Converting flexural rigidity to no load span rating

I would like to know if/how to convert flexural rigidity to a span rating.

for example, product has a given flexural rigidity; we want to install it as a headliner and want to limit the sag over that span

The product, let's say sags 1/16th inch in a one foot span and that is acceptable.

I want to calculate the deflection of the product to determine whether it is a foot or 14", etc.

I can probably figure it out, but a little help might be good.

 

Is it a flat or a curved panel?

 

Sly question.

 

That represents a typical application of the basic law of elasticity by Hooke. A strip of the panel is regarded a beam, suspended at the ends, either stiff or as a "hinge". With knowledge of the transverse shape of the strip, its modulus of elasticity, length between suspensions, type of end fixation and the applied load, the deformation can be calculated. Straightforward classical engineering.

 

For that you need to look at the definition of the term.
I assume you mean flexural 'strength' or sometimes modulus of rupture.
This is given by = 3.F.L/(2.b.h^2)

Where
F = force applied
L = span, or distance between the supports used in the test
b = width of section
h = depth of section

So, the 'span' is the "L" in the above eqn. Then just rearrange to suit your requirements.

 

So, here is what I did. Perhaps someone can tell me why it is wrong. I am trying to determine how much a pvc panel will sag when used as a headliner.

The section is a square section. The pvc is either 3mm or 6mm. The density is .65 g/cm^3

I setup a span of 24" to try to determine the sag factor, if you will, or more properly, deflection.

I used this calculation.

Deflection (inches) = 5/384*(weight of the panel in pounds)*width of the span ^3 /(moment of intertia * youngs modulus)
(think this is deflection of a beam and while this is not a beam, per se, it is similar if supported on both sides, no?)

For the moment of intertia, I used (width of span*thickness of panel^2)/6
For youngs modulus, I used 116,030 psi, 800MPa is from the datasheet

Units would be pounds*inches^3/(inches^3*pounds/inch), result is inches..

So, for my panel I get a sag of 0.040" for the 3mm panel. The weight is a little tricky, all I did was treated the weight as the weight of a square panel, so the example panel was 24" square; this is suspicious.

For Ad Hoc's formula(thank you), I wasn't sure which one was the deflection of the panel.

And for semantics, consider the panel supported on two sides fully, only the middle of the span would sag under its own weight..

Not rational. It is not enough. No way does a 3mm panel 24" wide not sag about a good 1/8"...

Perhaps my idea to treat the thing as beam is all off.

Anyhow, I realize it is sort of basic engineering, but I am an accountant by trade; not an engineer, and I never like to pretend to be otherwise.

 

Correct, it is the deflection of a beam...and NO, it is not similar.
Plate theory is rather more complex, especially when dealing with uniformly distributed loads.

So, rather than writing it all out, here an excerpt from Roark's book - the simple reference bible for such:-



With, if a square panel, boils down to:-

y = 0.044.q.b^4/(E.t^3)

That's it...

 

The theory of beam loading is that there must be a load for it to deflect. There are many kinds of load model and the one that suits our needs is the fixed support uniformly loaded beams (which translate to pressure).

It is usually the combination of fixed stiffeners with plate that is analyzed and the panel size is in the 1:2 ratio or larger. If it is less than 1:2 ratio, then an adjustment in the bending moment is required.

What do you want to test? A panel or the sub structure (a beam with a plate)?

 

I just want to determine how much a pvc headlining panel will sag.

For example, a 3mm panel, of strength a, supported 24" apart will sag y amount. If the y amount is too great and will become unsightly; then I would reduce the span to say 18".

I realize this sag is limited by the fact the panel is fastened, but we have all experienced how something will sag overtime if it is unsupported too far.

And by building a spreadsheet, I can also compare different materials. I am trying to choose between 3 different products. Plywood, pvc harder foam and a light pvc marine foam that I could glue headliner to...or glass one side with something ultralight..

So, mainly for headliner spans where the loading is limited to the material itself and the loading is mostly static, although boats bounce some...

 

y is my deflection
b is one distance
If I headline 8' pieces 2' apart; I might need to take the 8' into account, call it the a distance...this is need; despite thinking not earlier
q is an unknown (weight of panel I presume to cancel the modulus)
E is youngs modulus for the material
t is my material thickness

then I would change the calc to reflect the a dimension as the length of the panel is perhaps a factor....despite my pleas otherwise

Thanks. I will build an excel sheet tomorrow...

 

I could perhaps do something simple for the dynamic aspect of the loading and double the panel weight.

not very sound engineering, of course, but a bit more rational than a full discount

 

Ok, now I understand. At first i thought you wanted to know how much the panel will sag at the center. That is difficult as AH has pointed out.

The analysis would be that of a girder or a primary longitudinals, firmly fixed at the ends with transverse stiffeners to support the plate. What you should be looking for is if the primary beams will sag. The primary beam is an L shaped stiffener welded to wide flange or an H beam with unequal faces. The plate is the flange. You will be solving for the section modulus (how stiff) so it does not bend more than specified y given a static or live load.

Are we in the right direction?

 

Conceptually, one problem I have is let's say the panel is 16 miles long; at some point the length no longer matters; so hopefully the calc will resolve that.

 

yes, well, sort of, that is, and this goes along with my last comment about the panel getting longer and longer; eventually; the length does not matter I believe

the headliner panel is just a sheet good; a trim board will go up and support it port to star..the ends will also get a trim board, but those ends must float port to star for expansion

so, the L and H might be there wrt the trim

I want to determine for and aft spacing of the trim boards. The headliner will sit on furring strips at the spacing needed to prevent sag. The trim will ride on top or in this case under the sheet that is under the furring.

 

No matter how long, the formula will solve it. It is beam theory, uniformly loaded, fixed ends, except that the beam is an unequal width face. It is simple. But only if the materials are similar. If you start mixing wood and steel (engineered lumber), a different formula is used.