Stiffened panel - moments of inertia

[FEAnalyst]

TL;DR

How to determine the moments of inertia for stiffened panel calculations ?

Hi,

I'm looking for a way to find the maximum deflection of a simply-supported stiffened panel analytically (to validate the FEM result). I've found some literature on this topic and I have 3 different formulas from various sources. The problem is that neither of them gives me the expected results (I use an already solved example from a research paper to check my calculations). I think that it's mostly a matter of the moments of inertia. In the aforementioned literature, stiffeners are treated as a grillage. However, it's never mentioned for which segments of the plate the moments of inertia should be determined. The results are far from the expected values when I calculate the moments of inertia for just a single T section of a stiffener. Also, some formulas don't even include the thickness of the plate itself so I guess it should be part of the segment used to evaluate the moment of inertia. But maybe it should be extended further to the sides to also cover the gaps (or at least their parts) between the stiffeners.

Here's one of the simplest formulas (from Design of Ship Hull Structures - short fragment with this formula is available here): $$\delta=\frac{abp}{\frac{\pi^{6}E}{16} \left( \frac{I_{x}(m+1)}{a^{3}} + \frac{I_{y}(n+1)}{b^{3}} \right) }$$
where: $a$ and $b$ - length of a longer and shorter edge of the panel, respectively, $p$ - pressure, $E$ - Young's modulus, $I_{x}$ and $I_{y}$ - "sectional moments of inertia with effective breadths" (whatever they mean here - any idea ?), $n$ and $m$ - number of stiffeners along the longer and shorter edge, respectively.

What segments would you use to determine the moments of inertia in this case ? The deflection I got this way was approximately 2 times larger than it should be.

Here's the panel under consideration:

Close-up on the stiffeners:

I've also tried the formulas from Timoshenko's "Theory of Plates and Shells", among the others, but it didn't work either.

 

[QuarkyMeson]

So you tried chopping it up into sections and then applying the parallel axis theorem?

And a meta question? But why? Checking FEA with a prototype and an experiment might be a better route here since I don't imagine the plate is all that expensive to prototype. Unless there's just no tools around that can be used to experimentally check the FEA results.

Analytically checking FEA seems wrong to me, but I guess there could be valid reasons.

 

[FEAnalyst]

So you tried chopping it up into sections and then applying the parallel axis theorem?

For now, I've only tried using the moment of inertia calculate for a single stiffener with respect to its centroid. The formulas only require a single moment of inertia for each side (a and b or x and y). But I guess it should include some section of a plate.

Here's what I've found on this page:

It will be necessary to assume that the stiffener beam acts together with an effective section of the plate to resist the total bending moment. How much of the plate is assumed effective in resisting bending will depend on the particular application, the engineering discipline and in some cases the codes in use. The most conservative assumption would be to take only the section of plate to which the beam connects as being effective. In other not so conservative applications such as aeronautical applications it is common to assume that a considerable section of the plate either side of the stringer is also effective in resisting the bending.

But they describe a different approach - solving the panel as a beam with effective section.

My formulas treat the panel as a grillage instead but I can't figure out what section to use for moments of inertia.

And a meta question? But why? Checking FEA with a prototype and an experiment might be a better route here since I don't imagine the plate is all that expensive to prototype. Unless there's just no tools around that can be used to experimentally check the FEA results.

Well, this isn't real-life project. Just a simple made-up benchmark/educational case to compare FEA with hand calcs.

Analytically checking FEA seems wrong to me, but I guess there could be valid reasons.

It's usually advised to do that unless experiments are available. There are also aforementioned benchmark/verification problems used to check whether a given FEA code provides correct results for different cases. I'm evaluating open-source FEA solvers on this occasion.

 

[QuarkyMeson]

Well, this isn't real-life project. Just a simple made-up benchmark/educational case to compare FEA with hand calcs.

It's usually advised to do that unless experiments are available. There are also aforementioned benchmark/verification problems used to check whether a given FEA code provides correct results for different cases. I'm evaluating open-source FEA solvers on this occasion.

That all make sense then.

For now, I've only tried using the moment of inertia calculate for a single stiffener with respect to its centroid. The formulas only require a single moment of inertia for each side (a and b or x and y). But I guess it should include some section of a plate.

But they describe a different approach - solving the panel as a beam with effective section.

My formulas treat the panel as a grillage instead but I can't figure out what section to use for moments of inertia.

I mean grillage seems right.

Here's one of the simplest formulas (from Design of Ship Hull Structures - short fragment with this formula is available here): δ=abpπ6E16(Ix(m+1)a3+Iy(n+1)b3)
where: a and b - length of a longer and shorter edge of the panel, respectively, p - pressure, E - Young's modulus, Ix and Iy - "sectional moments of inertia with effective breadths" (whatever they mean here - any idea ?), n and m - number of stiffeners along the longer and shorter edge

Effective breadths would be how much of the system contributes to the stiffness of the system. Normally half the distance to the next stiffener is what I've found quoted in other articles. This goes into effective breadths a bit even though it isn't exactly related to what you're trying to do: Breadths (background and fig 2 esp)

 

[deajohn]

It is difficult to get effective section since both the panel and its ribs are rectangular and not square. When you say simple support is that on all four edges? The effective section is the inertia of the repeating cross section (including face sheet, ribs, and the bottom reduced flange). For repeating section width (a) use no more than 20 x face sheet thickness, then calculate equivalent thickness of cube root (12 I/a). Then use Timoshenko if supported 4 sides or beam theory if supported only two sides.