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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 2
T.E.T. BUTTIGNOL |
L.C ALMEIDA
Tension stiffening effect is the ultimate concrete tensile stress value
that contributes to prevent crack propagation, increasing the stiffness
of the structures. It is defined by the tension-stiffening factor (c
ts
).
In elastic state, concrete obeys Hookes’s Law. In post-cracking
state, structure’s rupture plane is determined by Drucker-Prag-
er’s plasticity model in compression and by Rankine’s theory in
tension. Specific Fracture Energy, determined through equation
1, is a concrete essential parameter that allows numerical simula-
tion of concrete structures and corresponds to the strain energy
deformation tax release that is stored in the system and is re-
leased according to crack opening and propagation. It describes
concrete behavior in post-cracking state and corresponds to the
internal graphic area of stress versus crack opening curve that is
shown in Table 3.
(1)
ef
t
f
'
000025
,0 G
F
×
=
[MN/m]
For steel bars, an elastic-perfectly plastic behavior is adopted.
Steel bar materials properties are listed in Table 4. The yielding
criterion is based in von Mises definitions.
Finally for steel plates pile supports an isotropic elastic material
was defined as shown in Table 5.
2.3 Geometric model and steel bars disposal
Figure 1 shows the geometric characteristics of the pile caps. The
distribution of pile caps is presented in Figures 1, 2, 3 and 4.
For the pile cap with splitting reinforcing bars (model 4) two 16mm
steel bars were used passing perpendicularly through the struts,
following Delalibera’s [1] recommendations. The reinforcing bars
were designed according to requirements proposed by [1], due to
equations 2, 3 and 4.
(2)
yd
ct
sf
f
R
A
min ,
min ,
=
studied are originated from Delalibera’s [1] experimental model nº
B35P25E25e0. The numerical models maintained the shape of the
structural elements, the distribution of the reinforcing bars and the
materials properties.
In the model 1, all vertical displacements at the base of the piles
were restrained (Table 1, model 1). In the models 2 and 3 verti-
cal displacements are restrained respectively on 50% and 25% of
the base of the piles, as shown in Table 1. The reduction of piles
supports (vertical displacements restrain) has the main purpose to
study its influence on pile caps stiffness.
Model 4 characteristics are equal to model 1 except for splitting steel
bar addition as shown in Table 1. The objective of this reinforcement
is to observe its contribution to pile caps load bearing capacity.
Another aspect analyzed is the concentration of the struts compres-
sive stress in the cross-section on the surfaces of the piles. High
compressive stress concentration was observed on cross-section
region of the pile closer to the column. Compressive stresses were
more intensive in the beginning of inferior nodal zone, being very
low in the end of the inferior nodal zone. To detect this phenom-
enon and observe the behavior of specimens (compressive stress
flow) a pile cap with the pile’s cross-section width reduced was
modeled (Table 1, model 5).
2.1 Computational software
Numerical analysis was carried out with computational software
ATENA 3D [2]. The software’s basic concept operation is based
on finite element theory and non-linear analysis of reinforced con-
crete structures.
The software simulates the behavior of real structures using either
linear or non-linear analysis. Maximum load is obtained by incre-
ments of time force integration, applying Arc-Length and Newton-
Rhapson methods. To determine concrete strain structural behav-
ior it is used the Lagrangian or Euler’s formulations.
2.2 Materials properties
A plastic fracture model is adopted for concrete as described by
Cervenka [2] and shown in Table 3. Concrete principal character-
istics are presented in Table 2.
Table 3 – Concrete constitutive laws (Cervenka [2])
Stress vs. strain constitutive law Stress vs. crack mouth opening curve