Composite moment of inertia question:

[jamesm]

Good morning PF,

I am having some difficulty finding info on calculating a composite moment of inertia. The section I am working with is 8 separate columns (WT5x15) arranged evenly spaced around the circumference of a circle. You can see what I'm taking about in the attached jpg. I found that the moment of inertia of each column is 9.28 in^4, is the total moment of inertia just the sum of each column or does their arrangement factor in also?

Thank you.

 

[Simon Bridge]

I'm going to be careful with my answer here:
The total moment of inertia about an axis is the sum of the individual moments of inertia about the same axis.

Their distance from the axis does affect this - but that should be included in the individual moment of inertia, i.e. by the parallel axis theorem.

Their arrangement in relation to each other affects the center of mass, which will affect the motion when the composite object is spun.

You appear to have made sure the center of mass coincides with the axis so you should be fine.

 

[jamesm]

Thanks for the answer.

That is similar to what I was thinking. I will have to calculate the moment of inertia for the columns myself since the cookbook answer (9.28 in^4) is most likely calculated using the columns own neutral axis.

 

[SteamKing]

It looks like you are trying to calculate the second moment of area, which would have units of L^4.

If you look at a table of steel sections for a WT5x15, you will notice that the moment of inertia is larger about the axis which is parallel to the flange of the T versus the axis which is parallel to the web. In addition to using the parallel axis theorem to calculate the MOI about an axis which doesn't pass thru the centroid of the section, you must also calculate how the MOI changes when the axis is rotated with respect to the principal centroidal axis.

 

[Simon Bridge]

When I went away I wondered if it was second moment... ;)

 

[jamesm]

The second moment of area is what I am looking for.

I understand the need for parallel axis theorem, but I do not follow this: "you must also calculate how the MOI changes when the axis is rotated with respect to the principal centroidal axis.".

This is what I'm using to get my data for the columns: http://www.webcivil.com/readusShapeWT.aspx

If I'm looking for second moment of area should I be using J as opposed to I_x? (0.31 in^4)

Haha sorry if I'm being slow, I appreciate the help.

 

[SteamKing]

No, I is what you are looking for. J is also known as the torsional constant of the section and is used only when analyzing torsion about the longitudinal axis of the beam.

However, that being said, the inertia of the individual T members is going to change from the table value as you go around the circle. The T section is not symmetric w.r.t. rotation, so for calculating the MOI about the x-axis (which I presume is the horizontal axis in the diagram) for the array of beams, you must not only account for the position of each of the 8 beams relative to the center of the circle, but you must also account for whether the web of the individual T beams is oriented vertically, horizontally, or at a 45-deg. angle. The table value of I for the T of 9.28 in^4 is true only when the web is oriented vertically w.r.t. the x-axis. When the T is oriented such that the web is horizontal, the I value is 8.35 in^4.

Pages 9-10 of this article:

http://ocw.nthu.edu.tw/ocw/upload/43/763/static_ch9.pdf

show how to calculate the inertia of a section about oblique axes when you have values of Ix, Iy, and Ixy about another axis which is rotated at an angle. For the T section, Ixy = 0, because the section has an axis of symmetry.

 

[jamesm]

Thanks for the help. I got the moment figured out finally, I calculated it with AutoCAD after calculating the moment of a few known sections to make sure I was doing it right.