Civil Engineering Reference
In-Depth Information
and the balancing tensile reinforcing and the part due to the nominal moment capacity of
the compression steel and the balancing amount of the additional tensile steel. This situa-
tion is illustrated in Figure 5.13. In the expressions developed here, the effect of the con-
crete in compression, which is replaced by the compressive steel is neglected. This
omission will cause us to overestimate
M
n
by a very small and negligible amount (less
than 1%). The first of the two resisting moments is illustrated in Figure 5.13(b).
s
,
A
a
2
M
n
1
A
s
1
f
y
d
The second resisting moment is that produced by the additional tensile and compres-
sive steel (
A
s
2
and
s
A
), which is presented in Figure 5.13(c).
s
f
y
(
d
d
)
M
n
2
A
Up to this point it has been assumed that the compression steel has reached its yield
stress. If such is the case, the values of
A
s
2
and will be equal because the addition to
T
of
A
s
2
f
y
must be equal to the addition to
C
of for equilibrium. If the compression steel
has not yielded, must be larger than
A
s
2
, as will be described later in this section.
Combining the two values, we obtain
s
A
s
f
y
A
s
A
a
2
M
n
A
s
1
f
y
d
A
s
2
f
y
(
d
d
)
a
2
M
n
A
s
1
f
y
d
A
s
2
f
y
(
d
d
)
The addition of compression steel only on the compression side of a beam will have
little effect on the nominal resisting moment of the section. The lever arm
z
of the internal
couple is not affected very much by the presence of the compression steel, and the value
of T will remain about the same. Thus the value
M
n
Tz
will change very little. To in-
crease the nominal resisting moment of a section, it is necessary to add reinforcing on
both the tension and the compression sides of the beam, thus providing another resisting
moment couple.
Examples 5.7 and 5.8 illustrate the calculations involved in determining the design
strengths of doubly reinforced sections. In each of these problems, the strain
s
)
(
f
in the
Figure 5.13
