Civil Engineering Reference
In-Depth Information
3.1.4 Bending Rigidities of Uncracked Sections
In a flexural member at service load, some parts of the member will be cracked and some
parts will remain uncracked, depending on the bending moment diagram. The deflection of
the member will be a function of both the cracked and the uncracked portions. The bending
rigidities of the
cracked
sections have been derived in Section 3.1.3 for singly reinforced
sections, doubly reinforced sections and flanged sections. The bending rigidity of
uncracked
sections will be derived in this section.
3.1.4.1 Rectangular Sections
The bending rigidities of uncracked rectangular sections (Figure 3.5a), will first be derived.
Before the cracking of a flexural member, the strain and the stress distributions in the cross-
section are shown in Figure 3.5 (b) and (c), respectively. First, the concrete stresses below the
neutral axis can no longer be neglected. Second, the full height of the cross-section
h
becomes
effective, rather than the effective depth
d
. The transformed area of the cross-section is shown
in Figure 3.5(d).
The internal resistance to the applied moment is contributed primarily by the concrete before
cracking. Assuming that the rebars have no effect on the position of the neutral axis, then the
neutral axis will lie at the mid-depth of the cross-section. As a result,
kd
=
h
/
2,
jd
=
(2
/
3)
h
and
C
=
T
=
(1
/
4)
f
c
bh
. Taking moments about the neutral axis,
1
6
bh
2
f
c
+
=
−
/
M
mA
s
f
cs
(
d
h
2)
(3.38)
where
m
n
- 1, because the original rebar area
A
s
has been included in the first term for
concrete and must be subtracted from the transformed rebar area in the second term.
Observing that
A
s
=
ρ
=
bd
and
f
cs
=
f
c
(
d
−
h
/
2)
/
(
h
/
2), Equation (3.38) becomes
1
2
d
h
d
1
6
bh
2
f
c
−
(
h
/
2)
M
=
+
3
m
ρ
(3.39)
(
h
/
2)
Figure 3.5
Uncracked rectangular sections (singly reinf.)
