# Finding Centroids

1. Dec 1, 2014

### Bluestribute

1. The problem statement, all variables and given/known data
Determine the product of inertia of this area with respect to the centroidal x′ and y′ axes.
Locate the centroid x bar of the beam's cross-sectional area. Set a = 120mm , b = 290mm , d = 30mm , h = 390mm .

2. Relevant equations
???

3. The attempt at a solution
So I can't find anything online that would help with finding the Ix'y' value except maybe Ixy = Ix'y' + ACxCy. But I don't have anything to help find Ixy, so I'm pretty much stuck in an inertia loop. Any explanation would be great as to how the answer is -1.21e8 mm^4!

2. Dec 1, 2014

### SteamKing

Staff Emeritus
It's not clear what 'this area' refers to. Is there a picture which didn't get attached?

3. Dec 1, 2014

### Bluestribute

Ah, right. There is indeed!

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4. Dec 1, 2014

### SteamKing

Staff Emeritus
The product of inertia Ixy of any area which has an axis of symmetry is equal to zero. After that, apply the parallel axis theorem to determine Ixy about the centroid of the entire area.

5. Dec 1, 2014

### Bluestribute

So Ixy is 0 in this case because each piece is a rectangle (aka symmetrical)?

So it's just -ACxCy?

If so, how do I find Cx and Cy? Is that the center of each piece of the object in the x and y directions respectively?

6. Dec 1, 2014

### SteamKing

Staff Emeritus
No, I don't think you understand. The Ixy of each rectangle forming the cross section is zero about its own centroid.

The section in the figure does not have an axis of symmetry because the areas are not distributed evenly side to side or top to bottom.

Just like you find the centroid of the whole section by calculating moments about a reference axis, so you'll have to calculate Ixy about this same axis using the PAT. Once all moments of area and moments of inertia have been computed about this reference axis, then you use the PAT again to find the MOI about the centroidal axes.

Yes. In these types of calculations, it's easy to set up a table of are information for all the individual pieces and make the calculations.

7. Dec 1, 2014

### Bluestribute

But the total Ixy is just all the Ixy's added together . . . so wouldn't that be 0?

And in the previous problems, I found X bar and Y bar of the shape, if that helps, since I'm lost as to the whole parallel axis thing. I know you can change your axis. Don't know why you do, don't know how you apply it to real life, and don't know how to appropriately use it to calculate Ix'y' unfortunately =\, but I know it exists.

8. Dec 1, 2014

### SteamKing

Staff Emeritus
Nope, you're ignoring the distance between the centroid of each rectangle and the centroid of the section as a whole. Because the centroids of each rectangle do not coincide with the centroid of the figure as a whole, this is why applying the PAT is necessary.

This slide show may illustrate why the PAT makes these calculations easier:

http://www.ce.memphis.edu/3322/Pdfs/PaulsPDFs/Centroids and Moment of Inertia Calculation.pdf

9. Dec 1, 2014

### Bluestribute

X bar = 73.2 mm
Y bar = 154 mm

(73.2 + (30 * 120 * 60^2)) + (73.2 + (30 * 390 * 15^2)) + (73.2 + (30 * 290 * 145^2)) = 1.985e8 = Ix
(154 + (30 * 120 * 15^2)) + (154 + 30 * 390 * 195^2)) + (154 + (30 * 290 * 15^2)) = 4.477e8 = Iy

Right track in relating that to Ixy (which is the next point I'm stuck at)?

10. Dec 2, 2014

### SteamKing

Staff Emeritus
This is why I suggested you put your calculations into a tabular form. It makes the calculations much easier to check.

I'm going to show you how to calculate Ix about the x-axis with a tabular form, but I am going to use the letter dimensions rather than actual numbers.

Code (Text):

ITEM                AREA    y-bar     A*y-bar        A*y-bar^2          Io
Lower piece          b*d     d/2      b*d*d/2       b*d*(d/2)^2      b*d^3/12
Vertical piece  (h-2d)*d     h/2   - Fill in the numbers here -  (d/12)*(h-2d)^3
Upper piece          a*d    h-d/2    a*d*(h-d/2)   a*d*(h-d/2)^2     a*d^3/12
------------------------------------------------------------------------------
Totals              Σ A    y'-bar     Σ A*y-bar    Σ A*(y-bar)^2       Σ Io

You can find the Total Area and the moment of Area about the x-axis by adding the appropriate columns in the table.
The moment of Inertia about the x-axis is the Σ Io + Σ A*y-bar^2.

y'-bar = (Σ A*y-bar) / (Σ A)

To find the moment of inertia about the centroidal x'-axis, you would apply the PAT to the totals above like so:

Ix' = Σ Io + Σ A*(y-bar)2 - ΣA*y'-bar2,

where y'-bar is the vertical distance from the x-axis to the x'-axis.

