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# Discontinuous Cross-Section Moment of Intertia Calculation

## Discontinuous Cross-Section Moment of Intertia Calculation

(OP)
Hi All,

I have a design that I'm trying to analyze by hand calcs before I use numerical methods (FEA). I'm working on beam calculations for max shear/moment so that I have something to validate against an FEA result. The design involves applying pressure to two flat plates that are joined to and separated by bars (see picture attached). The intended design is locate the bars such that the two plates act as a composite beam (which is a totally different discussion unto itself).

The place I'm running into trouble is in the calculation of the moment of inertia. I was trying to think of this design as a series of I-beams with the edges of the flanges welded together. Based on the orientation of the bars in relation to the load, I'm assuming the bars (ie. the web of the I-beams) won't add much, if any, stiffness to counteract the bending moment. Similar to an I-beam, all of the bending stiffness will come from the two flat plates (ie. the flanges of the I-beams). However, this is just my assumption.

It's easy to ignore the bars, calculate I based on two flat plates acting as a composite beam and say that is a conservative approach. However, I would like to see how much the value for I changes if the bars are included in I. The issue is, I can't figure out how to calculate I for a beam with a discontinuous cross-section. Can any one provide some guidance on how to do this? Is there another way to look at this setup to make calculating I easier?

Thanks in advance for the help!

-Steris

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

Is the lower diagram an idealization of the upper diagram? Or is it the upper diagram rotated 90 degrees?
For design purposes, you could make some checks and assumptions on the spacers and shear through them, buckling of the unsupported web, etc. But those aren't necessarily going to be very close to the stresses calculated from a finite element approach.
Keep in mind that in normal beam design assumptions, you're analyzing stress away from the supports.
For example, in your upper diagram, all of the shear has to be carried by the flanges at the support, which is exactly opposite the distribution assumed in a beam. If this is actually a rectangular plate supported on 4 edges, etc., then the difference may be considerable.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

(OP)
The lower diagram is a (maybe?) simplified way that I was thinking about the upper design. The upper diagram is the actual arrangement.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

Steris:
Jstephen alludes to this, but I’ll say it more directly, you will have a far more efficient and traditional bending structural system if you reorient the web elements 90̊ in plan, so that they span from reaction point (line) to reaction point and work with the t&b plates (skins) as part of the bending system. Then, you will be able to calc. a meaningful moment of inertia, etc. and the two t&b plates will take most of the bending forces/stresses and the webs will take most of the shear/shear stress at the reactions, where these are the greatest. The way you have the top sketch detailed right now, the bending strength will be pretty much dependent upon the t&b plates, with a deflected shape which has fairly straight line/flat sloped shape btwn. each of the ribs and a fairly concentrated rotations at each of the ribs. It is a stretch, but your top sketch will act kinda like a Vierendeel truss structure, depending upon how you detail it.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

is this a very wide (ie into the page) deck ?

why orient the webs that way ? for convenience in welding ?
as noted above the other way (spanning across the supports) is a much more natural way to carry the shear load.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

(OP)
Thanks for the feedback! I do realize that having the web elements rotated 90deg would make the structure stronger and easier to analyze. Unfortunately, the span is specifically designed as shown for a variety of reasons and other design factors. I completely understand that with the given orientation, the web isn't doing much, if anything, to resist the bending moment or the shear forces, however, I was looking to characterize what impact it may have.

To clarify, the top diagram is a cross-section of the design with the span running into the page.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

is this deck supported on 4 sides ?

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

(OP)
For the purpose of this analysis, no it is not.

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

Pretty sure the web elements add almost nothing. I would be surprised if your full FEA showed 10% better than a hand calc that completely ignores them.

-handleman, CSWP (The new, easy test)

### RE: Discontinuous Cross-Section Moment of Intertia Calculation

Q is this deck supported on 4 sides ?
A For the purpose of this analysis, no it is not.

ok then, how will it react shear ? through membrane loads in the caps ?? transverse shear in the caps ??

sure your hand calc shows that the caps can handle the moment; but what's the shear loadpath ?

how much did the FEM deflect ?

how did you constrain the caps in the FEM ? rigid constraint on both sides, both caps ?

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