Civil Engineering Reference
In-Depth Information
taken from the monotonic curves of Equations (6.42) and (6.45) with two modifications:
f
cT4
)
2
¯
2
¯
ε
ζε
o
ε
ζε
o
c
f
c
−
f
cT4
Stage C1
σ
=
(
D
ζ
−
+
ε
o
≤
ε<
¯
0
(6.100)
1
2
¯
ε/ε
o
−
1
c
f
c
Stage C2
σ
=
D
ζ
−
ε<ε
o
¯
(6.101)
4
/ζ
−
1
where
is the softening coefficient given by Equation (6.52). Also, when the cyclic load is in
the positive direction,
ζ
c
c
2
σ
=
σ
and ¯
ε
=
ε
2
. When the cyclic load is in the negative direction,
¯
c
c
1
however,
ε
1
.
The first modification in Equations (6.100) and (6.101) is the incorporation of a damage
coefficient
D
(Mansour, 2001; Mansour and Hsu, 2005b). This damage coefficient
D
takes
into account the effect of
cyclic shear
loading, where cyclic compression and tension occur in
both principal directions.
To be consistent with the concept of a softening coefficient due to tensile strain, the damage
coefficient
D
is taken as a linear function of the compression strain
σ
=
σ
and ¯
ε
=
¯
ε
c
:
−
ψ
ε
c
D
=
1
ε
o
≤
1
.
0
(6.102)
ε
c
(always negative) in Equation (6.102) is the maximum compression strain
normal to the compression direction under consideration, and occurred in the previous loading
cycles. The compressive strain
The strain
ε
o
(always negative) is the concrete cylinder compressive strain
at the peak cylinder stress
f
c
. The symbol
ψ
=
0.4 was chosen to best fit the test results of the cyclic shear stress-strain curves of the test
panels (Mansour, 2001). Because the damaging effect of the perpendicular tensile strain
ψ
is a constant taken as 0.4. The value of
ε
t
(or the uniaxial ¯
ε
t
) is taken care of by the softening coefficient
ζ
, the damage coefficient
D
in
ε
c
cannot be positive.
The second modification is the incorporation of a stress
f
cT4
in Equation (6.100).
f
cT4
is the
concrete stress of point TD on the vertical axis at the end of stage T4. Because the envelope
curve C1 starts from the point TD, rather than the origin, Equation (6.100) needs to be adjusted
accordingly.
Equation (6.102) cannot be greater than unity, and the strain
6.3.2.3 Unloading and Reloading Curves of Concrete (C3-C7 and T3-T4)
In Figure 6.28, stage C3-C7 and stage T3-T4 represent the stress-strain curves of unloading
and reloading. Starting with the cyclic load in tension from the origin, the response is elastic
as long as the tensile stress is less than the cracking stress at point TA (
f
cr
). The loading
and unloading behavior follows the straight line T1 given by Equation (6.98). If the load
is reversed from tension to compression before cracking, the compressive response follows
the dotted line with a slope equal to the initial modulus of concrete E
c
. Once the cracking
stress of concrete at point TA is exceeded, the tensile response follows a concave curve T2
given by Equation (6.99). As the load is reversed at point TB from the tensile direction to the
compressive direction, there is initially a region that represents the closure of the cracks (stage
T3) up to point TC. As the crack continues to close, the second region, stage T4, will be stiffer.
This stage T4 represents the increase in concrete stiffness before the complete closing of the
