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Moment of Inertia vs Theta

Mitchell S. King 09/27/2012. Moment of Inertia vs Theta. Supplement to Project Proposal. Y. [ X and A - axes both coming out of the page . ]. Y. ϴ. B. B. Z. Z. C. C. Reference Figure. Figure 1. I took your advice…. I looked into the case of a square beam.

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Moment of Inertia vs Theta

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  1. Mitchell S. King 09/27/2012 Moment of InertiavsTheta Supplement to Project Proposal

  2. Y [X and A-axes both coming out of the page. ] Y ϴ B B Z Z C C Reference Figure Figure 1

  3. 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.

  4. I’ve derived these equations: • (full derivation will be in 1st progress report) • See Figure 1 Equation A Equation B Equation C

  5. 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:

  6. Moment of Inertia of a Square Cross-Section, Rotated 90° • This is the same value as when the cross-section appears as a square. Figure 2 =h/√(2) Z Z h h

  7. 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.

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