Moment of Inertia Calculation for a Horizontal Element

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SUMMARY

The discussion focuses on the calculation of the moment of inertia for a horizontal element, specifically addressing the second moment about the x-axis (Ix) and y-axis (Iy). The correct approach for Ix involves using horizontal elements (dA) expressed in terms of y, leading to the integral of y² dA from 0 to a, resulting in a final answer of a⁴/28. The method criticized in the discussion involves using vertical strips, which complicates the calculation unnecessarily. For Iy, the correct method requires choosing vertical elements and expressing dA in terms of x.

PREREQUISITES
  • Understanding of moment of inertia concepts
  • Familiarity with integral calculus
  • Knowledge of geometric properties of shapes
  • Ability to manipulate differential area elements (dA)
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  • Study the derivation of moment of inertia for various geometric shapes
  • Learn about the application of integration in calculating area moments
  • Explore the differences between horizontal and vertical element methods in structural analysis
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kieranl
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can somebody please tell what i am doing wrong with this moment of area question?

question and working in the attached doc

thanks a lot
 

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To calculate the second moment about the x-axis, you have to choose a horizontal element dA, because only a horizontal element has a constant y. I don't know why you chose vertical strips and used their centers as the y value, but your method is not right.

To calculate Ix, express dA in terms of y and you'll get dA=(sqrt(ay)-y)dy. Integrate y^2 dA from 0 to a and you'll get the correct answer, a^4/28. I have no idea how your lecturer got to his equation for Ix, but it's probably unnecessarily complicated because the method I just described leads to the right answer very quickly.

To calculate Iy, choose vertical elements and express dA in terms of x. Then integrate x^2 dA from 0 to a.
 

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