Basic HTML Version
738
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
Numerical simulation of the mechanical performance of deep beam
1. Introduction
Among the pioneering procedures aimed, for the evaluation of the
mechanical performance of reinforced concrete deep beams, there
are two alternate segments. One of them is based on the model of
connecting-rods. The other is based on horizontal stress distribu-
tion diagrams along the height of the beam, in hypothetically criti-
cal sections [5].
In the first model conception, the reinforcement steel bars play
the role of tie rods by absorbing the tensile stresses. Concrete,
in turn, when loaded, presents cracking mechanisms charac-
terized by establishing the standard connecting rods, located
strategically in outlined regions, which transmit the compressive
stresses. The rest of the beam concrete mass remains in idle
condition.
In another procedure, the stress distribution diagrams, obtained
from the theory of elasticity, are taken as reference for the determi-
nation of efforts to design the structural member.
It should be emphasized that the mechanical behavior of Portland
cement concrete is not linear since from the low-intensity stress-
es, as revealed on experimental tests [1], [2],[3],[4],[6],[7],[9],[11].
The source of its non-linearity is, especially, cracking develop-
ment prior to loading, arising from the shrinkage phenomenon
and thermal changings associated to the cement hydration heat,
as well as to the cracking propagation during the load-deforma-
tion process [3] e [9].
Thus, on the perspective for adoption bolder solutions, the design
of concrete deep beams, from the modeling criteria presented
above, can be precarious. It is appropriate, therefore, to adopt
models more suitable to structures design on the base of better
use of material.
The nonlinear Orthotropic Model [9], in spite of its modest formu-
lation, describes, properly, the mechanical behavior of concrete,
providing effective resources to overcome the shortcomings of pio-
neer models.
The subject of this work is the numerical simulation of mechanical
behavior of reinforced concrete deep beams by finite element ap-
proximation on a nonlinear Orthotropic Model.
2. Modelling
The numerical analysis was performed from the employing of an
incremental iterative procedure and finite element approximation.
The adopted mathematical modeling was based in the non-linear
orthotropic formulation [9], according which the constitutive matrix
elements to be used are defined in the base on similar equations
to those employed in uniaxial state of stress, however, taking as a
reference, the equivalent strains, which are given by:
(1)
ii
j ij
i
ei
D D
/
The “i” and “j” indexes refer to principal plane directions. The pa-
rameters “Dij” represent the matrix constitutive elements.
For concrete in compression simulation the constitutive relations
proposed by Hognestad [6] was adopted, presented in the form:
(2)
ei
ip
ei
ip
ip
i
.
.2
1
.2
for
ip
<
ei
< 0; and
ip
cu
ip
ei
ip
i
20
3
1
for
cu
<
ei
<
ip.
The parameters”
“
e
ip
”
and
“
s
ip
”
represent the concrete peak strain
and the peak stress, respectively, on the principal direction “i”, and
“
e
cu
”,
the failure limit strain. These equations represent the hard-
ening and softening snippets, OA and AB segments, respectively,
of the curve in Figure 1.
To represent the behavior of concrete in tension it was ad-
opted the smeared crack model, whose advantages are
consider the continuity of the displacement fields, and to
dispense topological character modifications in finite ele-
ment mesh, in the course of its calculation procedures [12].
In addition, it will adopt the multidirectional cracking pat-
tern, represented by a mutually orthogonal system of rotat-
ing cracks, in which its plane slope is conditioned upon the
state of current stresses and may change depending on the
stage of loading.
The concrete behavior, for strains of magnitude lower than one
corresponding to tensile strength, stretch OC on the curve of Fig-
ure 1, is considered linear elastic. For strains of magnitude higher
than that, its behavior is plastic softening, and it is represented by
the straight line segment CD that is defined from the ultimate strain
value, “
ε
o
”.
The cracking on the mass of the concrete disturbs, significantly,
its uniformity and continuity, and the larger the finite element will
be the greater stiffness variation in its interior. In order to com-
pensate the errors arising from such disturbance and minimize
the loss of quality of the results, in this work, it is used the fea-
Figure 1 – Stress strain curve for concrete