Civil Engineering Reference
In-Depth Information
Y
y
y
y
Cross sectional
area,
A
a
A
A
z
O
b
O
O
z
z
Z
G
F
IGURE
9.24
Product second
moment of area
(a)
(b)
(c)
Summation (i.e. integration) over the entire section of the product second moment of
area of all such pairs of elements results in a zero value for
I
zy
.
We have shown previously that the parallel axes theoremmay be used to calculate sec-
ond moments of area of beam sections comprising geometrically simple components.
The theorem can be extended to the calculation of product second moments of area.
Let us suppose that we wish to calculate the product second moment of area,
I
zy
,of
the section shown in Fig. 9.24(c) about axes
zy
when
I
ZY
about its own, say centroidal,
axes system G
ZY
is known. From Eq. (9.43)
I
zy
=
zy
d
A
A
or
I
zy
=
(
Z
−
a
)(
Y
−
b
)d
A
A
which, on expanding, gives
b
a
ab
I
zy
=
ZY
d
A
−
Z
d
A
−
Y
d
A
+
d
A
A
A
A
A
If
Z
and
Y
are centroidal axes then
A
Z
d
A
=
A
Y
d
A
=
0
.
Hence
I
zy
=
I
ZY
+
abA
(9.44)
It can be seen from Eq. (9.44) that if either G
Z
or G
Y
is an axis of symmetry, i.e.
I
ZY
=
0, then
I
zy
=
abA
(9.45)
Thus for a section component having an axis of symmetry that is parallel to either of
the section reference axes the product second moment of area is the product of the
coordinates of its centroid multiplied by its area.
A table of the properties of a range of beam sections is given in Appendix A.
