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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Numerical-computational analysis of reinforced concrete structures considering the damage,
fracture and failure criterion
The constitutive model of the exponential softening is character-
ized by the curve in which the tensile stiffness decreases exponen-
tially in relation to the relative displacement. The contribution of the
tangential stiffness component is disregarded in this model. Matrix
D
can be written by:
(17)
D=
[
[
{
0 0
0 K
0
'
w≤w
c
→K
0
'
=K
0
e
-θw
w>w
c
→K
0
'
=0
Where
θ
is the exponential softening coefficient. The fracture en-
ergy (G
f
) for the exponential softening can be obtained by integrat-
ing the constitutive law and varying the opening of the crack from
0 to
∝
, where we have:
(18)
G
f
=
σ
u
Θ
The bilinear constitutive model is characterized by a curve with
two different slopes, considering that the material loses its strength
from the beginning of the stress. Matrix
D
can be written the fol-
lowing way:
(19)
[
[
{
D=
0 0
0 K
0
'
w
≤
w
1
→
K
0
'
=K
0
+
(
K
1
-K
0
)
w
w
1
w
1
<w=w
c
→
K
0
'
=
K
1
(
w-w
c
)
w
1
-w
c
w>w
c
→
K
0
'
=0
Where
w
1
is the opening of the crack and
K
1
is the tensile stiffness
from which the relation stiffness-opening follows the other consti-
tutive law. In case of bilinear softening, the critical opening of the
crack
w
c
is obtained by:
(20)
w
c
=
2G
F
U
-w
1
-
2
U
1
Where
σ
1
is the tensile stress of the material for an opening equal
to
w
1
.
5. Tsai and Wu’s strength criterion
The procedure proposed by Tsai and Wu [18] was to increase the
number of terms in Hill’s [6] failure criterion equation for a better
approximation of the experimental data obtained for several ma-
terials. The failure of a certain material is interpreted as the oc-
currence of any discontinuity in the material response to the me-
chanical stimuli (Nicolas
et al
. [12]). Some of the discontinuities of
interest are: the beginning of the non-linearity in the relation stress
versus
strain, the occurrence of irreversible strains and material
rupture. The conditions for the occurrence of these phenomena
as a function of the relative displacements. The relative displacements
(
∆w
) of the element are calculated through the following relation:
(12)
w
=
Bu
D
being the matrix of the material properties and, considering that
the line interface element has no dimension in the direction η and
that the thickness
e
is constant along the length of the material, the
stiffness matrix
K
is obtained by:
(13)
K=e
∫
B
T
DB
L
d
=1
=-1
Where
L
is the length of the element. The constitutive matrix
D
is
given by:
(14)
D=
[
[
K
S
0
0 K
0
Where
K
S
and
K
0
denote the horizontal stiffness components (tan-
gential stiffness) and vertical stiffness (tensile stiffness), respec-
tively. When calculating the matrix stiffness components,
D
, the
softening phenomenon – linear, bilinear or exponential - can be
considered in the constitutive model.
The linear softening model disregards the tangential cohesive ef-
fects and simplifies the tensile stiffness curve, considering that the
material loses its strength from the beginning. Matrix
D
can be writ-
ten the following way:
(15)
D=
[
[
0 0
0 K
0
'
{
w≤w
c
→K
0
'
=K
0
(
w
c
-w
w
c
)
w>w
c
→K
0
'
=0
Where
w
c
is the critical relative displacement from which there is no
transmission of stresses between the crack faces,
K
0
is the initial
tensile stiffness and,
w
is the opening between the nodes of the
element that has a normal interface in relation to the crack faces.
The opening of the critical crack (
w
c
), in the case of linear soften-
ing, is obtained from the fracture energy (
G
f
) and it is given by:
(16)
w
c
=
2G
F
U
Where σ
u
is the ultimate tensile stress of the material. This model is
in agreement with the Fracture Mechanics principles, for the area
limited by the stress curve transmitted through the crack
versus
the opening of the crack (σ x
w
) is equal to the fracture energy of
the material (
G
f
).