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# Finding moment of inertias for complex geometries

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## Finding moment of inertias for complex geometries

(OP)
Hi,
The moment of inertia for a hollow cylinder is given as follows: pi(outer radius^4 - inner radius^4)/4. This hand calculated moment of inertia has been proven to work in calculating bending deflection of the hollow cylinder when comparing the results to outputs in softwares like solidworks FEA. If this hollow cylinder were to be broken up into disks that are connected by I-beams what would the moment of inertia equation be or if you have an idea on how you could possibly adjust it.
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### RE: Finding moment of inertias for complex geometries

So you are trying to work out the deflection of a beam with a varying cross section?

In the linear range M/I=E/R=(sigma /y) always applies at every slice through the beam. So a piecewise approach should get you there.

Cheers

Greg Locock

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### RE: Finding moment of inertias for complex geometries

For a multiple segment beam there won't be a single moment of inertia, you have variable moment of inertia along the length of the beam, and if it changes abruptly you could think of it like a piecewise function as GegLocock suggests (i.e. non-prismatic beam).

You could however calculate the actual beam deflection by accounting for the variations in cross section along the length, then calculate an effective moment of inertia that would apply to a prismatic beam section and give the same midspan deflection as the non-prismatic beam.

There are many textbook examples and it has been hashed out here as well. https://www.eng-tips.com/viewthread.cfm?qid=360332

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