Civil Engineering Reference
In-Depth Information
6.3.3 Cyclic Stress-Strain Curves of Mild Steel
The cyclic constitutive relationships of reinforcing steel bars embedded in concrete and sub-
jected to uniaxial strains (Mansour
et al
., 2001b) are summarized in Figure 6.29. The solid
curves represent the smeared stress-strain curves of steel bars, while the dotted curves are the
monotonic stress-strain relationship of a bare bar. In the smeared stress-strain curves, stage 1,
2T and 2C are the envelope curves and stages 3 and 4 are the unloading and reloading curves.
6.3.3.1 Envelope Curves of Mild Steel (Stages 1, 2T and 2C)
When the envelope curves of embedded steel bars under cyclic loading are compared with the
monotonic curves, Belarbi and Hsu (1994, 1995) and Hsu and Zhang (1996) found that the
envelope curves for the cyclic stress-strain curves closely resemble the monotonic stress-strain
curves given in Section 6.1.9. In other words, stages 1 and 2T can be described by Equations
(6.85) and (6-86) in Section 6.1.9.4 for a bilinear model:
ε
s
≤
ε
y
)
(Stage 1)
f
s
=
E
s
ε
s
(¯
(6.105)
ε
s
>ε
y
)
(Stage 2T)
f
s
=
(0
.
91
−
2B)
f
y
+
(0
.
02
+
0
.
25B)
E
s
¯
ε
s
(¯
(6.106)
If the steel stress progresses in the compression region, the smeared maximum stress
f
s
is
limited to the compressive yield stress
−
f
y
, as indicated in stage 2C:
(Stage 2C)
f
s
=−
f
y
(
f
s
≤−
f
y
)
(6.107)
6.3.3.2 Unloading and Reloading Curves of Mild Steel (Stages 3 and 4)
The unloading and reloading stress vs. strain curves of embedded steel bars, stages 3 and 4,
take into account the Bauschinger effect, as shown in Figure 6.29. When the unloading branch
starts from a point beyond the smeared yield point, the unloading stress vs. strain curve of a
steel bar follows essentially a straight line in the tension region, and then becomes curved in
the compression region. This curve was found by Mansour
et al.
(2001b) to be well represented
by the Ramberg-Osgood type of expression first used by Yokoo and Nakamura (1977):
1
1
A
−
R
R
−
f
s
−
f
i
f
s
−
f
i
(Stages 3 and 4)
ε
s
−
¯
¯
ε
si
=
+
(6.108)
E
s
f
y
where
f
s
and ¯
ε
s
are the smeared stress and smeared uniaxial strain of an embedded steel bar;
ε
si
are the smeared stress and smeared uniaxial strain of steel bars at the initial load
reversal point.
The coefficients
A
and
R
in Equation (6.108) were determined from the reversed cyclic
loading tests to best fit the test results:
A
f
i
and ¯
10
k
−
0
.
p
. The parameter in the
coefficients
A
and
R
is the plastic strain ratio
k
p
which is defined as the ratio ¯
9
k
−
0
.
1
p
=
1
.
,
R
=
ε
p
/ε
y
=
ε
si
−
ε
y
)
/ε
y
. In this expression ¯
ε
y
(¯
ε
p
is the smeared plastic strain, and
is the smeared yield
strain.
The coefficients
A
and
R
and the ratio
k
p
control the shape of the stress-strain curves of
an embedded steel bar to best fit the test data. The coefficient
A
controls the strains at which
