# Converting flexural rigidity to no load span rating

Discussion in 'Materials' started by fallguy, Dec 12, 2019.

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### rxcompositeSenior Member

Can you make a sketch of what you are going to do? When you say end must float, it becomes a cantelever design. Once we have a fairly good idea of your design, I will post the formula(s) to be used.

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### fallguySenior Member

Nothing is cantilevered.

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### Jolly MonJunior Member

It might be nice to get a sample of the material and do a quick test. You could support the two ends and measure the “sag” against a straight edge....a level for example.

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### fallguySenior Member

That is happening as well, but it is also good to calculate it for expectations and like I said, I could simply double the weight for the dynamic aspect. It is a headliner and no lives are at risk. The other reality is that this will probably not sag immediately; it will occur over a time period and changing temperatures, etc. I will build a spreadsheet today.

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### fallguySenior Member

y (inches) != K*(pounds)*(inches^4)/(pounds/inch*inches^3)

This result is in inches squared, unless q is supposed to be in psi..

When I use psi, the thing has the right units and is more rational. According to my math, a 24" x 24" panel would sag 0.195" then in 3mm material and the 6mm material would sag 0.048".

In order to get the 3mm to sag under 1/16", I need to support it at 17". This makes good sense on the gutcheck.

Let me know if I have interpreted the equation correctly on the further look.

I will try to make a nicer Excel spreadsheet and perhaps upload it here.

Again, thanks.

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### fallguySenior Member

Hello Gonzo, there would be some minor curvature in the panel, but I am disregarding that for this effort. I realize an arch has some different properties, but the arch is a couple inches in a hundred inches or less than 5%.

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### TANSLSenior Member

The theory of plates gives several formulas that depend, among other things, on the shape of the plate, the fastening of its edges and the type of load applied to the plate. It is very important to define these points correctly before knowing what formula to apply. See, for example, a very normal case for hull plates: flat plate with straight boundaries and constant thickness, all its edges fixed and subjected to a uniform "q" load (load per unit area), over entire plate.

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I get very similar values to you, not exactly the same but consistent, based on the values you provided.
3mm = 5.37mm
6mm = 1.34mm.

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### fallguySenior Member

Thanks a bunch. During the course of the efforts, I realized raw corecell, (san) is 7 times lighter than pvc komatex. Of course, raw corecell is also not as stiff and about three times the cost and not finished. But I could glue headliner to corecell and not add much weight if I can resolve the stiffness. That could be done with a one sided lamination or two sides light.

So, as with many boat things; this has morphed a bit.

For interior cabin walls that are not intended to support a roof, but may indeed support a roof to some degree; does anyone have a typical lamination? Using 12mm corecell...

Then for kitchen cabinets, same question. Although the kitchen cabinets are super inviting to build in komatex for the easy factor; at least the faces.

Then for my headliner panels, is there a way to determine the E for a foam panel with a lamination using math or will I need to test? For example, can I compare the stiffness of the komatex and the san core with amendment? The raw san core's E is 72 vs 800MPa for the komatex. So, the komatex is 11 times stiffer. So, san core will really sag on its own. But I am thinking something like 6 oz glass n epoxy one side might get it closer. Someone must have this data somewhere, but perhaps it is closely held. I was thinking about using 6mm corecell with a lamination of sorts.

Thanks for the replies. Really saved me a lot of experimenting and guessing. Still not sure which way I'll go.

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Well, at the heart of your question is really the basics of structural theory and understanding. Since which material structural arrangement is better?...only a simple analysis will tell you that…

For example, take a flat bat, let’s say it is 50 x 5mm.

If you have this FB orientated vertically (long side is 50mm) the inertia is 5.208cm4, with a modulus of 2.08cm3.
If this same flat bar is rotated 90 degrees, so the width is 50mm and its depth is 5mm, the values become 0.052cm4 and 0.21cm3.

By orientating the same structural element you can change the stiffness, its inertia/modulus, by a factor of 100 and 10 respectively.

So, in your case all you are doing, or beginning to realise, is that by adding skins, moving them apart with a material that weights nothing (the purpose of a core), you increase the stiffness.

So, taking that same flat bar which is now 50mm wide and 5mm thick, if you split this into 2 skins of 2.5mm each and separate them by 10mm, the inertia becomes 0.9896cm4 which doesn’t sound much, but has increased its stiffness by a factor of 19 – which is massive! And the modulus becomes 1.32cm3, an increase of over 6 times.

This then leads into real structural understanding and how to use and manipulate materials and structural arrangements to achieve an objective. When designing a structure, especially with low modulus materials, the design driver is the deflection, not the stress. So, the metric of measure becomes what is called Flexural Rigidity or Flexural Stiffness – which is pretty much where you started this thread. It is the “EI” in the bending moment equation which is integrated to become the deflection.

The E = Young’s modulus and the I = 2nd moment of inertia.

So, now back to the 2 skins that are 2.5mm and 10mm apart.

If the E, the young’s modulus increases, in simple words, if you select a much stronger material for the skins, the value of EI increases; the higher the EI the less deflection. And the by-product of this is a lowering of the stress too.

So you can select a material, cheap that has a low E. But to satisfy a design criterion results in having either a very thick single skin which ends up heavy, or 2 skins separated at a large distance apart – which may be too thick to sue effectively.
Or
If you select a material that is very expensive but very strong, you could end up with a thin single skin, that is to say thinner than the low E version and be lighter, or 2 skins that are also thinner with much less distance between them, but ends up way beyond your budget for the whole project.

How you decide which you want and which arrangement is simply dictated by your budget and the geometric dimensional constraints that you may or may not have.

That’s it….

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### fallguySenior Member

I also forgot that the q in the calculation drops dramatically for the san core, so when I entered the data for the san core; despite having 10% of the modulus, it also has 10% of the weight and they offset ftmp.

Many thanks. I have some more work to do..

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### Jolly MonJunior Member

Fallguy, I would recommend 1/2” XPS foam for this application to stiffen the PVC headliner. It is lightweight and stiff. The XPS needs to be sanded with 36 grit sandpaper or something similar, then vacuumed. I would put the xps down on a flat surface first, then coat the xps and pvc with epoxy and add the pvc to the top. Put some weights on top of the pvc. You can use plywood or small bags of sand whatever. This will help insure a good bond.

I have made panels like this using 3 mm ply on both sides and the finished panel is incredibly lightweight and strong. On a beam 12” wide and 24” long I can stand on the center and there is no measurable deflection. I weigh 230 lbs. I use neat epoxy with excellent results, although the Gougeon bro’s would probably recommend thickening to a ketchup consistency.

my only concern is that it may be too stiff to accomplish the bending that you require. Since you’re only adding a skin to one side and there’s minimal bending it just might work for you.

all the best

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### fallguySenior Member

too heavy, my friend, but

the xps alone with a headliner is probably a good option, but

it would not be very strong under a screw and finish washer as it has poor compressive strength which would mean the need for laminating with light ply or pvc and back to more weight

the best solution is probably 6mm san with a headliner glued on

if I leverage some structural laminations; I can easily make the spans and perhaps use velcro to reduce the number of screws and trims...

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