Civil Engineering Reference
In-Depth Information
6.3.2.1 Envelope Curves of Concrete (C1, C2, T1 and T2)
Tensile envelope curves T1 and T2
In theory, the tensile stress-strain curve of concrete subjected to biaxial stresses should be
different from those subjected to uniaxial stresses. However, the tensile envelope curves, T1
and T2 in the biaxial condition, were found to be close to the monotonic curves given in
Section 6.1.8. Since the tensile stress is very small compared with the compressive stress, the
tensile envelope curves T1 and T2 are taken, for simplicity, to be the same as the monotonic
curves expressed by Equations (6.54) and (6.55):
c
Stage T1
σ
=
E
c
¯
ε
0
≤
ε
≤
ε
cr
¯
(6.98)
f
cr
ε
cr
¯
0
.
4
c
Stage T2
σ
=
ε>ε
cr
¯
(6.99)
ε
c
and ¯
of concrete
in compression do not have a subscript. This is because they can be applied to either the
horizontal 1-direction or the vertical 2-direction. When the cyclic load is in the positive
direction,
Notice in Equations (6.98) and (6.99) that the stress and strain symbols
σ
ε
c
1
and ¯
c
σ
=
σ
ε
=
ε
1
. When the cyclic load is in the negative direction, however,
¯
σ
=
2
and ¯
σ
ε
=
ε
2
.
¯
Compressive envelope curves C1 and C2
In the case of concrete cylinders under uniaxial cyclic compression, Karsan and Jirsa (1968)
showed that the hysteretic loops of the compression stress-strain curves produced an envelope
curve that was virtually identical to the curve for monotonic compressive loading. In a more
general case of RC 2-D elements subjected to a cyclic normal stress in one principal direction
and a contant tensile strain in the other principal direction, Mansour
et al.
(2001b) showed
that the
envelope
compression curves of the hysteretic loops were similar to the
monotonic
compression curves (Equation 6.42 and 6.45), proposed by Belarbi and Hsu (1994, 1995) in
Section 6.1.6.
The difference between the above two experiments is the fact that the strain normal to the
cyclic compression direction was zero in Karsan and Jirsa's tests, while a constant tensile
strain was applied normal to the cyclic compression direction in the 2-D elements of Mansour
et al
. (2001b). This constant tensile strain in the orthogonal direction caused a 'softening' of
the concrete compressive strength. The tests carried out by Mansour
et al
. showed that the
'softening coefficient'
given by Equations (6.46)-(6.49) in Section 6.1.7 was valid not only
for the monotonic loading curves (Zhang and Hsu, 1998), but also for the envelope curves of
cyclic loading.
ζ
6.3.2.2 Damage Coefficient D for Compression Envelope
When a 2-D element is subjected to
cyclic shear
loading, however, an additional phenomenon
needs to be considered when modeling the constitutive relationships of concrete in compres-
sion. Since the horizontal and vertical principal applied stresses are subjected to out-of-phase
compression-tension stresses, a damage coefficient
D
needs to be incorporated in the envelope
compression stress-strain curves of concrete to take into account the damage caused by the
history of tensile and compressive stress reversal normal to the compression direction being
considered. As such, the
compressive envelope curves
for the cyclic stress-strain curves are
