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# Moment of Inertia of Rounded Hollow Rectangle

## Moment of Inertia of Rounded Hollow Rectangle

(OP)
Hello,

I can't seem to find how to calculate the second moment of inertia (Ixx and Iyy) of a rectangle WITH rounded corners.

The below link is great for hollow rectangles, but does anyone know how to calculate it if the corners have radii?
http://www.amesweb.info/SectionalPropertiesTabs/Se...

### RE: Moment of Inertia of Rounded Hollow Rectangle

mjgarrin, doubt it is much different between rounded and square corners, but if you need to know that accurately:

Get out your strength of materials textbook. Calculate the centroid and MOI of each segment about its own centroid. Then calculate the centroid and MOI of the composite shape per the parallel axis theorem.

Or, if it is a standard shape, you may be able to just look it up.

Regards,

Mike

The problem with sloppy work is that the supply FAR EXCEEDS the demand

### RE: Moment of Inertia of Rounded Hollow Rectangle

Depending on the exact geometry, it may be easier to calculate the properties of the area removed by the radii and subtract that from the rectangle, rather than adding stuff up.
Also note that tabulated values may not show the same section properties that you'd get using tabulated radii. I think on beams, they use the largest radius for clearance reasons and the smaller radius or some average radius for area properties, etc., so they don't always correspond.

### RE: Moment of Inertia of Rounded Hollow Rectangle

Do you have access to a CAD program? Most have the ability to calculate the sectional properties of an arbitrary shape that you create.

### RE: Moment of Inertia of Rounded Hollow Rectangle

separate the corners from the overall section as 1/4 circles, and do the simple math ...

### RE: Moment of Inertia of Rounded Hollow Rectangle

If it's not in my copy of the Manual of Steel Construction, I just use Solidworks...

I've done enough parallel axis theorem calculations over the years to know the approach, and yes it could be done this way, but it would be slow and painful.

### RE: Moment of Inertia of Rounded Hollow Rectangle

Well, yeah, but the OP did ask how to calculate it...

Regards,

Mike

The problem with sloppy work is that the supply FAR EXCEEDS the demand

### RE: Moment of Inertia of Rounded Hollow Rectangle

I suppose if you have a CAD package sitting on your computer, and you are efficient using it, that's a reasonable way to do it, but it only takes a few minutes to set the calculation up on a spreadsheet, and once set up it then takes a few seconds to update it for a different section.

See sample spreadsheet attached. I have split the section into two vertical rectangles, two horizontal rectangles and two semi-annuli for the corners. The section properties for the semi-annulus come from my spreadsheet previously linked by Pete (https://newtonexcelbach.wordpress.com/2014/11/02/s...).

As always, please assume that both spreadsheets give incorrect results, until proved otherwise.

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

### RE: Moment of Inertia of Rounded Hollow Rectangle

and a hollow section is just the outside minus the inside.

### RE: Moment of Inertia of Rounded Hollow Rectangle

An easier way is to draw the object in 3D using ACAD and then check its mass property.

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