Civil Engineering Reference
In-Depth Information
The maximum bending moment that can be applied such that the permissible stress
in the concrete is not exceeded is given by Eq. (12.15). Thus
163
.
5
400
7
2
×
163
.
5
3
10
−
6
M
=
250
×
−
×
=
49
.
4kNm
Similarly, from Eq. (12.16) the stress in the steel limits the applied moment to
942
.
5
400
163
.
5
3
10
−
6
M
=
140
×
−
×
=
45
.
6kNm
The steel is therefore the limiting material and the moment of resistance of the beam
is 45.6 kNm.
E
XAMPLE
12.5
A rectangular section reinforced concrete beam is required to
support a bending moment of 40 kNm and is to have dimensions of breadth 250mm
and effective depth 400mm. The maximum allowable stresses in the steel and concrete
are 120N/mm
2
and 6.5N/mm
2
, respectively; the modular ratio is 15. Determine the
required area of reinforcement such that the limiting stresses in the steel and concrete
are attained simultaneously.
Using Eq. (12.13) we have
15
400
1
120
=
6
.
5
×
n
−
from which
n
=
179
.
3mm.
The required area of steel is now obtained from Eq. (12.16); hence
M
σ
s
(
d
1
−
A
s
=
n
/
3)
i.e.
10
6
40
×
979
.
7mm
2
A
s
=
179
.
3
/
3)
=
120(400
−
It may be seen from Ex. 12.4 that for a beam of given cross-sectional dimensions,
increases in the area of steel reinforcement do not result in increases in the moment
of resistance after a certain value has been attained. When this stage is reached the
concrete becomes the limiting material, so that additional steel reinforcement only
serves to reduce the stress in the steel. However, the moment of resistance of a beam
of a given cross section may be increased above the value corresponding to the limiting
concrete stress by the addition of steel in the compression zone of the beam.
Figure 12.7(a) shows a concrete beam reinforced in both its tension and compression
zones. The centroid of the compression steel of area
A
sc
is at a depth
d
2
below the upper
surface of the beam, while the tension steel of area
A
st
is at a depth
d
1
. The section
may again be transformed into an equivalent concrete section as shown in Fig. 12.7(b).
