10.4 Moments of Inertia About Inclined Axes; Principal Moments
10.4 Moments of Inertia About Inclined Axes; Principal Moments Procedures and Strategies, page 1 of 2
Procedures and Strategies for Solving Problems Involving
Moments of Inertia About Inclined Axes; Principal Moments
y
Begin by calculating Ix, Iy, and Ixy. Then to determine moments of inertia about
axes rotated with respect to the x and y axes, use the formulas
Ix + Iy
Ix  Iy
Iu = 2
+
cos 2Ixy sin 2

2
Iv =
Ix + Iy
Ix  Iy

cos 2Ixy sin 2
2
2
Iuv =
Ix  Iy
sin 2Ixy cos 2
2
x
Determine Ix, Iy, and Ixy first.

1. Formulas:
Ix + Iy

2


Determine the principal (maximum and minimum) moments of inertia either by
using formulas or by using Mohr's circle.
I max, min =
v
u
( Ix  Iy )2 + I2xy
2
The axes corresponding to Imax and Imin are given by the roots  and 2 of
the equation
2Ixy
tan 2 =
Ix  Iy
To determine which root, 1 or , goes with Imax and which goes with Imin,
substitute 1 into Eq. 1 and see whether the formula gives Imax or Imin.
10.4 Moments of Inertia About Inclined Axes; Principal Moments Procedures and Strategies, page 2 of 2
2. Mohr's circle:
a) Construct a coordinate system in which
the horizontal axis is labeled I (for values of
the moment of inertia, Ix and Iy), and the
vertical axis is labeled Ixy (for values of the
product of inertia).
Ixy
Ixy
X
X
R
Ixy
I
I
Iave = (Ix + Iy)/2
b) Plot the point (Ix, Ixy) and label it "X."
Ix  Iave = (Ix  Iy)/2
c) Plot the center of the circle, (Iave, 0),
where Iave = (Ix + Iy)/2 = average moment of
inertia.
Radius R =
Ixy
d) Draw a straight line connecting X and the
center, use geometrical relations to calculate
the value, R, of the radius, and then draw the
circle.
( Ix  Iy )2 + I2xy
2
Iave
X
Imax = Iave + R
Imin
R
e) Imax and Imin are the far left and far right
points on the circle.
R
Imax
I
Imin = Iave  R
y
f) The principal axes (axes corresponding to
Imax and Imin) can be found by rotating the
x and y axes half the amount of the rotation
in Mohr's circle.
y'
Ixy
X
p Imax
I
p/2
x'
x
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 1
1. Determine the moments of inertia of the standard rolled-steel
angle section with respect to the u and v axes.
y
0.987 in.
0.5 in.
Ix = 17.40 in4
v
Iy = 6.27 in4
Ixy = 6.08 in.4
6 in.
x
C
45°
u
1.99 in.
0.5 in.
4 in.
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 2
2. Determine the moments of inertia of the crosshatched area with
respect to the u and v axes for a) = 25° and b) = 90°
y
40 mm
v
220 mm
u

x
100 mm
100 mm
20 mm
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 3
3. Determine the value of for which the product of inertia of the
crosshatched area with respect to the u and v axes is zero. Calculate
Iu and Iv for this value of and compare Iu and Iv to Imax and Imin
v
y
x2 + y2 = 1002
Ix = 1.9635 x 107 mm4
u
Iy = 1.9635 x 107 mm4

x
Ixy = 1.2500 x 107 mm4
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 4
4. Determine the principal moments of inertia with respect to all
possible rectanglular coordinate systems with their origin at the
centroid C.
y
20 mm
50 mm
C
x
50 mm
20 mm
60 mm
60 mm
10 mm
Ix = 2.2013 x 107 mm4
Iy = 0.9213 x 107 mm4
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 5
5. Determine the principal moments of inertia and
principal axes having their origin at point O.
y
150 mm
15 mm
Ix = 1.1714  107 mm4
30 mm
Iy = 8.9083  107 mm4
Ixy = 2.3971  107 mm4
O
x
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 6
6. Determine the principal moments of inertia
and principal axes having their origin at point O.
y
6 in.
x
O
4 in.
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 7
7. Use Mohr's circle to determine the principal moments of
inertia and principal axes having their origin at the centroid C
of the standard rolled-steel channel section.
y
15.3 mm
Ix = 32.6  106 mm4
Iy = 1.14  106 mm4
127 mm
C
x
127 mm
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 8
8. Use Mohr's circle to determine the principal moments of
inertia and principal axes having their origin at the centroid C
of the standard rolled-steel angle section.
y
0.987 in.
0.5 in.
Ix = 17.40 in4
Iy = 6.27 in4
Ixy = 6.08 in.4
6 in.
x
C
1.99 in.
0.5 in.
4 in.
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 9
9. Use Mohr's circle to determine the principal moments of inertia
and principal axes having their origin at the centroid C.
105 mm
y
15 m
Ix = 6.7245  106 mm4
52.5 mm
Iy = 6.3520  106 mm4
80 mm
C
x
15 mm
7.5 mm
80 mm
15 mm
Ixy = 5.1300  106 mm4
10.4 Moments of Inertia About Inclined Axes; Principal Moments Problem Statement for Example 10
10. Use Mohr's circle to determine the principal moments of
inertia and principal axes having their origin at point O
y
2 in.
2 in.
2 in.
2 in.
2 in.
2 in.
O
x