1
4.18. A beam of circular crosssection, radius
3
a
has two eccentric holes of ra
dius
a
as shown in Figure P4.18.
2
3
a
2
3
a
O
3
a
x
y
a
a
Figure P4.18
Find the centroid of the section and the second moments of area,
I
x
,
I
y
,
I
xy
about centroidal axes.
2
It follows that
A
=
9
π
a
2

π
a
2

π
a
2
=
7
π
a
2
A
¯
x
=
9
π
a
2
(
0
)

π
a
2
(
0
)

π
a
2
parenleftbigg
3
a
2
parenrightbigg
=

3
π
a
3
2
A
¯
y
=
9
π
a
2
(
0
)

π
a
2
parenleftbigg
3
a
2
parenrightbigg

π
a
2
(
0
) =

3
π
a
3
2
and hence the centroid has the coordinates
¯
x
=
¯
y
=

3
π
a
3
2
×
1
7
π
a
2
=

3
a
14
.
We can now construct the last two rows of the table, noting for example that
¯
x
3

¯
x
=
3
a
2

parenleftbigg

3
a
14
parenrightbigg
=
12
a
7
.
The second moments of area are then given by equations (4.36–4.38) as
I
x
=
I
y
=
81
π
a
4
4

π
a
4
4

π
a
4
4
+
9
π
a
2