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103
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
L. A. F. de Souza | R. D. Machado
model, the steel is represented as an elastoplastic material and it
has the same behavior in tensile and compression stresses. The
schematic is given by a bilinear stress-strain diagram. Thus, the
stress in steel is determined by (Tiago
et al
. [17]):
(7)
σ =
{
E
a
ε
E
at
ε
, -ε
sy
≤
ε
≤
ε
sy
,otherwise
Where
E
a
is the initial longitudinal modulus of elasticity for steel,
ε
sy
is the yielding extension and
E
at
=
k
a
E
a
is the longitudinal modulus
of elasticity after the steel yielding.
4. Line interface element
The geometrical discontinuities can be successfully modeled us-
ing interface finite elements. These elements aim to transmit the
stresses between either bodies or parts of the same body among
which it is found.
Line interface elements were used in this paper in order to simu-
late pre-established cracks in the part, assuming cracking mode I.
Mode I effect is represented by the transmission of stresses that
are normal to the crack faces.
The line interface element is based on the work of Schellekens
[16]. This one-dimensional element is isoparametric, with four nod-
al points (two degrees of freedom per node -
u
,
v
), linear shape
functions and zero thickness.
The nodal displacement vector
u
is given by:
(8)
u=
[
u
1
v
1
u
2
v
2
u
3
v
3
u
4
v
4
]
T
Where
u
i
and
v
i
,
i
= 1,...,4, are the nodal displacements in the di-
rection
ξ
and
η
, respectively. Operator
B
, which relates the nodal
displacements to the displacement field regarding the element, is:
(9)
B=
[
[
h
1
0 h
2
0 h
3
0 h
4
0
0 h
1
0 h
2
0 h
3
0 h
4
Where
h
i
,
i
= 1,...,4, are the shape functions given by:
(10)
h
1
=h
4
=
1
2
(
1-
)
(11)
h
2
=h
3
=
1
2
(
1
+
)
Usually, the stresses are evaluated as a function of the strains; how-
ever, in the case of the stresses in the interface, those are determined
Where
ε
i
,
i
= 1,...,3, are main strain components and <
ε
i
>
+
,
i
=
1,...,3, are the positive parts defined by:
(3)
<
ϵ
i
>
+
=
1
2
(ϵ | |)
i
+ ϵ
i
The concrete, with regards to the rupture modes, presents a dis-
tinct behavior in relation to the tensile and compression stresses.
The concrete rupture by tensile strengths happens due to crack
formation and the consequent loss of normal strength in the direc-
tion of the crack. As for the damage during compression, the con-
crete presents a behavior that can be considered plastic, which is
the crushing caused by the internal cohesion being overcome due
to the shear stress characterized by a large quantity of microcracks
(Leonel
et al
. [9]).
Considering a continuously increasing or radial loading of the
stress-strain curves obtained in tensile and compression uniaxial
tests, the damage variables
D
T
and
D
C
can be explicitly determined
the following way, respectively:
(4)
D
T
(ε)=1-
ε
d0
1(
(
(
( -A
T
ε
-
A
T
e
B
T
ε-ε
d0
~
~
~
(5)
D
c
(ε)=1-
ε
d0
1(
(
(
( -A
c
ε
-
A
c
e
B
T
ε-ε
d0
~
~
~
Where
A
T
and
B
T
are characteristic parameters of the material in
uniaxial tensile stress,
A
C
and
B
C
are parameters of the material in
uniaxial compression stress and
ε
d0
is the limiting elastic strain. The
subindexes
T
and
C
mean tensile and compression, respectively.
For complex stress states, the variable for damage can be deter-
mined by a linear combination of
D
T
and
D
C
by means of the follow-
ing condition (Pituba and Proença [13]):
(6)
D=α
T
D
T
(
(
ε
ε
~
~
)
)
+α
C
D
c
,
α
T
+α
C
=1
Where the coefficients
α
T
and
α
C
take on values in the closed inter-
val [0,1], and represent the contribution of the tensile and compres-
sion stresses to the local extension state, respectively. Mazars [10]
proposed the following variation limits for the parameters
A
T
,
B
T
,
A
C
and
B
C
, obtained from the calibration with experimental results:
0,7
≤
A
T
≤
1 10
4
≤
B
T
≤
10
5
1
≤
A
C
≤
1,5 10
3
≤
B
C
≤
2 10
3
10
-5
≤
≤
2 10
-4
3. Constitutive model for steel
In this work, a uniaxial model is used to describe the behavior of
armatures, once, in reinforced concrete structures, the steel bars
essentially resist axial strains. In the implemented computational