A separate table can be set up to calculate Iy and Ixy, or the table shown above can be extended by adding columns to cover the additional properties desired.

11. Dec 2, 2014

### Bluestribute

Ok, so this definitely helps when I have to calculate the Ix and Iy (which I'll be doing tomorrow while studying). How exactly do you relate this information to Ixy though?

12. Dec 2, 2014

### SteamKing

Staff Emeritus
The same principle applies to calculating Ixy as it does to calculating Ix or Iy. The Ixyo values for each rectangle are all zero (because the rectangle has an axis of symmetry), and you can use the table to compile the values of A*x*y for the transfer quantities of each rectangle about the x-axis and the y-axis. Once the location of the centroidal x'-y' axes have been established relative to the x-y axis, then the PAT can be applied to determine Ixy for the entire section relative to these centroidal axes, thus

Ixy = Σ Ixyo + Σ A*x*y - Σ A*(x-bar)*(y-bar)

In this case, Σ Ixyo = 0, leaving

Ixy = Σ A*x*y - Σ A*(x-bar)*(y-bar)

where x-bar and y-bar are the location of the centroid of the entire cross section relative to the x-y axis.

13. Dec 2, 2014

### Bluestribute

Right right. What I mean though, is like Y bar is just half the height of each piece (in this particular case), I0 is bd3/12, etc.

But what's Ixy in terms of math (since I know physically it isn't anything)?

14. Dec 2, 2014

### SteamKing

Staff Emeritus
Ixy is just as physically real as Ix or Iy, it's just not dealt with often unless the cross section doesn't have an axis of symmetry.

Maybe this article can better illustrate what I've been trying to explain:

http://m-sudo.sakura.ne.jp/soft_data/kikaisekkei_data/mo-ru eN_Kansei_Momennto5.pdf

15. Dec 3, 2014

### Bluestribute

Ok. So basically, unless you're applying a force at the centroid of the object, you'll essentially be bending on a different axis? Is there a simple equation for Ixy? Using what I think I know about the other two, is it just XYA (x bar, y bar, and area)?

16. Dec 3, 2014

### SteamKing

Staff Emeritus
Think of Ixy as a measure of how evenly area or mass is distributed about the centroidal axes.

If Ixy = 0, then there is at least one axis of symmetry.

Unlike Ix and Iy , it is possible for Ixy to take on negative values.

There is no simple formula for calculating Ixy except those I have tried to explain.

Ixy does not have any influence in the bending of a beam.

17. Dec 3, 2014

### Bluestribute

Ah, so I just had false hope with that XY part. Dammit! Oh well, I think I can apply the rest (but we'll find out!)

18. Dec 3, 2014

### Bluestribute

Ok, so I want to see if I'm messing up somewhere or if the answer key is wrong. I've done this multiple times, I've used a table, and I'm getting 2.65 inches or .942 inches. The key says 2.25 inches . . .

TOP: Y bar = 0.5 | Area = 1.3
MID: Y bar = 1.9 | Area = 1.9
BOT: Y bar = .25 | Area = 1.8

I'm summing AY as my numerator and just A as my denominator, so my fraction becomes: 4.71/5
Summing up Y bars only, which gets me closer to the key, is the 2.65. That's not right, but it's closer . . .

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19. Dec 3, 2014

### SteamKing

Staff Emeritus
Your answer isn't coming out right because you're not measuring the y-bar values for each piece from the same reference.

For instance, either pick to top or the bottom of the section and measure all of the y-bar values from that reference.

For example, the bottom piece is 0.5/2 = 0.25" from the bottom of the section, the middle piece is 3.8/2 + 0.5 = 2.4" from the bottom,
and the top piece is 1.0/2 + 3.8 + 0.5 = 4.8" from the bottom. These are the y-bar values you put into the table in order to calculate the centroid of the entire section.

20. Dec 3, 2014

### Bluestribute

You know, some people feel stupid because the right method was, so to speak, on the "tip of their brain".

I'm more of just plain stupid since I was looking at the axis in the center of the picture going "How do I find how high up these axis's are? Everything needs to be from these axis's" haha. Oh man, yes, yes you are correct and I got the answer.

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