Civil Engineering Reference
In-Depth Information
y
y
1
z
A
z
1
y
1
z
1
y
f
f
z
O
F
IGURE
9.31
Principal axes in
a beam of arbitrary section
Substituting for
y
1
in the first of Eq. (9.47)
z
sin
φ
)
2
d
A
I
z
(1)
=
(
y
cos
φ
−
A
Expanding, we obtain
cos
2
φ
sin
2
φ
2 cos
φ
sin
φ
y
2
d
A
z
2
d
A
I
z
(1)
=
+
−
zy
d
A
A
A
A
which gives, using Eq. (9.46)
I
z
cos
2
φ
I
y
sin
2
φ
I
z
(1)
=
+
−
I
zy
sin 2
φ
(9.48)
Similarly
I
y
cos
2
φ
I
z
sin
2
φ
I
y
(1)
=
+
+
I
zy
sin 2
φ
(9.49)
and
I
z
−
sin 2
φ
I
y
I
z
(1),
y
(1)
=
+
I
zy
cos 2
φ
(9.50)
2
Equations (9.48)-(9.50) give the secondmoments of area and product secondmoment
of area about axes inclined at an angle
φ
to the
x
axis. In the special case where O
z
1
y
1
are principal axes, O
z
p
,
y
p
,
I
z
(p),
y
(p)
=
0,
φ
=
φ
p
and Eqs (9.48) and (9.49) become
I
y
sin
2
φ
p
−
I
z
cos
2
φ
p
+
I
z
(p)
=
I
zy
sin 2
φ
p
(9.51)
and
I
y
cos
2
φ
p
+
I
z
sin
2
φ
p
+
I
y
(p)
=
I
zy
sin 2
φ
p
(9.52)
respectively. Furthermore, since
I
z
(1),
y
(1)
=
I
z
(p),
y
(p)
=
0, Eq. (9.50) gives
2
I
zy
I
y
−
tan 2
φ
p
=
(9.53)
I
z
