# Tensor of inertia of a curved beam

I need help calculating the inertia tensor for a curved beam.

I found the formulas for this in the article https://hal.archives-ouvertes.fr/hal-01084693/document[page 20, Appendix A] and decided to check their correctness. And found the following:

1. The formula for calculating the distance $$\chi$$ from point $$O$$ to the center of mass works correctly (tested in SolidWorks using mass analysis);

2. The integration result does not correspond to the original integration formula for the inertia tensor components $$I_{yy}$$ and $$I_{zz}$$. $$I_{xx}$$ component works well when I align the $$r$$ vector (shown in the picture) with the $$x$$-axis. This corresponds to my tasks and there are no questions here yet.

1. As far as I understood from the text, this tensor was searched for by placing the origin of coordinates in the center of mass.

My question is: How to find the inertia tensor during the rotation of a curved arc around the point O and, if possible, recalculate it for any point?

• Is the cross section of the beam always a rectangle (i.e. is the beam generated by sweeping a rectangle along a curved path)? Commented Jan 1, 2023 at 1:27
• @Zegpi The cross section can have any shape (square, round).
– ayr
Commented Jan 1, 2023 at 4:19