Moment of Inertia of an L shaped beam

[Roomie]

Homework Statement

I'm trying to work out the moment of inertia for this L shaped beam:

http://dl.dropbox.com/u/630750/Screen%20Shot%202011-12-31%20at%2017.54.34.png

Homework Equations

(Included on my sheets below(

The Attempt at a Solution

I've made two attempts, the first one I think I did wrong as I didn't take into account that Rectangle A wasn't rotating around it's center?

Attempt 1: http://dl.dropbox.com/u/630750/Sheet1%202.jpeg

Attempt 2 I tried to correct that but Ix and Iy have came out pretty much the same, so that can't be correct, can it?

Attempt 2: http://dl.dropbox.com/u/630750/Sheet1%203.jpeg

Thanks for your help.

Also, how would I work out Ixy from this?

 

[nvn]

Roomie: Both attempts 1 and 2 are currently incorrect. You forgot to first compute the centroidal (neutral) axis location, which is the first step for any asymmetric cross section. Check any textbook to learn this first step, in an example problem having an asymmetric cross section. Yes, Ix equals Iy for this particular cross section, because notice your cross section is (essentially) identical in the x and y directions. Ixy does not come from Ix nor Iy. Ixy is a different equation. You must provide relevant equations yourself. Check textbooks for relevant equations.

 

[Roomie]

Roomie: Both attempts 1 and 2 are currently incorrect. You forgot to first compute the centroidal (neutral) axis location, which is the first step for any asymmetric cross section. Check any textbook to learn this first step, in an example problem having an asymmetric cross section. Yes, Ix equals Iy for this particular cross section, because notice your cross section is (essentially) identical in the x and y directions. Ixy does not come from Ix nor Iy. Ixy is a different equation. You must provide relevant equations yourself. Check textbooks for relevant equations.

Thanks for your help. Just spent a while studying how to find the centroid, and then redoing the moment of inertia. Hopefully this is correct. Would you mind please checking?

Attempt 3: http://dl.dropbox.com/u/630750/attempt3.jpg

Thanks!

 

[nvn]

Roomie: In attempt 3, Ix is correct, except you rounded numbers slightly too much.

(1) Generally always maintain at least four significant digits throughout all your intermediate calculations, then round only the final answer to three significant digits, unless the first significant digit of the final answer is 1, in which case round the final answer to four significant digits.​

Iy is currently incorrect. When Iy is computed correctly, it will match Ix, for this particular cross section.

By the way, you do not need to convert everything to meters, if you do not want to. You can perform all these calculations in mm, if you wish. Your choice.

(2) For long numbers having five or more digits (on either side of the decimal point), the international standard says you can write the digits in groups of three, separated by spaces. E.g., 0.000 045 365 m^3, instead of 0.000045365 m^3. Or e.g., 8 735 400 000 mm^4, instead of 8735400000 mm^4.​

 

[Roomie]

Roomie: In attempt 3, Ix is correct, except you rounded numbers slightly too much.

(1) Generally always maintain at least four significant digits throughout all your intermediate calculations, then round only the final answer to three significant digits, unless the first significant digit of the final answer is 1, in which case round the final answer to four significant digits.​

Iy is currently incorrect. When Iy is computed correctly, it will match Ix, for this particular cross section.

By the way, you do not need to convert everything to meters, if you do not want to. You can perform all these calculations in mm, if you wish. Your choice.

(2) For long numbers having five or more digits (on either side of the decimal point), the international standard says you can write the digits in groups of three, separated by spaces. E.g., 0.000 045 365 m^3, instead of 0.000045365 m^3. Or e.g., 8 735 400 000 mm^4, instead of 8735400000 mm^4.​

Thanks, are they definitely supposed to be the same? As I have some experimental results that I need to compare these to and they are not the same. And later I have to construct a Mohr's circle with Ixy to converted these to principle second moments of inertia, that wouldn't work if Ix=Iy would it?

Could you please give me a hint with where I've gone wrong with Iy please :)

Thanks!

 

[nvn]

Roomie: Yes, Iy equals Ix, for this particular cross section. And this does not cause a problem. In attempt 3, under the heading "Rectangle 2," dx is currently wrong.

 

[Roomie]

Roomie: Yes, Iy equals Ix, for this particular cross section. And this does not cause a problem. In attempt 3, under the heading "Rectangle 2," dx is currently wrong.

I checked my dx and it was completely wrong, redid it and you're right Ix=Iy.

I then tried to work out Ixy, I've never done this before for a shape where the axis isn't the center of the shape though so I may have done this completely wrong.

Here is my working for Ixy, I would really appreciate it if you would also give this a check:

http://dl.dropbox.com/u/630750/3.png

Thank you so much for your help!

 

[nvn]

Roomie: Your answer in post 7 is correct, but inaccurate, because you rounded most numbers too much in your calculations. I.e., during your calculations, you listed (used) some numbers with three significant digits, instead of four significant digits. If you follow the advice in item 1 of post 4, you will get accurate answers.

 

[Roomie]

Roomie: Your answer in post 7 is correct, but inaccurate, because you rounded most numbers too much in your calculations. I.e., you listed some numbers with three significant digits, instead of four significant digits. If you follow the advice in item 1 of post 4, you will get accurate answers.

Great! Thank you for your help, I will redo the working more accurately when I write it up properly.

Thanks again.

 

[asb]

I'm working on the same problem as Roomie and I understand everything. Although, in post 7 for the product moment of inertia is the correction term missing because you have to use the parallel axis theorem? Or do I not need a correction term (Adydx)?