PROBLEM 9.101
Using Mohr’s circle, determine for the area indicated the
orientation of the principal centroidal axes and the corresponding
values of the moments of inertia.
Area of Problem 9.74.
SOLUTION
4
PROBLEM 9.101 (Continued)
PROBLEM 9.102
Using Mohr’s circle, determine for the area indicated the orientation of
the principal centroidal axes and the corresponding values of the
moments of inertia.
Area of Problem 9.77
(The moments of inertia
x
I
and
y
I
of the area of Problem 9.102
were determined in Problem 9.44).
SOLUTION
4
PROBLEM 9.102 (Continued)
PROBLEM 9.103
The moments and product of inertia of an L4 3
1
4
-in. angle cross section with respect to two
rectangular axes x and y through C are, respectively,
x
I
1.33 in
4
,
y
I
2.75 in
4
, and
0,
xy
I
with the
minimum value of the moment of inertia of the area with respect to any axis through C being
min
I
0.692
in
4
. Using Mohr’s circle, determine (a) the product of inertia
x
y
I
of the area, (b) the orientation of the
principal axes, (c) the value of
max
.I
x
when the 4-in. leg of the angle is parallel to the x axis. Further, for
0,
xy
I
the angle must be oriented as
shown.)
Now
4
ave
11
()(1.332.75)2.040 in
22
xy
III
and
min ave
or 2.040 0.692 IIR R
4
1.348 in
Using
ave
I
and R, the Mohr’s circle is then drawn as shown; note that for the diameter XY,
(1.33, )
xy
XI
and
(2.75, | |).
xy
YI
(a) We have
2
22
1()
2
xy xy
RIII
or
2
22
1
1.348 (1.33 2.75)
2
xy
I
Solving for
xy
I
and taking the negative root (since
0)
xy
I
yields 4
1.14586 in .
xy
I
4
1.146 in
xy
I
(b) We have
22( 1.14586)
tan 2 1.33 2.75
1.61389
xy
mxy
I
II
or
2 58.217 29.1
mm
The principal axes are obtained by rotating the xy axes through
29.1° clockwise
about C.
(c) We have
max ave
2.040 1.348 IIR
or
4
max
3.39 inI
PROBLEM 9.104
Using Mohr’s circle, determine for the cross section of the
rolled-steel angle shown the orientation of the principal
centroidal axes and the corresponding values of the moments of
inertia. (Properties of the cross sections are given in Figure 9.13.)
PROBLEM 9.104 (Continued)
PROBLEM 9.105
Using Mohr’s circle, determine for the cross section of the rolled-
steel angle shown the orientation of the principal centroidal axes
and the corresponding values of the moments of inertia.
(Properties of the cross sections are given in Figure 9.13.)
PROBLEM 9.105 (Continued)
PROBLEM 9.106*
For a given area the moments of inertia with respect to two rectangular centroidal
x
and
y
axes are
x
I
1200 in
4
and
y
I
300 in
4
, respectively. Knowing that after rotating the
x
and
y
axes about the
centroid 30
counterclockwise, the moment of inertia relative to the rotated
x
axis is 1450 in
4
, use Mohr’s
circle to determine (
a
) the orientation of the principal axes, (
b
) the principal centroidal moments of inertia.