model of behavior.
It is pointed out that the values or the expressions suggested have only one indicative value, and that
arbitrary combination can give inaccurate and unexpected results compared to the behavior
hoped for connection; in other words, a bad choice of the parameters can produce a strong rigidity
or a weak response of the interface steel concrete.
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Law of behavior (in 2D) for the steel-concrete connection: JOINT_BA
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Value
Variables
Parameter Unit
Analytical expression
proposed
concerned
Diameter of
dB
the bar
H
H
= D
PEN
mm -
PEN
B
Sr
Relative surface
Sr
veins
Coefficient of
Carm correction by
reinforcements
G
G
= C
G
bound
MPa -
bound
ARM
concrete
Modulate
Gbeton rigidity of
concrete
1
min 1.0x10-4
T
-
max 1.5x10-3
Resistance to
f' C,
compression
concrete
(MPa)
1
F 'C
E
With
min 1.0
has
With
=
1
1
Modulate
DT
-
max 5.0
DT
(1+
30
E
SR)
B
Ea
of elasticity of
steel
Modulate
Eb
of elasticity of
concrete
B
min 0.1
1
DT
-
max 0.5
With
2
1
(1 TD) 4
2
=
1 -
T
- - T
(H
With
PEN) 2
+ (1 T
D)
1.0
9
4
With
min 1.0x10-4
2
DT
MPa-1 max 9.x10-2
B
min 0.8
2
DT
-
max 1.5
MPa
max 10.0
has
min 0.01
MPa-1
max 1.0
C
- 1.0
(value
recommended)
1
min 10-4
-
NR
max 0.9 10-3
With
(value recommended, not gauged)
DNN
MPa-1 min 1.0x10-1
B
(value recommended, not gauged)
DNN
- 1.
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Law of behavior (in 2D) for the steel-concrete connection: JOINT_BA
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6 Bibliography
[1]
ARMSTRONG, P.J. & FREDERICK, C.O. : In Mathematical Representation off the Multiaxial
Bauschinger Effect. G.E.G.B. ; Carryforward RD/B/NR, 731, 1966.
[2]
BERTERO V.V.: Concrete Seismic behavior off structural linear elements (beams and
columns) and to their connections. Committee Euro-International of Béton (CEB); Bulletin
of information No. 131; Paris, France, 1979.
[3]
ELIGEHAUSEN R., POPOV E.P. & BERTERO V.V.: Room jump stress-slipway relationships off
deformed bars under generalized excitations. University off California; Carryforward No.
UCB/EERC-83/23 off the National Science Foundation, 1983.
[4]
ORTIZ Mr. & SIMO J.C. : Year analysis off has new class off integration algorithms for elastoplastic
constitutive relations. International Journal for Numerical Methods in Engineering; Vol. 23,
pp. 353 366, 1986.
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Law of behavior (in 2D) for the steel-concrete connection: JOINT_BA
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