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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
E. L. MADUREIRA | J.I.S.L. ÁVILA
where
'
1 1
/
c
f
s
β
=
,
'
2 2
/
c
f
s
β
=
. “
s
1
” e “
s
2
”
are principal
stresses so that 0 >
s
1
>
s
2
. “
'
c
f
” is the concrete uniaxial compres-
sive strength. Considering
2 1
/
s
s
α
=
, then the biaxial com-
pressive strength is given by:
(7)
'
2
2
)
1(
.65.31
c
c
f
and
c
cu
c
2
1
.
In compression-tension state of stress, compressive strength is
given by the fashion proposed by Darwin and Pecknold, defined in
[3] and [9], in the form:
(8)
'
2
2
)
1(
.28.31
c
c
f
The tensile strength, in turn, can be obtained from the equation:
(9)
tu
c
pt
f
.
8.01
'
2
in which “
s
tu
” represents the uniaxial tensile strength.
For elements under tension-tension, the tensile strength is con-
stant and equal to the concrete uniaxial tensile strength.
The strains related to peak stresses in biaxial compression state is
obtained according to the relationships:
(10)
2 3
2
2
co
p
and
1
2
1
3
1
1
35.0
25.2 6.1
co
p
where
c
p
f
1
1
s
β
=
,
c
p
f
2
2
s
β
=
and “
e
co
” is the strain
corresponding to the compressive stress peak on uniaxial state of
stress.
For concrete subjected to plane state of stresses it is adopted the
constitutive relationship on incremental fashion proposed by Desai
and Siriwardance, (apud [9]), written by:
(11)
.
).G -(1
0
0
0
E
.E
0
E
1
1
12
22
11
2
2
2 1
2 1
1
2
12
22
11
d
d
d
E
EE
d
d
d
where the “
E
i
’s
“ are the deformation modules for each one
of the principal directions, which are considered as they are
oriented according the crack directions. The shear module is
obtained from:
ture proposed by Kwak and Filippou [9], that define the ultimate
concrete strain by:
(3)
) 3.(
) /3 ln( .
.2
b
f
b
G
t
f
o
to which “b” is the finite element length, expressed in inches. The
parameters “
f
t
”and “
G
f
” represent, respectively, the tensile strength
and the concrete fracturing energy per unit of area. The latter is
defined according to the CEB-FIP model code 90 criteria [4].
In reference to concrete deformation modulus it was adopted its
reduced version, obtained from:
(4)
0
85,0
E
E
cs
where “E
o
” is the initial deformation modulus, expressed, from [13],
in the form:
(5)
) MPa ( f
5600
E
ck
o
The concrete ultimate stresses are defined by the failure envelope
proposed by Kupfer and Gerstle [8], Figure 2, described by:
(6)
0
65.3
)
(
1
2
2
2 1
Figure 2 – Stress envelope of concrete
on biaxial state of loading