Civil Engineering Reference
In-Depth Information
member at the midspan of simple and continuous beams or at the end of a cantilevers is
2
K
5
M
=
(3.48)
48
EI
where
where
M
=
moment at midspan for simple and continuous beams or moment at support for
cantilever beams;
=
clear span of a beam;
K
=
1 for simple beam, 12/5 for cantilever beam, 1.20 - 0.20
M
o
/
M
c
for continuous beam,
2
where
M
o
=
simple span moment
=
w
/
8, and
M
c
is the net midspan moment.
2
For a beam with both ends fixed
M
c
=
w
/
24 and
K
=
0.60; and for a beam with one
2
end fixed and the other end hinged
M
c
=
w
0.8.
The difficulty of applying Equation (3.48) to reinforced concrete beams with both cracked
and uncracked sections is the evaluation of the bending rigidity
EI
. To overcome this difficulty
we introduce an effective bending rigidity
E
c
I
e
, which can be used in connection with Equation
(3.48) to calculate the deflections.
/
16 and
K
=
3.1.5.2 Effective Bending Rigidities
The effective bending rigidity
E
c
I
e
is the product of two quantities
E
c
and
I
e
.
E
c
is the modulus
of elasticity of concrete defined by the ACI Code (ACI 318-08) to be
1
.
5
f
c
(
MPa
)or
E
c
=
1
.
5
f
c
(
psi
)
E
c
=
44
w
33
w
(3.49)
where
E
c
and
f
c
m
3
(or lb
ft
3
). In
the case
are in MPa (or psi) and the unit weight
w
is in kN
/
/
4730
f
c
(MPa)
m
3
(or 144 lb
ft
3
) and
E
c
=
of normal w
eight c
oncrete
w
is taken as 22
.
6kN
/
/
(or 57000
f
c
(psi)).
The effective moment of inertia
I
e
will now be derived. A uniform moment
M
actingona
length of beam is expected to create a curvature
φ
=
/
d
d
x
, as shown in Figure 3.7(a). The
moment-curvature relationship (
M
-
curve) is plotted in Figure 3.7(b). Up to the cracking
moment
M
cr
, the curve follows approximately a slope calculated from the bending rigidity
E
c
I
g
of the gross uncracked section. After cracking, however, the curve changes direction
drastically. When the nominal moment
M
n
is reached, the slope approaches the bending
rigidity
E
c
I
cr
, calculated by the cracked section.
The trend described above can be clearly illustrated by plotting the bending rigidity
EI
against the moment
M
in Figure 3.7(c). Below the cracking moment
M
cr
, the experimental
curve is roughly horizontal at the level of the calculated
E
c
I
g
. After cracking, however, the
curve drops drastically. When the nominal moment
M
n
is approached, the curve becomes
asymptotic to the horizontal line of the calculated
E
c
I
cr
. A theoretical equation to express the
moment of inertia, therefore, must vary from
I
g
at cracking to
I
cr
at ultimate.
The trend of the experimental curve, Figure 3.7(c), can be closely approximated by a
theoretical curve suggested by Branson (1965):
φ
1
4
I
cr
M
cr
M
4
M
cr
M
I
=
I
g
+
−
≤
I
g
(3.50)
