Second moment area of symmetrical shape

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SUMMARY

The discussion focuses on the second moment area (I) of a square cross-section beam with side lengths b x b when rotated 45 degrees. It is established that the second moment area remains the same despite the rotation, as the formula I = b^4/12 for a cube applies. Participants suggest deriving the formula for a diamond shape using integral calculus or applying the parallel axis theorem for triangular shapes. The relevance of the integral I = integral(r^2dr) is questioned, indicating it may not directly pertain to the problem at hand.

PREREQUISITES
  • Understanding of second moment area (I) calculations
  • Familiarity with the formula I = b^4/12 for a cube
  • Knowledge of the parallel axis theorem
  • Basic integral calculus concepts
NEXT STEPS
  • Research the derivation of the second moment area for various shapes
  • Study the application of the parallel axis theorem in structural analysis
  • Explore integral calculus techniques for calculating moments of inertia
  • Learn about the properties of symmetrical shapes in engineering mechanics
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Engineering students, structural analysts, and professionals involved in beam design and analysis will benefit from this discussion.

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Homework Statement



-a beam with square cross section, side lengths bXb is rotated 45 degree.
-can we assume the second moment area (I) is the same if this is done?

Homework Equations



I=b^4/12 (cube)
I=integral(r^2dr)

The Attempt at a Solution

 
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If you know how the formula I=b^4/12 (cube) has been derived from an integral, then you should be able to do the same for the diamond shape. Or, if you accept the formula for a triangle and apply the parallel axis theorem, you can answer the question yourself. I don't see the relevance of I=integral(r^2dr).
 

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