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Math Help - Second moment of area - Need walkthrough

  1. #1
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    Sep 2007
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    81

    Second moment of area - Need walkthrough

    Hi all,

    I'm at uni starting off engineering, and I'm looking for a walk through in this algebra involved in the second moment of area. While I know it's probably pretty basic I'm undertaking a bridging course to try and keep up with my maths.

    http://home.exetel.com.au/peleus/smoa.jpg

    Here's a picture of the most relevant lecture slide discussing the problem.

    I'll type out the steps they undertook to get the final formula for the second moment of area for a rectangle.

    On the next page we take the integral of this, which I can do fine.

    This gives

    1. I = \frac{b}{3}[y^3] with limits +h/2 and -h/2

    Taking it further we end up with

    2. I = \frac{b}{3}[\frac{h^3}{8}-(-\frac{h^3}{8})]

    and finally we take it to the step

    I = \frac{bh^3}{12}

    Ok, I can understand a bit about this but here are my questions.

    - Why are the limits h/2 and -h/2, isn't this simply the middle of the rectangle?
    - Why is dI = y^2 dA, where does the y^2 come from?
    - I understand that the \int y^2 is \frac{1}{3}y^3, but where / why does the b jump in with step 1? why are we multiplying it by b?

    Any help is greatly appreciated.
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  2. #2
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    Joined
    Mar 2009
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    1
    Hello,

    1. The limits are the extents of the rectangle around its centroid. I'm sure you remember that a centroid for a rectangle is in the middle, so h/2 is the distance from the centre to the outsides. You are integrating the whole rectangle. The integral sign should be read as 'integrated from -h/2 to h/2'.

    2. y is the distance from the centroid of the rectangle to the centroid of the strip. it is y^2 because we want the second moment of inertia, not the first (which would be just y and would end up giving you the centroid again). By squaring it you are giving more importance to parts of the rectangle that are far away from the centroid. A real life example of this is a ruler. Try bending your ruler the easy way, with it flat. Now turn it on its side and try again - it's a lot harder to bend it on its side because the material of the ruler is further away from the centroid!

    3. the b doesn't 'jump' in. When you integrate a value, in this case y, you have to integrate it with respect to itself. You start with dA, but this has no reference to y, so you substitute dA for yb (both describe the area of the strip, but now you have a relationship of the area with y). b never changes for the rectangle (is a constant) and so can come out of the equation with the 3.


    I recommend that to truly appreciate the ideas you calculate the second moment of area using a spread sheet. All you need to do is change size of bdy and you can get a good approximation and understand better what integrating really is.
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