Civil Engineering Reference
In-Depth Information
Once
bd
2
is obtained, either
b
or
d
has to be assumed to determine the remaining concrete
dimension. The steel area
A
s
can then be calculated from
bd
. However, because the dimen-
sions
b
and
d
are always adjusted to become round numbers, an accurate calculation of the
steel area
A
s
can be obtained using the 'first type of design' method in which the cross-section
of concrete (
b
and
d
) is given.
Second Method.
Assume a suitable reinforcement index
ρ
f
c
. From Equation (3.97)
ω
=
ρ
f
y
/
it can be seen that this index
ω
is directly proportional to the depth ratio,
a
/
d
. Substituting
a
/
d
=
ω/
0
.
85 into the moment equilibrium equation, (3.96), gives
M
u
bd
2
=
(3.99)
ϕ
f
c
ω
(
1
−
.
ω
)
0
59
Equation (3.99) is tabulated and used in the ACI Design Handbook (ACI Committee 340,
1973).
3.2.2.7 Curvature and Tensile Steel Strain
In conclusion, once a mild steel reinforced concrete beam is found to be ductile, it is analyzed
and designed by the equilibrium condition without using Bernoulli's compatibility condition
and the stress-strain curve of steel. It would appear that the analysis and design of ductile
beams are based on the equilibrium (plasticity) truss model presented in Chapter 2, Section
2.1.1. This similarity is true if we are interested only in the bending strength. If we are also
interested in deformations, however, we must then rely on Bernoulli's compatibility condition,
Figure 3.11(b), to determine the ultimate curvature
φ
u
:
φ
u
=
ε
u
c
(3.100)
where
c
/β
1
and the depth
a
is one of the two unknowns solved by the two equilibrium
equations in either the analysis or the design problems. If the steel strain
=
a
ε
s
is desired at the
ultimate load stage, Bernoulli's compatibility condition is also required to give:
d
−
c
ε
s
=
ε
u
(3.101)
c
3.2.3 Doubly Reinforced Rectangular Beams
A doubly reinforced beam has compression steel
A
s
in addition to the tension steel
A
s
(Figure
3.12a). The compression steel could be employed for various purposes. First, to increase the
moment capacity of a beam when the cross-section is limited. Second, in a continuous beam
(Figure 3.2) the ACI code requires that a portion of the bottom positive steel in the center region
of a beam must be extended into the supports. These extended bars provide the compression
steel for the rectangular support sections which are subjected to negative moment. Third,
compression steel could be used to reduce deflections. Fourth, the ductility of a beam could
be enhanced by adding compression steel (see Section 3.3.5).
The additional compression steel introduces three additional variables, namely,
A
s
,
f
s
and
ε
s
for the area, stress and strain of the compression steel, respectively, Figure 3.12(a)-(c).
Therefore, the analysis and design of doubly reinforced rectangular concrete beams involve
12 variables, namely,
b
,
d
,
A
s
,
A
s
,
M
u
,
f
s
,
f
s
,
f
c
,
ε
s
,
ε
s
,
ε
u
and
c
(or
a
). At the same time,
