Moment of Inertia Clarification

In summary, the conversation was about determining the resistance and stiffness of a long thin plate with C channels welded along its length when subjected to an upwards pressure load. The individual components were treated as multiple rectangular bodies and the MOI was computed about their centroids using the parallel axis theorem. There was some confusion about which axis to use and the role of Ix and Iy in resisting bending and buckling. It was clarified that Ix resists rotation of the vertical strip about the x axis, while the top of the strip moves toward the viewer and the bottom moves away. The structure will deflect into a rainbow shape mainly due to Iz, but Ix would be more important if the structure was constrained along its front and back edge. It
  • #1
Hi all. It's been a while since I've taken mechanics of solids, I was hoping I could get some confusion ironed out here.

I have a long thin plate with C channels welded along its length, below is a cross sectional view.
[PLAIN]http://img375.imageshack.us/img375/6672/channelexcel.jpg [Broken]
As you can see, I have placed my axis about the neutral axis of the plate as well as its left edge. In my scenario I have an upwards pressure load on the bottom face of the plate.

I treated this entire setup as multiple rectangular bodies, computed the MOI about their centroids then used the parallel axis theorem to relate them to my chosen axis and combined them for the overall Ix and Iy about that origin.

My problem arises when I don't comprehend what I have found. Is Ix the resistance of this entire cross section bending in and out of the page about the x axis? So for example, a vertical strip would become a 3 dimensional U shape with the "prongs" facing the viewer?

If so, this is not what I am looking for. I need the resistance of the above shape transforming into a "rainbow" shape; it's resistance to buckling. Is this Ixy?
 
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  • #2
laforzadiment: No, you need Ix about the cross section neutral axis, not about your xy coordinate system axes. Did you do this correctly, to already obtain Ix about the cross section neutral axis? If so, that is what you need.

You do not need product of inertia, Ixy.

No, Ix will not cause a vertical strip to become a U shape with the prongs facing the viewer. It will instead cause a vertical strip to rotate about the horizontal neutral axis. We cannot really say what shape your structure will transform into, because we do not know your boundary conditions. But in general, Ix will allow your cross section to deflect vertically upward.
 
  • #3
nvn said:
laforzadiment: No, you need Ix about the cross section neutral axis, not about your xy coordinate system axes. Did you do this correctly, to already obtain Ix about the cross section neutral axis? If so, that is what you need.

You do not need Ixy. Product of inertia is zero here.

No, Ix will not cause a vertical strip to become a U shape with the prongs facing the viewer. It will instead cause a vertical strip to rotate about the horizontal neutral axis. We cannot really say what shape your structure will transform into, because we do not know your boundary conditions. But in general, Ix will allow your cross section to deflect vertically upward.

Thanks for your reply. Ok, so you're saying that I cannot arbitrarilly place the axis and that it must go at the centroid of the mass as a whole? Alright, I'll update my spreadsheet.

Consider the left and right end to be fixed in place so any upward uniformly distributed load will exclusively cause first order bowing until failure.

In the text you wrote it seems like you were saying Ix resists this motion:
[PLAIN]http://img806.imageshack.us/img806/9324/85217762.jpg [Broken]
If this is the case, wouldn't I need Iy?
 
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  • #4
laforzadiment: No, what you drew in post 3 is bending rotation about the z axis. Ix resists rotation of the vertical strip about the x axis; therefore, the top of the strip would move toward the viewer, and the bottom of the strip would move away from the viewer.

Because the left and right edges of your structure are fixed, your structure will deflect into somewhat of a rainbow shape, in the view you showed in post 1. But it will do this mainly because of Iz, instead of Ix. Ix would be more important if your structure were constrained along its front and back edge. So you need Iz of the yz cross section about its z neutral axis.
 
  • #5
Ah I gotcha now. Awesome, thanks for the help!
 
  • #6
laforzadiment said:
I treated this entire setup as multiple rectangular bodies, computed the MOI about their centroids then used the parallel axis theorem to relate them to my chosen axis and combined them for the overall Ix and Iy about that origin.

Parallel axis theorem WON'T work in this case. The theorem only works when the sectional property is uniform over the span of the bending member. You have non uniformity, as each segment has different properties from its adjacent neighbors.
 
  • #7
skeleton said:
Parallel axis theorem WON'T work in this case. The theorem only works when the sectional property is uniform over the span of the bending member. You have non uniformity, as each segment has different properties from its adjacent neighbors.

I don't follow. The area is uniform in and out of the page, I follow the same method used for a T or I-Beam just with more components.
 
  • #8
Oops. I mistakenly thought you were trying to determine the stiffness from left to right of the picture. Evidently from your comment your interest is perpendicular. In that case, I agree the theorem does apply.
 
  • #9
skeleton said:
Oops. I mistakenly thought you were trying to determine the stiffness from left to right of the picture. Evidently from your comment your interest is perpendicular. In that case, I agree the theorem does apply.

No prob :) it can be difficult to convey the original problem setup only through text.
 

1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass and distribution of mass within an object and is typically represented by the symbol "I".

2. How is moment of inertia different from mass?

While mass is a measure of the amount of matter in an object, moment of inertia is a measure of how that matter is distributed and its resistance to rotational motion.

3. How is moment of inertia calculated?

Moment of inertia is calculated by multiplying the mass of each particle in an object by its distance from the axis of rotation squared and then summing these values for all particles in the object.

4. What are the units of moment of inertia?

The units of moment of inertia depend on the system of measurement being used. In the SI system, it is measured in kilograms per meter squared (kg/m^2), while in the English system, it is measured in slug-square feet (slug-ft^2).

5. How is moment of inertia used in physics?

Moment of inertia is an important quantity in rotational motion and is used in equations to calculate the angular acceleration, torque, and kinetic energy of rotating objects. It also plays a role in the stability and behavior of objects in motion.

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