# Reporting second moment of area in a paper

Hi Everyone,

I am reading the literature of different projects that have used Bonej for data analysis and I came with the question of why some researchers report the second moment of area as a ratio between Imax/Imin? Is this the best way to describe the resistance to bending? I know many of them explain it as a descriptor of the shape of the cross section of the bone, however I wonder whether is better to report the IPM or Product moment of Inertia that BoneJ gives when we run slice geometry.

Also from Michael Doube papers I see he reports Imax and Imin separately, so I just wanted to get some advice of the best way to report second moment of area.

If they are, then they are wrong. Second moment of area is just I. The subscript indicates in which direction I has been calculated. Lots of anthropology people orient their bones’ anatomical axes to the x and y axes of the image and then report Ix and Iy. It seems more probable that people are reporting a measure of eccentricity or circularity in calculating Imax / Imin. The closer the cross-section is to an annulus, the closer that ratio gets to 1, and the closer it gets to a long and pointy ellipse the closer the ratio gets to infinity. Alternatively you can invert the ratio to get a circularity measure that goes from 0 (long and pointy) to 1 (circular).

Bear in mind that polar moment of inertia J is calculated as Imax + Imin but is meaningful in relation to estimating resistance to torsional stress only when the cross-section is annular (i.e. very close to circular and ImaxImin).

Just report Imax and Imin. You can also set your own axes using the Orientation tool.

Yes, and instead I am reporting the Zpol value to inform torsion resistance on the bones that I am measuring, as they do not accomplish that requirement. However I am still curious about the Product moment of Inertia or IPM from BoneJ. is this the same as J (Imax+Imin)?

I am sorry if my questions sound too obvious, I am still learning about the mechanical properties of the bones, and the terminology is sometimes confusing to me.

Thank you again!

Zpol is just J divided by the radius of the annulus, so shouldn’t really be used except on circular cross-sections.

I don’t remember precisely what product moment of area Ipm is meant to represent, just that it approaches 0 when there are few pixellation errors. In other words the better the estimate of I, the smaller is Ipm. So far I haven’t found a practical use for Ipm except to help determine that the maths is working on test images of rotated rectangles.

Thank you for making this more clear to me. According to your explanation I am not going to be able to use Zpol to report the resistance to torsion on the cross sections of the bones that I am studying. I wonder if you could recommend any tool or alternative that I can use to measure the resistance to torsion on non circular bones?

thanks again

Reporting Imin and Imax will be sufficient.

Section modulus Z is distinct from second moment of area I in that by including a chord length Z takes into account the most highly strained element of the cross section and thus is better suited to an analysis of failure, rather than simply resistance to stress.

If you really want to know how your specimens respond to torque you should load them and image them using a speckle pattern, and measure the surface strains. Alternatively you could do that experiment in a CT (or XMT) scanner to get 3D strain data.

The next best thing would be finite elements analysis. FEA has the advantage of estimating strains in locations throughout the specimen. Strains might concentrate in unexpected places.

Thank you very much for this information Michael. I really appreciate it!