Civil Engineering Reference
In-Depth Information
(a) The stress distribution in the steel between two adjacent cracks is assumed to follow a full
cosine curve.
(b) The smeared stress-strain relationship of concrete in tension (Equation 6.55), is valid both
before and after yielding. That is to say, Equation (6.55), which had been calibrated to fit
the test results before yielding, remains valid after yielding.
From the first assumption we can write
a
s
cos
2
π
x
f
so
=
f
s
+
(6.73)
L
where
a
s
is the amplitude of the cosine curve. At the cracked sections,
f
s
(
x
)
=
f
so
and
cos
2
π
x
=
1 (i.e.
x
equals to 0 or
L
). Therefore
L
a
s
=
f
so
−
f
s
(6.74)
Substituting (
f
so
−
f
s
) from Equation (6.65) into (6.74) gives
a
s
=
σ
c
ρ
(6.75)
Substituting
a
s
from Equation (6.75) into (6.73) gives
f
s
+
σ
c
ρ
cos
2
π
x
f
s
(
x
)
=
(6.76)
L
c
1
Now, using the second assumption and substituting
σ
c
=
σ
and ¯
ε
1
=
ε
s
from Equation
¯
(6.55) into (6.76) gives:
f
cr
ε
cr
¯
0
.
4
1
ρ
cos
2
π
x
f
s
(
x
)
=
f
s
+
(6.77)
ε
s
L
With these two assumptions, the averaging process is summarized as follows:
1. Select a value of the smeared steel stress
f
s
.
2. Assume a smeared steel strain ¯
ε
s
.
3. Calculate the distribution of steel stress
f
s
(
x
) from Equation (6.77).
4. Determine the corresponding distribution of steel strain ¯
ε
s
(
x
) according to the stress-strain
curve of bare bars (Equations 6.67-6.68).
5. Calculate the smeared steel strain ¯
ε
s
by numerical integration of the following integral:
L
1
L
ε
s
=
¯
¯
ε
s
(
x
)d
x
(6.78)
0
6. If ¯
ε
s
calculated from Equation (6.78) is not the same as that assumed, repeat steps 2-5
until the calculated ¯
ε
s
and
the selected value
f
s
provide one point on the smeared stress-strain curve in the post-yield
range.
7. By selecting a series of
f
s
values and find their corresponding ¯
ε
s
is sufficiently close to the assumed value. The calculated ¯
ε
s
values from steps 2-6,
the whole smeared stress-strain curve in the post-yielding range can be plotted.
