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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
Simplified damage models applied in the numerical analysis of reinforced concrete structures
Assuming a general situation of damaged medium for dom-
inant tension states, the criterion for the identification of
damage increments is represented by the fol lowing rela-
tionship:
(19)
f
T
(
s
) =
0
0
£ -
*
+
*
T
e
Y W
where the reference value
∗
T
Y
0
is defined by the maximum com-
plementary elastic energy computed throughout the damage pro-
cess up to the current state. Analogous expressions are valid for
dominant compression states.
In the loading case, i. e., when
0
¹
T
D
.
or
0
¹
C
D
.
, it is nec-
essary to update the values of the scalar damage variables that
appear in the
D
T
and
D
C
tensors, considering their evolution laws.
Considering just the case of monotonic loading, the evolution laws
proposed for the scalar damage variables are resulting of fittings
on experimental results and present similar characteristics to those
one described in Mazars [1] and Berthaud [4] works. The general
form proposed is:
(20)
[
]
)
(
exp
i
i
i
i
i
i
Y YB A
A
D
0
1
1
-
+
+
-=
with i = 1, 2
where A
i
, B
i
and Y
0i
are parameters that must be identified. The
parameters Y
0i
are understood as initial limits for the damage
activation, the same ones used in Eq. (16). The parametric
identification of the model is accomplished by uniaxial tension
tests in order to obtain A
1
, B
1
and Y
01
= Y
0T
, by uniaxial com-
pression tests for the identification of the parameters A
2
, B
2
and Y
02
, and finally by biaxial compression tests in order to
obtain A
3
, B
3
and Y
03
= Y
02
= Y
0C
. On the other hand, the identi-
fication of the parameters for the evolution laws corresponding
to the damage variables D
4
and D
5
, which influence the shear
concrete behavior, it won’t be studied in this version of the
model because the experimental tests are not available yet
to allow the parameter calibration or, even, the proposition of
more realistic evolution laws.
When the damage process is activated, the formulation starts
to involve the tensor
A
that depends on the normal to the trans-
verse isotropy plane. Therefore, it is necessary to establish
some rules to identify its location for an actual strain state. Ini-
tially, it is established a general criterion for the existence of
the transverse isotropy plane. It is proposed that the transverse
isotropy due to damage only arises if positive strain rates ex-
ist at least in one of the principal directions, Pituba [6]. After
assuming such proposition as valid, some rules to identify its
location must be defined. First of all, considering a strain state
in which one of the strain rates is no-null or has sign contrary to
the others, the following rule is applied:
“In the principal strain space, if two of the three strain rates are
extension, shortening or null, the plane defined by them will be
the transverse isotropy local plane of the material.”
The uniaxial tension is an example of this case where the
transverse isotropy plane is perpendicular to the tension stress
direction. Obviously, it can be suggested criteria based on oth-
ers formulations, such as for instance, the microplanes theory
developed by Bazant [14].
Figure 1 – Geometry details of
the plain concrete beam
All dimensions in metres
P
P
Figure 2 – Parametric identification in uniaxial compression and tension tests for plain concrete beam