Civil Engineering Reference
In-Depth Information
E
XAMPLE
9.7
Determine the second moments of area
I
z
and
I
y
of the I-section
shown in Fig. 9.22.
b
y
t
f
t
w
O
z
d
w
d
t
f
F
IGURE
9.22
Second moments of area of
an I-section
Using Eq. (9.36)
t
w
)
d
w
12
Alternatively, using the parallel axes theorem in conjunction with Eq. (9.36)
bd
3
12
−
(
b
−
I
z
=
2
bt
f
2
bt
f
d
w
+
t
w
d
w
12
t
f
I
z
=
12
+
+
2
The equivalence of these two expressions for
I
z
is most easily demonstrated by a
numerical example.
Also, from Eq. (9.37)
2
t
f
b
3
12
d
w
t
w
12
I
y
=
+
It is also useful to determine the secondmoment of area, about a diameter, of a circular
section. In Fig. 9.23 where the
z
and
y
axes pass through the centroid of the section
2
d
2
cos
θ
y
2
d
y
d
/
2
y
2
d
A
I
z
=
=
(9.39)
A
−
d
/
2
Integration of Eq. (9.39) is simplified if an angular variable,
θ
, is used. Thus
d
cos
θ
d
2
sin
θ
2
d
π/
2
I
z
=
2
cos
θ
d
θ
−
π/
2
i.e.
π/
2
d
4
8
cos
2
θ
sin
2
θ
d
θ
I
z
=
−
π/
2
