Civil Engineering Reference
In-Depth Information
γ
(
)
2
2
k
=ρ
2(
n
)1- (
k
+ρ
n
)
2-1
(1 -)
k
(5.57)
f
1
f
f
2
f
γ
where
ρ
f
1
=
A
f
1
/
bd
and ρ
f
2
=
A
f
2
/
bd
(See Figure 5.3 for the definitions of
A
f
1
and
A
f
2
.) and γ
=
d
/
h.
Note that the last term of Equation (5.57) accounts for the presence of
the two side layers of reinforcement. Although this equation can be solved
for the exact value of
k,
the final closed-form solution is rather unwieldy.
Since the exact value of
k
is of little practical importance, a simpler itera-
tive solution is suggested as an alternative. Defining the “effective” tensile
reinforcement ratio, ρ
f
,
as
ρ=ρ+
γ
(1 -)
2-1
k
ρ
(5.58)
f
f
1
f
2
γ
and starting with an initial guess for
k
(
k
= 0.2 is recommended), Equation
(5.56) can be used repetitively with Equation (5.58) until convergence to
calculate
k.
Equation (5.55) provides a conservative estimate of the concrete contri-
bution,
V
c
,
if the column is subjected to a compressive axial force. In the
uncommon case of tensile axial force, the use of this equation remains valid
with the appropriate value of
kd.
To attain this, Equation (5.59) must be
employed, which, with negligible approximation, predicts the location of
the neutral axis,
c,
under the combined effects of axial load and flexure
based on the value of
k
calculated from Equation (5.57):
k
2
k
1-
3
2
c
d
Pd
M
u
(5.59)
0
≤=+
k
≤
0.4
u
where
P
u
and
M
u
are the ultimate axial load and moment corresponding
to
V
u
.
Compressive axial force is positive and tensile axial force is nega-
tive while the moment is always positive. With
c
from Equation (5.59) and
noting that
c
=
kd,
Equation (5.55) may be used to achieve an accurate
assessment of the shear strength provided by concrete when the section is
subjected to a tensile axial force.
5.9.2 Shear reinforcement contribution,
V
f
,
for rectangular sections
Once
V
c
is calculated, the shear strength provided by ties (or spiral) of
the column,
V
f
,
can be calculated as discussed in Chapter 4 and shown in
Equation (5.60):
Afd
s
fv
fv
V
=
(5.60)
f
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