Civil Engineering Reference
In-Depth Information
s
s
u
y
r
z
t
F
IGURE
9.30
Second moment of area of a
semicircular thin-walled section
Properties of thin-walled curved sections are found in a similar manner. Thus
I
z
for
the semicircular section of Fig. 9.30 is
π
r
ty
2
d
s
I
z
=
0
Expressing
y
and
s
in terms of a single variable
θ
simplifies the integration; hence
π
r
cos
θ
)
2
r
d
θ
I
z
=
t
(
−
0
from which
π
r
3
t
2
I
z
=
9.7 P
RINCIPAL
A
XES AND
P
RINCIPAL
S
ECOND
M
OMENTS OF
A
REA
In any beam section there is a set of axes, neither of which need necessarily be an
axis of symmetry, for which the product second moment of area is zero. Such axes are
known as
principal axes
and the second moments of area about these axes are termed
principal second moments of area.
Consider the arbitrary beam section shown in Fig. 9.31. Suppose that the second
moments of area
I
z
,
I
y
and the product second moment of area,
I
zy
, about arbitrary
axes O
zy
are known. By definition
y
2
d
A
y
=
z
2
d
A
zy
=
I
z
=
zy
d
A
(9.46)
A
A
A
The corresponding second moments of area about axes O
z
1
y
1
are
y
1
d
A
y
(1)
=
z
1
d
A
z
(1),
y
(1)
=
I
z
(1)
=
z
1
y
1
d
A
(9.47)
A
A
A
From Fig. 9.31
z
1
=
z
cos
φ
+
y
sin
φ
y
1
=
y
cos
φ
−
z
sin
φ
