Section Modulus Calculation

Can someone give me the formula's for calculating the section modulus (metric, cm^3) and moment of inertia (cm^4) of a 1 cm wide strip (FRP sandwich panels in this case...)
Thanks

 

For moment of inertia (technically, more correctly termed the second moment of area, I) you need to depth of the material as well. Ignoring (for now) the fact that you have a sandwich construction, I for a rectangular cross section is given by:

I = (bd^3)/12

where b is breadth (1cm in your case) and d is depth

Section modulus, Z, is:

Z = I/y

where y is the normal distance of the outer edge of the section to the neutral axis. For a rectangular section y is just d/2.

Now, because you have a sandwich section, where the core is not structurally significant (just used as a spacer for the skins), you have to modify this slightly. You can ignore the core and imagine the skins as two, less deep, rectangles of thickness t. I (for one skin) is (bt^3)/12, but there is also an additional term, based on how far the skin is from the neutral axis of the overall section, to add on. As implied earlier, the overall neutral axis of a rectangle is on the centreline, so the distance from the centre of one skin to the NA is:

h = d/2 - t/2 where t is the thickness of the skin.

To calculate the overall I of the total section (ie two skins):

I = ((bt^3)/12 + Ah^2) * 2

where b is breadth of skin (1cm in your case)
t is depth of one skin
A is the cross sectional area of one skin ie b*t
h is as described above.

Section modulus is the same:

Z = I/y

where y = d/2 (d is overall section depth)


Hope that helps.

 

It is a very opened ended question. It will depend on the reinforcement and the number of layers.

I assume you want to do some strength and rigidity calculations for a panel. This link has some useful information on densities, UTS and elastic modulus for different reinforcements:
http://www.fgi.com.au/products/reinforcements/Reinforcements.pdf

I treat the reinforcement skins as thin and determine their effective thickness based on cloth weight and material density. I neglect any contribution from the core other than providing a controlled separation distance.

So for a FRP strip w wide, T thick with a skin of t on either side, the second moment of area I = T^2 / 4 * t * w.

Note that in determining t you have to allow for material on two axes for normal cloth or three if it is a triaxial weave. So the effective thickness t will usually be half of the value derived from the material density and mass per unit area.

This may seem complicated but it is very simple if you make a sketch of the composite and apply a load and reactions.

Rick W.

 

Yes,

If t is small (which it is) then t^3 is very small, so the first term in my equation (bt^3/12) is negligible. The second term (Ah^2) is the same as your equation for I (where you have approximated h as T/2, not T/2 - t/2, which is fair enough given the small t).

Good point about the effective t value. Presumably 0.5t is for a 0/90 weave, with 0.707t if 45/45?

 

Thanks PI, Rick
I'm working my way through the ABS scantling rules (motor pleasure craft). Pretty confusing 1st time through! I'm up to the bit where I've already determined the thickness of a single skin laminate, now onto the sandwich. There are 2 formula's for the minimum required section modulus of the skins and one for the min moment of inertia of the core. But I don't have their thicknesses yet!
I'm assuming that both I & Z need to be greater than or equal to that of the solid laminate, so I figured if I could calculate the Z & I for that, then I could establish the required Z & I for the sandwich - thus allowing me to calculate thickness...

 

I have not seen the rules you refer to.

Your last statement on the Z & I may not hold. I can understand the logic for the higher Z as this lower the maximum stress. It does not hold that the I would be higher.

You also refer to a formula for the moment of inertia for the core. Is this just the core or do you mean the whole sandwich. The only things that I thought would be important for the core are compressive strength and durability of the bond.

My practical outlook on the design issue is that there is a practical limit to just making thicker cores and thinner skins. You actually need the skin to resist puncturing.

Rick W.

 

My thoughts are exactly the same as Rick's. Do you mean I for the core? I'm not familiar with those ABS rules (we use Lloyd's), but they are confusing aren't they?!

 

A bit of friendly caution: be careful.

Not trying to knock you or be denegrating, but if you are asking how to calculate section modulus of a simple strip, you might not have the best understanding of structures. And you might want to get some elementary background or consult someone who works in this area before you design your own.

 

Guys - sorry - I've been away for a while.
That was a rather confusing post wasn't it... I lumped both Z & I into the same sentence... and yes - you both have it right.

Water Addict - thanks for the advice. This is all theory for me at the moment, though I do have at least a basic understanding of the engineering involved. I guess that's what scantling rules were designed for to some extent: so that people are able to design safe scantlings without having to hold an engineering degree, but your point is well taken.

 

Remember that Scantling rules are published for experienced designers and with the expectation that the final design will be scrutinized . There are pitfalls and the rules are not always overly specific, without experience they can be hard to interpret at times. Best to get someone to review your first few designs.