Civil Engineering Reference
In-Depth Information
where
M
u
is the factored moment
=
ϕ
M
n
, and
ϕ
is the reduction factor for material.
ϕ
=
0.9
for bending in the ACI Code.
3.2.2 Singly Reinforced Rectangular Beams
A singly reinforced rectangular section Figure 3.9(a), is subjected to a nominal bending
moment
M
n
at the ultimate load stage. The cross-section has a width of
b
, and an effective
depth of
d
. The mild steel reinforcing bars have a total area of
A
s
. The moment
M
n
induces a
linear distribution of strains, as indicated in the strain diagram of Figure 3.9(b). The maximum
concrete strain at the top surface
ε
u
is specified by the ACI Code (ACI 318-08) to be 0.003
at ultimate. The moment
M
n
is resisted by a
C
T
couple as shown in Figure 3.9(c) and (d).
The resultant
C
is statically equivalent to the compression stress block with real stress-strain
curve shown in Figure 3.9(c) and to an equivalent stress block of rectangular shape shown in
Figure 3.9(d).
−
3.2.2.1 Modes of Failure
A singly reinforced beam may fail in three different modes:
Insufficient amount of steel
When the beam is reinforced with a very small amount of steel, the steel will yield at cracking
and the beam will collapse suddenly. To exclude this undesirable mode of failure, the ACI
Code provides a minimum amount of flexural steel as follows:
25
f
c
(MPa)
f
y
3
f
c
(psi)
f
y
0
.
ρ
min
=
or
ρ
min
=
(3.71)
Under-reinforced beams
When the beam is reinforced with a moderate amount of steel, the steel will yield first, followed
by a secondary crushing of concrete. Consequently, failure is preceded by a large deflection,
and the failure mode is ductile and desirable.
Over-reinforced beams
When the beam is reinforced with an excessive amount of steel, the concrete will crush first,
before the yielding of steel. Consequently, failure is preceded by a very small deflection, and
the failure mode is brittle and undesirable.
To differentiate the under-reinforced beams from the over-reinforced beams, we will derive
a balanced percentage of steel
ρ
b
, which is defined as the percentage of steel that causes the
yielding of the steel and the crushing of concrete to occur simultaneously. Therefore:
ρ<ρ
b
gives under-reinforced beams
ρ>ρ
b
gives over-reinforced beams
3.2.2.2 Balanced Condition
The 'balanced condition' is the condition when the steel reaches the yield point, i.e.
ε
s
=
ε
y
,
simultaneously with the crushing of concrete, i.e.
0.003.
The bending of singly reinforced beams involves nine variables
b
,
d
,
A
s
,
M
u
,
f
s
,
f
c
,
ε
u
=
ε
s
,
ε
u
,
and
c
(or
a
), as shown in Figure 3.9(a)-(d). The coefficients
β
1
, are not considered variables
because they are determined from the given compression stress-strain curve (Figure 3.9e),
and listed in Section 3.2.1.2. A total of four equations are available from the Bernoulli's
