Civil Engineering Reference
In-Depth Information
Figure 6.17
Smeared stress-strain curves of mild steel-theories and tests
ε
s
) curve of steel obtained using the model of Tamai
et al.
is compared with an experimental stress-strain curve of steel in Figure 6.17. The agreement
is acceptable.
The prediction of this theoretical method can be improved if the stress distribution of the
steel bar is expressed by a function closer to the actual condition. Belarbi (1991) modified the
cosine function in Eq. (6.73) by adding more terms in the following form:
A smeared stress-strain (
f
s
−
¯
b
s
sin
3
1358
a
s
cos
2
π
x
π
x
6sin
5
π
x
f
so
=
f
s
+
+
−
0
.
−
0
.
(6.79)
L
L
L
where
a
s
is given by Equation (6.75). In the added terms
b
s
, 0.6 and 0.1358 were chosen
to satisfy the boundary conditions at the cracks and at the midpoint between the cracks.
The additional sinusoidal and constant terms and the resulting stress distribution are given in
Figure 6.18. The post-yield
f
s
−
ε
s
curve can then be calculated using the seven-step iteration
procedure used for solving Equation (6.77).
Belarbi's improved
f
s
−
¯
ε
s
curve based on the stress distribution of Equation (6.79) is also
shown in Figure 6.17. It can be seen that the improved method is closer to the test points.
Notice also that the post-yield theoretical curve is very close to the experimental curve of the
bare bar in the strain hardening region. This is because the smeared tensile stress of concrete
becomes negligible in this region of very large strains.
A parametric study using Belarbi's method was carried out to examine the effect of the per-
centage of steel (
¯
) and the cracking strength of concrete (
f
cr
) on the post-yielding stress-strain
curves of steel. Figure 6.19(a) gives a family of stress-strain curves using the percentage of
steel (
ρ
) as parameter. Figure 6.19(b) gives a family of stress-strain curves using the cracking
strength of concrete (
f
cr
) as parameter. It can be seen that the smeared stress-strain curves
move downward when
ρ
ρ
is decreased and when
f
cr
is increased.
