Civil Engineering Reference
In-Depth Information
layer of longitudinal tension steel. The depth
d
t
is somewhat different from the well-known
effective depth
d
, which is defined as the distance from the extreme compression fiber to the
centroid of longitudinal tensile reinforcement. If only one layer of steel bars is used, then
d
t
=
d
. If more than one layer of steel bars are used, then
d
t
is somewhat larger than
d
.For
simplicity, one layer of steel bars is used in Figure 3.10 and
d
t
=
d
.
Figure 3.10 (a), (b), (c) and (d) from left to right show four strain distributions at ultimate
load stage of four beams reinforced with decreasing percentages of steel. The first diagram
(a) shows the strain distribution of a beam with a balanced steel percentage
ρ
b
. That means
when the concrete reaches the ultimate strain of
0.003 specified by the ACI Code, the
steel bar reaches the yield strain of 0.002 for a mild steel bar with a yield stress of about
f
y
=
ε
u
=
413 MPa (60 000 psi). In this case, the position of the neutral axis gives a balanced value
of
c
b
=
ρ
b
, the beam is considered to be under
'compression control' because concrete will crush before the yielding of steel. Such behavior
is only allowed in columns, which must be designed with a low reduction factor
0
.
6
d
t
. When the steel percentage
ρ
is greater
ϕ
=
0.7 for
spiral columns and
0.65 for tied columns.
The third diagram (c) shows the strain distribution of a beam with the maximum steel
percentage
ϕ
=
ρ
max
. In this case, when the concrete reaches the ultimate strain of
ε
u
=
0.003, the
steel bar reaches the ACI specified strain limit of
ε
t
=
0.005. The position of the neutral axis
gives a maximum value of
c
max
=
β
1
d
t
.
The fourth diagram (d) shows the strain distribution of a beam with the steel percentage
0
.
375
d
t
,or
a
max
=
0
.
375
ρ
less than
ρ
max
. That means that when the concrete reaches the ultimate strain of
ε
u
=
0.003,
the steel bar will have a strain
375
d
t
. A beam
located in this region will be called a 'ductile beam', and the behavior of the beam is called
'tension control.' A ductile beam has the privilege of using a high reduction factor of
ε
t
greater than 0.005, while
c
is less than 0
.
ϕ
=
0.9.
A typical ductile beam could be designed for
ρ
=
0.5
ρ
max
. In this case, the neutral axis is
located at c
=
0.5(0
.
375
d
t
)
=
0.1875
d
t
, and the tensile strain
ε
t
=
ε
u
(
d
t
-c)
/
c
=
0.003[(
d
t
/c) -
1]
=
0.003[(1/0.1875) - 1]
=
0.013
>
0.005, ductility, OK. For a beam with
b
=
200 mm,
d
=
500 mm, and
f
c
=
41.4 MPa (6000 psi),
a
=
β
1
c
=
0.75(0.1875
×
500)
=
70.3 mm. The
85
f
c
ba
(
d
factored moment
M
u
=
ϕ
0
.
−
a
/
2)
=
(0.9)(0.85)(41.4)(200)(70.3)(500 - 70.3/2)
=
207 kN m.
The second diagram (b) shows the strain distribution of a beam with the steel percentage
ρ
ρ
max
, but less than the balanced
ρ
b
(
ρ
max
≤
ρ
≤
ρ
b
). When the
greater than the maximum
ε
u
=
ε
t
less than
concrete reaches the ultimate strain of
0.003, the steel bar reaches a strain of
0.005, but greater than 0.002 (0.005
≥
ε
t
≥
0.002), while
c
is larger than 0
.
375
d
t
, but less than
0
.
6
d
t
(0
.
375
d
t
≤
c
≤
0
.
6
d
t
). In this range, the reduction factor
ϕ
will decrease from 0.9 to
0.7 (or 0.65) with decreasing
ε
t
from 0.005 to 0.002. The analytical expressions for this
transition are shown in Figure 3.10(e).
In practice, it is recommended to use 'ductile beams', rather than nonductile beams, for
two reasons. First, ductile beams are more economical because they can be designed with a
ϕ
factor of 0.9 without penalty. Second, ductile beams are much simpler to design because the
procedure to penalize the
ϕ
factor is complicated.
3.2.2.5 Analysis of Ductile Sections
The analysis and design of ductile beams will be presented separately. The analysis of ductile
beams is included in this section, while the design of ductile beams will be discussed in Section
3.2.2.6.
