MOMENT OF INERTIAVS

THETA

Mitchell S. King09/27/2012

Supplement to Project Proposal

Reference Figure

Z

Y

B

C

Z

Y

B

C

ϴ

[ X and A-axes both coming out of the page. ]

Figure 1

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

Equation A

Equation B

Equation C

For a square cross-section… IYY = 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 cross-section is rotated about the X-axis.

I thought at first it to be bologna, but see next slide:

Moment of Inertia of a Square Cross-Section, Rotated 90°

This is the same value as when the cross-section appears as a square.

Z Z

h h

=h/√(2)

Figure 2

In conclusion to the square… It seems as though the square cross-

section 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.

0 10 20 30 40 50 60 70 80 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Moment of Inertia Variation for a 1.5" x 0.5" Rectangular Beamwith a 90° Twist

IbbIccIyyIzz

Theta (Degrees)

Mom

ent o

f Ine

rtia

(in4)

0 90 180 270 3600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Moment of Inertia Variation for a 1.5" x 0.5" Rectangular Beamwith a 360° Twist

IbbIccIyyIzz

Theta (Degrees)

Mom

ent o

f Ine

rtia

(in4)