Page 108 - Riem-Vol6_nº1

102 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Numerical-computational analysis of reinforced concrete structures considering the damage,
fracture and failure criterion
1. Introduction
Concrete has been the most used structural material in the world
in the past sixty years. It is employed in all types of construction for
its several advantages: easily malleable, adapting itself to any kind
of form; fire resistant, resistant to the atmospheric influences and
to mechanical wear; and it is an economic alternative for it is built
with low cost raw material.
One of the difficulties in the concrete structures computational mod-
eling is the definition of the constitutive relationships that consider
the non-linear behavior of the material and, the potential cracking
and the different responses to tensile and compression stresses. It
is very difficult to separate the deformation and rupture phenomena
in the concrete for the microcracks and the hollows that exist even
before any stress is applied, interfere directly in the initial response
of the moving material (Lemaitre and Chaboche [8]).
Analyses of reinforced concrete structures based on elastic mate-
rial models (linear or non-linear) are widely used in design offices
nowadays and the results are employed in the sizing and evalua-
tion of their global behavior. When these structures are subjected
to loadings that cause the beginning of cracks in the concrete in
tensile stress, the elastic analyses are not able to properly simulate
this behavior (Leonel et al. [9]).
The Plasticity, Damage Mechanics and Fracture Mechanics theo-
ries are widely used in structure analysis of reinforced concrete,
each one of them being appropriate to simulate such phenome-
non. As there is no complete constitutive model for the concrete
yet, the trend is to employ a set of these theories for representing
the phenomena related to the material behavior.
A natural evolution are the models that couple more than one the-
ory, creating formulations that are almost always complex. How-
ever, aiming to decrease the complexity of the formulations, but
still taking into account the coupling of the effects in the formula-
tion, and due to theories, a great highlight has been given to the so
called simplified constitutive models (Álvares et al. [1]).
In general, the concrete destruction can be divided into two types:
the first one happens by tensile stress and it is characterized by
crack formation and loss of tensile stress resistance in the normal
direction to the cracks formed; the second one is due to compres-
sion, and it is characterized by the formation of multiple cracks
parallel to the compression force. The cracks from the later have
a smaller size which make the concrete lose a great part of its
strength.
A strength criterion aims to establish laws through which it is pos-
sible to predict the rupture condition under any type of stress or
strain combination through the material behavior in the simple
tensile and compression tests (Nicolas et al. [12]). Many of the
existing strength criteria present restrictions in relation to the appli-
cation to heterogeneous and anisotropic materials with directional
strength and elasticity properties such as concrete. Therefore, the
investigation of a strength criterion that allows a proper evaluation
of the rupture of this material to an axial or biaxial stress state be-
comes important and necessary.
This article presents two modeling proposals for reinforced
concrete structures through two numerical computational stud-
ies using the Finite Element Method aiming to analyze the
aspects involved in computational modeling including items
related to constitutive models of the materials. The non-linear
analyses are carried out considering the arc-length method
with the modified Newton-Raphson iterative process. This
technique is characterized by presenting a concomitant load
and displacement control.
The first study consists of the one-dimensional structural analysis
of a beam proposed by Jarek et al. [7]. The modeling of this struc-
tural element is made with the Scilab program, version 5.3.3. The
concrete behavior is simulated by the damage constitutive model
proposed by Mazars [10] and, the steel by a bilinear elastoplastic
constitutive model. Tsai and Wu’s [18] failure criterion is also incor-
porated to the model.
From the two-dimensional problem of an adapted reinforced pull
rod from Mazars and Pijaudier-Cabot [11], the second study analy-
ses the structural response considering the damage constitutive
model coupling proposed by Mazars [10] and a rupture model in
Mode I based on the work of Schellekens [16]. This model was
implemented in Fortran code – Compaq Visual Fortran Edition 6.5.
The crack is simulated though a line interface element and the
softening phenomenon can be considered – linear, bilinear or ex-
ponential - in the constitutive model.
2. Constitutive model for the concrete
The damage model proposed by Mazars [10] is based on a couple
of experimental evidences observed in uniaxial tests of concrete
specimens, having as basic hypotheses (Proença [14]):
nlocally, the damage is due to extensions (elongations) evi-
denced by positive signs; at least by one of them, the main
strain components (εi > 0);
nthe damage is represented by a scalar variable D ∈ [0.1] which
evolution happens when a reference value for the ‘equivalent
elongation’ is overcome;
nit is considered, however, that the damage is isotropic although
the experimental analyses show that the damage leads, in gen-
eral, to a an anisotropy of the concrete (which can be initially
considered as isotropic), and
nthe damaged concrete behaves as an elastic medium. Thus,
the permanent strains that were experimentally evidenced in an
unloading situation are disregarded.
In this model, it is supposed that the damage begins when the
equivalent strain reaches a reference strain value εd0, deter-
mined in uniaxial tensile tests in relation to the maximum stress.
The constitutive relation is given by (Tiago et al. [17]):
(1) σ =(I -DI)Cº
Where I is the identity tensor and C 0 is the elastic tensor of the
undamaged material. The extension state is locally character-
ized by an equivalent strain that is expressed by (Pituba and
Proença [13]):
(2) ϵ~= <ϵ1 > +
2+<ϵ 2 >+
2 +<ϵ3 > +
2