Civil Engineering Reference
In-Depth Information
2.2
CRACKING MOMENT
The area of reinforcing as a percentage of the total cross-sectional area of a beam is quite
small (usually 2% or less), and its effect on the beam properties is almost negligible as
long as the beam is uncracked. Therefore an approximate calculation of the bending
stresses in such a beam can be obtained based on the gross properties of the beam's cross
section. The stress in the concrete at any point a distance
y
from the neutral axis of the
cross section can be determined from the following flexure formula in which
M
is the
bending moment, which is equal to or less than the cracking moment of the section, and
I
g
is the gross moment of inertia of the cross section:
My
I
g
f
Section 9.5.2.3 of the ACI Code states that the cracking moment of a section may be
determined with ACI Equation 9-9, in which
f
r
is the modulus of rupture of the concrete
and
y
t
is the distance from the centroidal axis of the section to its extreme fiber in tensio
n.
In this section, with its equation 9-10, the Code states that
f
z
may be taken equal to
for normal weight concrete with
c
7.5
f
e
f
in psi.
N
mm
2
or MPa,
f
r
c
in
c
.
Or in SI units with
f
0.7
f
Other values are provided in the same code section for all-lightweight and sand-
lightweight concretes. The cracking moment is as follows:
f
r
I
g
y
t
(ACI Equation 9-9)
M
cr
Example 2.1 presents calculations for a reinforced concrete beam where tensile
stresses are less than its modulus of rupture. As a result, no tensile cracks are assumed to
be present, and the stresses are similar to those occurring in a beam constructed with a ho-
mogeneous material.
EXAMPLE 2.1
(a)
Assuming the concrete is uncracked, compute the bending stresses in the extreme fibers of the
beam of Figure 2.5 for a
bendi
ng moment of 25 ft-k. The concrete has an
c
f
of 4000 psi and a mod-
474 psi.
(b)
Determine the cracking moment of the section.
ulus of rupture
f
r
7.5
4000
SOLUTION
(a)
Bending stresses:
I
g
1
12
bh
3
with
b
12 in. and
h
18 in.
1
12
(12)(18)
3
5832 in.
4
I
g
My
I
g
f
with
M
25 ft- k
25,000 ft- lb
