What is Product of inertia of a crossection 2

[omairzali]

Guys,

I understand the logic moment of inertia of a crossesction (which is basically a measure of resistance towards bending about a particular axis). But i cannot visualise or even understand the product of inertia of a crossection (ABOUT THE CENTROIDAL AXIS) from the perspective of solid mechanics.
Can someone share their understanding about product of inertia? (say the crossection i am referring to is a channel section)

 

[rb1957]

the only time i've used the product of inertia was to determine the principal axes of a section. a doubly symmetric section has a zero product of inertia, the principal axes align with the x- and y- axes of the section (ie the two CLs).

 

[GregLocock]

The handwavy explanation is that it is a measure of the degree of asymmetry of the section, when bending about that axis. this asymmetry causes an additional deflection normal to the applied load.

Cheers

Greg Locock


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[Lion06]

Here is a thread where I have an example calc posted.
thread507-241589

 

[BAretired]

I lifted this from a physics forum:

The product of inertia is related to the moments of inertia.

Imagine you have a ceiling fan which is operating correctly (no wobble). The fan has a specific non-zero moment of inertia about the center of rotation. That value for the moment of inertia does not relate any information about the balance of the object.

For example, if you attached a weight to one of the fan blades, the moment of inertia would go up, but there would be no way to distinguish it from an evenly distributed fan with heavier blades.

The product of inertia is what you need to determine rotational stabilities. In the case of the balanced fan, the product of inertia is zero, and there is no wobble. With the weight on one blade, the product of inertia becomes non-zero and it affects the rotational stability of the fan.

I like GregLocock's definition too.

BA

 

[omairzali]

thanks everyone for your responses.
i am now beginning to get the idea behind this terminology.

Greglock i agree that this provides a measure of assymetry. However say that now you are applying the load on this asymmetric section in such a manner that it is ONLY deflecting in the direction of the applied load (in this case it is a skewed load passing through the centroidal axis of the section). So in this case will the product of inertia be zero?

 

[BAretired]

The product of inertia, moment of inerta, section modulus, area and radius of gyration are properties of a cross section based on geometry alone. None are affected by loading.

BA

 

[rb1957]

"provides a measure of assymetry. However say that now you are applying the load on this asymmetric section in such a manner that it is ONLY deflecting in the direction of the applied load (in this case it is a skewed load passing through the centroidal axis of the section). So in this case will the product of inertia be zero?"

it sounds like your load is aligned to the principal axes of the section. ie, your section is asymmetric, like a Zed section; this will deflect about the principal axes, which you can determine from basic section properties.

 

[EdR]

The product of inertia, just like the moment of inertia, is a purely mathematically defined quantity i.e. integral( x y dA).

Just as with stresses when the product of inertia goes to zero the principal axes of a cross section are obtained....

Ed.R.