Civil Engineering Reference
In-Depth Information
Figure 3.6
Uncracked T-sections
the top surface gives:
b
w
−
1
h
h
2
1
+
h
2
b
w
−
1
h
h
x
=
(3.44)
1
+
The second step is to find the gross moment of inertia about the centroidal axis by the
transfer axis theorem:
b
w
h
h
x
2
b
w
)
h
f
x
2
b
w
)
h
3
f
12
(
b
−
b
w
h
3
12
h
f
2
I
g
=
+
2
−
+
(
b
−
−
+
(3.45)
Inserting
x
from Equation (3.44) into (3.45), simplifying and grouping the terms results in:
⎡
⎤
3
b
w
−
1
h
h
1
h
2
b
b
w
1
h
f
h
3
h
f
−
b
w
h
3
12
⎣
⎦
I
g
=
1
+
−
+
b
w
−
1
h
h
(3.46)
1
+
Let the expression within the bracket of Equation (3.46) be denoted as
K
i
4
, then Equation
(3.46) can be written simply as
I
g
=
K
i
4
(
b
w
h
3
/
12), where
K
i
4
is a function of
b
/
b
w
and
h
f
/
h
.
K
i
4
is plotted as a function of
b
/
b
w
and
h
f
/
h
in a graph in the ACI Special Publication
SP-17(73) and in other textbooks.
3.1.5 Bending Deflections of Reinforced Concrete Members
3.1.5.1 Deflections of Homogeneous Elastic Members
The bending deformations, i.e. deflections and rotations, of a member are, in general, calculated
from the basic moment-curvature relationships of the sections along the member, as shown
in Figure 3.7. In the simple case of a homogeneous elastic member, however, the curvature
φ
is proportional to the moment
M
and the proportionality constant is the bending rigidity
EI
. Since the curvature
φ
is defined as the bending rotation per unit length (d
/
d
x
), the
