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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
J. J. C. PITUBA | M. M. S. LACERDA
tween the numerical responses is mainly due to the excessive
stiffness reduction presented by isotropic model, what does not
happen with the anisotropic model. Being an isotropic damage
model, the Mazars’ model degrades the stiffness exaggerat-
edly in all directions. Besides, one believes that the quality of
results related to the Pituba’s model can be better if the dam-
age process related to the shear behavior of the concrete is
considered, because it can produce an important contribution
to the released energy.
Note that in the one-dimensional analysis the difference be-
tween numerical responses of damage models is mainly due
to more aggressive reduction of the stiffness in the anisotro-
pic model. While the Mazars’ model presents a linear reduc-
tion way of the material stiffness (Strain Equivalence - Eq.
(8)), the Pituba’s model reduces the material stiffness in a
quadratic way (Energy Equivalence - Eq. (11)). In this analy-
sis, it is possible to reproduce a stiffness break about 45 kN.
Soon after, a subsequent strength recovery of the structural
element is illustrated by the models. Then, the difference
between the numerical responses is increased. Finally, note
that in 1D and 2D numerical analyses, both models present
the same qualitative behaviors.
In order to visualize the damage distribution in the beam,
Figure [6] presents the isodamage curves for Mazars´model,
Proença [16]. Figure [7] shows the damage distribution for 1D
and 2D numerical analyses with Pituba´s model considering a
loading stage about 50 kN. These curves have been obtained
by interpolation of the damage variables along the adopted
integration points.
In the context of 1D analysis, when the Proença’s response
is compared with the Pituba´s model, the damage configura-
tion is very similar, mainly due to damage in tension. Note
that Mazars’ model presents only one damage variable rep-
resenting a combination of tension and compression damage
processes. For the concrete structures, both models assume
that locally the damage is due to extensions. On the other
hand, the 2D numerical responses show a more intense dam-
age process in compression given by D
2
and D
3
variables. In
fact, the variable D
3
tries to simulate a crushing process of
the concrete near of the supports and in upper middle zone
of the beam.
4.2 Reinforced Concrete Bar Structure
The second numerical simulation deals with a reinforced concrete
bar structure. Figure [8] shows the geometry of the structure. A
reinforcement bar with 10.0 mm is placed in the centre of the sec-
tion where a force P is applied. This numerical application was per-
formed previously by Mazars [1]. Table [2] presents the parameter
values of each damage model. These parameters were obtained
through procedure similar to the previous example with E
c
=30000
MPa, where the parameters suggested for the Mazars’ model were
taken as reference values, see Figure [9]. For the steel, E
a
= 210
GPa has been adopted.
Figure 8 – Geometry of the reinforced concrete bar structure
All dimensions in meters
P
Figure 9 – Parametric identification in uniaxial compression and tension tests
for reinforced concrete bar structure