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# ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia2

## ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

(OP)
ACI 318 R8.4.4.2.3, and CSA 13.3.5.5 make reference to a property of the critical shear section that is called "analoguous to the polar moment of inertia". The polar moment of inertia should be the radial distance from centroid multiplied by an increment of area, integrated over the area.

I'm at an absolute loss as to how they've calculated the formulas they use to determine the polar moment of inertia for an interior column, and so can't create my own for the case of an exterior column, or a circular column. Can anyone shed some light on this?

### RE: ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

Yeah, it's based on the work of Ghali at Calgary I believe. I asked him about it personally back in 2011. Walk with KootK...

1) Forget about your problem and consider a new one. Two squares on a page in front of you separated by a distance. You want Ix.

2) One component of Ix is a x d^2.

3) Another component of Ix is each square's bd^3/12.

4) This pseudo polar moment of inertia is identical but for two differences. Firstly "a" is an area on a vertical plane rather than a horizontally. Secondly, it turns out that the bd^3 component in this scenario is best excluded. It adds little and, apparently, correlates better with testing when it's ignored. Plus it's easier to calc this way which is peachy.

So the paramter that you want is just the areas of all the vertical shear planes involved times the square of their distances to the centriold of the perimeter.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

### RE: ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

(OP)
That aligns closely with what I've done in the past, and IMO best represents the concept of "polar moment of inertia", even a bit conservatively. The problem is I can't see the formula from AISC aligning with this. On top of that, as polar moment of inertia is not defined by an axis but rather a radial distance from the centroid, it should be the same irrespective of orientation, but the AISC formula should return different values depending on which axis is being analyzed. Photo attached -

### RE: ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

#### Quote (op)

On top of that, as polar moment of inertia is not defined by an axis but rather a radial distance from the centroid, it should be the same irrespective of orientation, but the AISC formula should return different values depending on which axis is being analyzed. Photo attached -

This may be true of the real polar moment of inertia but it is absolutely not true of the faux version used in punching shear. The punching shear approximation is very much axis dependent.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

### RE: ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

I used the references below while verifying the results of some proprietary studrail software a few years ago. The math was tricky, particularly when dealing with openings in the critical section.

### RE: ACI 318/CSA A23 - Property of critical section analoguous to the polar moment of inertia

Link 1 didn't work. - update; worked with ctrl-click

Other 2 look great thanks.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

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