Mitchell S. King
09/27/2012
Reference Figure
Y
Y
[ X and A-axes both coming out of the page. ]
B
B
Z
ϴ
Z
C
C
I took your advice…
I looked into the case of a square beam.
 Turns out that for a square beam of
rotating cross-section, the moment of
inertia does not change.
 See next slide.

I’ve derived these equations:

(full derivation will be in 1st progress report)
 See
Figure 1
For a square cross-section…
= IZZ , so the 2nd and 3rd terms in
Equation A and B are always zero.
 For a symmetrical cross-section whose
centroid lies on its neutral axis, IYZ = 0.
 Therefore, the moment of inertia of a
square beam will not change as its crosssection is rotated about the X-axis.
 I thought at first it to be bologna, but see
next slide:
 IYY
Moment of Inertia of a Square
Cross-Section, Rotated 90°
=h/√(2)
Z
Z
h

h
This is the same value as when the
cross-section appears as a square.
In conclusion to the square…

It seems as though the square crosssection with twists will not yield any
useful conclusions for me, though I’m
glad I discovered this now.

The next slides show Equations A and B
plotted for a 1.5” x 0.5” rectangular
cross-section. Refer to Figure 1 for
bending axis directions.
Moment of Inertia Variation for a 1.5" x 0.5" Rectangular Beam
with a 90° Twist
0.09
0.08
Moment of Inertia (in4)
0.07
0.06
0.05
Ibb
Icc
0.04
Iyy
Izz
0.03
0.02
0.01
0
0
10
20
30
40
50
Theta (Degrees)
60
70
80
90
Moment of Inertia Variation for a 1.5" x 0.5" Rectangular Beam
with a 360° Twist
0.09
0.08
Moment of Inertia (in4)
0.07
0.06
0.05
Ibb
Icc
0.04
Iyy
Izz
0.03
0.02
0.01
0
0
90
180
Theta (Degrees)
270
360