Moment of inertia for pipe

i think you are using product of Inertia- see moment of Inertia represented as a Tensor -about an axis which is not fixed in the body.

 

You apparently are talking about the second moment of area of the pipe, at least, that's what your formula calculates.

http://www.engineeringtoolbox.com/area-moment-inertia-d_1328.html

For a pipe with a circular cross section, Ix = Iy and Ixy = 0. The polar moment Ip = Ix + Iy

It's not clear what you would be using Iz or Ixy for.

If you are trying to calculate the mass moment of inertia of a pipe, then different formulas are required.

 

The second moment of area is used primarily to calculate the bending stress in a beam, so Iz would have no use for that calculation, assuming that the cross section of the pipe lies in the x-y plane.

Because a circular cross section is symmetric about any axis which passes thru the center, Ix or Iy is going to be the same value. For other types of cross sections, like I-beams for instance, Ix and Iy will have different values.

It all comes down to: what are you trying to do with the moments of inertia?

 

If you are using dynamic equations, there could possibly be a mix of area moments of inertia and mass moment of inertia for the different sections of pipe. If the piping assembly is fixed so that there are no gross rotations about a fixed point, then you are probably dealing just with area moments of inertia.

The stiffness matrices K will require the area moments of inertia, while the mass matrix M will generally require only the masses of the individual elements.

Because the stiffness matrices K will presumably be assembled from individual elements, the area moments of inertia will probably be referred to some local element coordinate system, so it would be unwise to specify the values of the individual element inertias until this point is established.