Civil Engineering Reference
In-Depth Information
The stress in the tension steel and in the compression steel are obtained from the
strain diagram of Fig. 12.7(c). Hence
σ
sc
/
E
s
n
σ
c
/
E
c
n
d
2
=
(both strains have the same sign)
(12.19)
−
so that
m
(
n
−
d
2
)
mM
(
n
−
d
2
)
σ
sc
=
σ
c
=−
(12.20)
n
I
c
and
mM
I
c
σ
st
=
(
d
1
−
n
) as before
(12.21)
An alternative expression for the moment of resistance of the beam is derived by
taking moments of the resultant steel and concrete loads about the compressive
reinforcement. Therefore from the stress diagram of Fig. 12.7(d)
C
c
n
d
2
M
=
T
(
d
1
−
d
2
)
−
3
−
whence
bn
n
d
2
σ
c
2
M
=
σ
st
A
st
(
d
1
−
d
2
)
−
3
−
(12.22)
E
XAMPLE
12.6
A rectangular section concrete beam is 180mm wide and has a
depth of 360mm to its tensile reinforcement. It is subjected to a bending moment
of 45 kNm and carries additional steel reinforcement in its compression zone at a
depth of 40mm from the upper surface of the beam. Determine the necessary areas
of reinforcement if the stress in the concrete is limited to 8
.
5N
/
mm
2
and that in the
steel to 140N/mm
2
. The modular ratio
E
s
/
E
c
=
15.
Assuming that the stress in the tensile reinforcement and that in the concrete attain
their limiting values we can determine the position of the neutral axis using Eq. (12.13).
Thus
15
360
1
140
=
8
.
5
×
n
−
from which
n
=
171
.
6mm
Substituting this value of
n
in Eq. (12.22) we have
171
.
6
171
.
6
3
40
8
.
5
2
10
6
×
=
140
A
st
(360
−
+
×
×
−
45
40)
180
which gives
954mm
2
A
st
=
