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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Numerical-computational analysis of reinforced concrete structures considering the damage,
fracture and failure criterion
Where the numerator is the standard Euclidean norm of the dis-
placement increment vector ∆
u
i
correspondent to the
i
iteration and
the denominator is the Euclidian norm of the total displacement
vector
u
i
=
u
i-1
+ ∆
u
i
of the i-th iteration.
The convergence criteria of the forces must obey the relation:
(29)
‖
Q
i
‖ ‖
φ
i
R
0
‖
≤
Q
tol
Where the numerator is the Euclidian norm of the non-balanced load
increment correspondent to the
i
iteration and the denominator is the
Euclidian norm of the force increment for the solution step.
7. Results and discussions
7.1 Simulation 1
In this example, adapted from Jarek
et al
. [7], a one-dimension-
al non-linear analysis is carried out using the finite elements of
a reinforced concrete beam, considering the damage model from
Mazars [10] to simulate the concrete and the bilinear elastoplastic
model for the steel. The criterion of Tsai and Wu [18] was adopted
as the failure criterion for the concrete. The beam supported by
two points is 6 m in length, rectangular cross-section of (20 x 40)
cm
2
, subject to a condensed force applied in the middle of the gap.
The lower longitudinal armatures (
A
st
) and the upper longitudinal
armatures (
A
sc
) of the beam are constituted by 3
φ
12.5 mm, with a
0.02 m coating. The material parameters and strength coefficients
for the Tsai and Wu’s [18] criterion are presented in Table 1.
In the finite element discretization, 100 beam elements with 2 nodes
and 2 degrees of freedom/node were employed using the geometry
and loading symmetry, analyzing thus, only half of the beam (Figure 1).
The modified Newton-Raphson method combined with the arc-length
technique was used in the solution of the non-linear equations. The load
increment used was taken as equal to 0.5 kN. The maximum admitted
errors at the end of each increment were
u
tol
= 10
-3
and
Q
tol
= 10
-2
.
In the analysis, the equivalent strain is evaluated the following
way (Tiago
et al
. [17]):
(30)
ε
~
=
{
ε
x
-ν 2ε
x
, ε
x
≥
0
, ε
x
<
0
Where
ν
is the Poisson coefficient for the concrete.
The equivalent bending stiffness for the beam (
EI
eq
) is determined
considering two parcels. The first one refers to the equivalent bend-
ing stiffness for the concrete (
EI
eqc
) and it is obtained by dividing
the cross-section of the beam in
n
layers. The moment of inertia
I
i
for the i-th
layer is calculated through the parallel axis theorem by:
(31)
I
i
=– –
b
(
y
i
-
y
i-1
)
3
12
+
b
(
y
i
-y
i-1
)
(
y
i-1
+
y
i
-y
i-1
2
)
2
,
i=1,
...
,n
Where
n
is the total number of layers,
b
is the width of the rectan-
gular cross-section and
y
i
is the coordinate for the i-th layer from
the section centroid. The portion
EI
eqc
is obtained by:
(32)
EI
eqc
=Σ
i=1
n
E
ci
I
i
Table 1 – Parameters of the model
Concrete
Steel
Tsai and Wu [18]
Coefficient
Equation
Adopted values
(MPa) (Gagliardo
et al. [3])
E = 30.2 GPa
c0
E = 210 GPa
a
F
1
0.224
= 0.2
= 0.3
F
2
0
A = 0.995
T
k = 0.85
a
F
11
-0.0288
5
B = 10
T
F
22
0
A = 1.1
C
F
44
0.0305
3
B = 8 10
C
F
12
±0.00385
1 1
f
t1
f
c1
f
c2
1 1
f
t2
1
f
t1
f
c1
1
f
t2
f
c2
1
2
f
v4
±
F
11
F
22