Civil Engineering Reference
In-Depth Information
In Example Problem 5.4, Section 5.4.6, the three
m
-coefficients were found to be
m
=
0.5,
m
t
=−
0.5 and
m
t
=
0.866. The
ρ
/ρ
t
ratio under the equal strain condition is
ρ
ρ
t
=
0
.
5
+
0
.
866
866
=
3
.
73
−
0
.
5
+
0
.
ρ
/ρ
t
ratio of 3.73 is represented by the dotted straight line OA in Figure 5.28. Because
the 2-D element in Example Problem 5.4 was designed to have a
The
ρ
/ρ
t
ratio of unity, which
is much less than 3.73, the yielding of the steel is expected to occur much earlier in the
longitudinal bars than in the transverse bars. The
ρ
/ρ
t
ratio of unity is represented by the
dotted straight line OC in Figure 5.28.
5.4.7.2 Balanced Condition
Now that the steel in a 2-D element can be designed to yield simultaneously under the equal
strain condition, we can proceed to determine the balanced condition between the steel and the
concrete. The balanced condition defines a mode of failure where both the longitudinal and the
transverse steel yield simultaneouly with the crushing of concrete. The balanced percentage
of steel, therefore, divides the under-reinforced 2-D element from the over-renforced 2-D
element under the equal strain condition.
Adding Equations (5.143) and (5.144), and taking
f
=
f
t
=
f
s
result in
m
+
ρ
+
ρ
t
=
−
σ
d
f
s
2
m
t
+
m
t
(5.146)
2
m
t
Equation (5.146) is a parametric representation of a point on the equal strain line with the
ratio
σ
d
and
the steel stress
f
s
will have to be determined from the strain compatibility condition and the
stress-strain relationships of concrete and steel. The balanced condition is obtained when the
normalized concrete stress,
σ
d
/
f
s
as an unknown parameter. The relationship between the concrete stress
f
c
, peaks at the same time the steel stress
f
s
reaches the yield
σ
d
/
point.
Assuming the yielding of steel under the equal strain condition, i.e.
α
r
=
45
◦
, both the strain compatibility equations, (5.97) or 4 and (5.98) or 5 , are reduced to
ε
=
ε
t
=
ε
y
and
ε
r
=
2
ε
y
−
ε
d
(5.147)
Inserting
ε
r
from Equation (5.147) into Equation (5.102) or
8
for the softening coefficient
of concrete:
0
.
9
ζ
=
1
(5.148)
+
600(2
ε
y
−
ε
d
)
The yield strain
ε
y
in Equation (5.148) is equal to 0.00207 for a mild steel with
f
y
=
413
MPa (60 000 psi).
Using the softened stress-strain relationship of concrete represented by Eqs.
7
a
and
7
b
,
and the softening coefficient expressed by Eq. (5.148), we can now trace the normalized
concrete stress,
f
c
, with the increase of the concrete strain
σ
d
/
ε
d
(in an absolute sense) as
shown in Table 5.4.
