Civil Engineering Reference
In-Depth Information
where
A
s
f
y
=
a
(3.125)
85
f
c
b
0
.
Since the lever arm (
d
−
kd
/
3) at yielding (Equation 3.120) must be less than the lever arm
(
d
2) at ultimate (Equation 3.123), then
M
y
must be less than
M
n
. Therefore, if
M
y
is
found to be greater than
M
n
, as in the rare case of large percentage of tension steel, then
M
y
should be taken as equal to
M
n
.
−
a
/
Doubly Reinforced Beams
The yield moment
M
y
and the yield curvature
φ
y
can be calculated by the linear bending
theory explained in Section 3.1.3.2. Changing
f
s
to
f
y
in the stress diagram of Figure 3.3(c),
we can write from the equilibrium of moments about the concrete compression resultant the
following equations:
A
s
f
y
d
1
A
s
f
s
kd
d
k
3
M
y
=
−
+
3
−
(3.126)
ε
y
in the strain diagram of Figure 3.3(b), we can also obtain the following
compatibility equation for the compression steel strain
Changing
ε
s
to
ε
s
:
−
d
ε
s
=
ε
y
kd
(3.127)
−
d
kd
The compression steel stress
f
s
in Equation (3.126) can now be written using Hooke's law:
k
ε
y
d
d
f
s
=
−
≤
E
s
f
y
(3.128)
1
−
k
The coefficient
k
in Equations (3.126) and (3.128) is obtained from Equation (3.31). The
ratio
β
c
in Equation (3.31) is defined in Equation (3.30). Once the coefficient
k
is obtained,
the yield curvature
φ
y
is computed from Equation (3.121).
The nominal moment
M
n
and the ultimate curvature
φ
u
can be computed by the principles
enunciated in Section 3.2.3. The moment
M
n
and the depth
a
can be obtained by solving
Equations (3.102)-(3.105) according to method 1 or method 2. Once the depth
a
is obtained,
the curvature
φ
u
is computed from Equation (3.124).
3.2.5.2 Bending Ductility
Because the bilinear
M
-
curve in Figure 3.14(c) captures the basic force-deformation char-
acteristic of bending action, it will be used to define the ductility of a reinforced concrete
member. The bending ductility ratio,
φ
µ
, is defined as the ratio of the ultimate curvature
φ
u
to
the yield curvature
φ
y
, i.e.
φ
u
φ
y
µ
=
(3.129)
