Section Modulus of almost rectangular sandwich beam 1

[kribri]

Hi,
I'm trying to calculate the Section Modulus for the beam cross-section presented in the attachment.

Can anyone please assist?

krish

 

[rstupplebeen]

http://en.wikipedia.org/wiki/Second_moment_of_area

The trick is to determine where you neutral axis is. I usually run with 2 cases one ignoring the skin and one with a 'large' section of skin.

Rob Stupplebeen

 

[gwolf2]

Split it into chunks and sum the chunks.

 

[desertfox]

Hi kribri

First convert the beam to a equivalent section of one material example core material by:-

E face/ E core = 200/9 = 2.2222

now the width of the beam should be multiplied by this figure:-

2.2222*100=222.22

so your equivalent beam made solely of core material would be 200 high by 222.22 wide from which you can calculate the second moment of area :-

I = b*d^3/12 = 222.22 * 200^3/12 = 148.15*10^6

section modulus Z= I/y = 148.15*10^6/ 100 = 1.4815*10^6

regards

desertfox

 

[SWComposites]

The core modulus is so low the core can be neglected in the calculations.

When calculating section properties where portions of the section have different moduli, it is simpler to perform all of the calculations using (EA) and (EI) instead of A and I, rather than converting widths/thicknesses to an equivalent section with one material (can get way too complicated).

 

[desertfox]

hi SWComposites

But the beam in this case is relatively simple thats why I chose to convert to one material.
Perhaps for the OP you could expand on your method then he will have a choice.

regards

desertfox

 

[RPstress]

Like SW, I'd ignore the core unless desperate.

If you must include the core and want the conceptually simple width-factoring method à la Desertfox then I don't quite get the same numbers...Desert, you were converting the beam section to be all core material? And thus multiplying the skin X dims (widths in the pic) by the modulus factor-difference and adding the core and skin widths together?

 

[PanelGuy]

hmmmm...seems that we do not know how this is loaded. My first thought would be to consider it a u shaped beam and go with that, however, the core will mitigate a ton of buckling so is worth something.

My gut feel would be to work it as a u-channel and after reviewing those results determine whether anything needs to be reviewed concerning the sides, the core and the ability to prevent buckling. The core will act to maintain the U shape and keep it stiffer...actually falling in line with the basic premise of the equations you are using...that the shape does not change.

For just reviewing bending and shear, you could probably consider only the laminate but run the calculations as it being a tee shaped beam...axis and all should be the same. If you ran FEA on it the machine should handle the U fine.

Like SW and RP indicated...start with the easy stuff...

 

[desertfox]

Hi kribri

My apologies, looking back at my calculation I made some silly errors, however look at this site which explains it far better then I can:-

http://www.freestudy.co.uk/statics/beams/beam tut1.pdf

go to page 11 for a good example,

desertfox

 

[kribri]

Hi guys,
thanks for the help.

SWComposites:
It would be great if you could expand on your method.

Panelguy:
The beam is loaded with a vertical load.
It is modeled in abaqus with a uniform pressure over the whole top surface.
How would you model it in FEA? I have modelled it as a conventional shell with a stringer in the bottom free edge.
(Top edge is connected to the ceiling).
I'm using Abaqus.

desertfox:
thanks for the article


Krish

 

[SWComposites]

kribri -

do you know how to calculate the section properties for a homogeneous material section? such a steel I-beam?

why do you need section properties for a shell FE model? section properties are typically with beam/bar elements. or are you referring to the modulus value to enter into the property cards for the shell elements?

perhaps you could post a drawing of what you are analyzing? with loads and boundary conditions shown.

rhetorical questions to the world: does any one these days do a simple hand analysis? or is every problem bludgeoned with a FE model?

 

[kribri]

Hi SWComposites,
thanks for the quick reply.

I know how to calculate the section modulus for a homogeneous material section. But this beam/girder is a bit different, so I'm a bit confused.

I need the section properties to check the FE model with my hand calculations. There are some deviation between the results, so i would like to check.

I have attached a picture of what I'm modelling.
The material is a glassfibre and balsa sandwich composite.
The beam is almost a boxed beam (but does not have the top face).

I actually do some hand calculation for verification reasons. But I guess there aren't many people doing this anymore.

Thanks for all the help

 

[SWComposites]

- ignore the core for the section properties

- the remaining u-channel is a homogeneous material (presumably) so just calculate the section properties as you would normally

- how to model it? well, it depends on what answer you want from the model - deflection, static strength, buckling, ?? also depends on the loads and boundary conditions.

- for basic static linear strength/deflection, a) model it as 3 plates (ignoring the core), b) use your web + beam idealization (the web will be 2X the web thickness of the u-channel webs, c) not sure why you would model it as an inverted Tee, but that could work also, with appropriate properties

- for a buckling analysis, you will likely need to model it as a u-channel, using plates for the webs and lower flange, with several nodes along the width of each to provide sufficient degrees of freedom, and model the core with solid elements to provide the support to the web and flange.

- can you post a complete free-body diagram?

 

[PanelGuy]

Pro - I think that is where my answer is a such a throwback. I would figure the material properties, run it by hand (OK, I have an Excel spread sheet for this), decide the U channel works then build it with the core and physically test if it holds the buckling.

I did get tired of running all my lamination theory calcs by hand and wrote a DOS program followed by an Excel spreadsheet.

Once you have the laminate properties, simple beam equations are great. You use the core props everywhere it says shear and the skins props where it is tension.

Pretty much a summary of Rourke and Young, they just get in to the complicated examples more...

I would probably try to model it in FEA as a u shaped beam with the laminae properties and an an element of sorts down the middle that is the right compression and tension but only surface mated on the opposing vertical surfaces. you should be able to see tension and compression peaks in the joining and use those to determine if the core or bondline is compromised.

 

[RPstress]

To reply to an earlier query by kribri on how to handle compound sections without using the width-factoring method:

A very short introduction to general handling of compound sections.
[tt]
t1
__
| |
| |
| |
| | d1
Matl. | |<---Matl. with E1
w. E2 | |______________________NA_______
| | | |
____V____|__|_________ |-------distance to NA, Yx1
X1---|______________________| t2 ---------------'--X1#
d2

# Using centreline of flange 2 saves time calculating the 1st mom area, compared with using X1-X1 = the bottom surface or similar.


EA = E1.d1.t1 + E2.d2.t2

E.1st Mom area_X1-X1 = E1.d1.t1.(d1+t2)/2

Yx1 = E.1st Mom area_X1-X1 / EA = E1.d1.t1.(d1+t2)/2 / (E1.d1.t1 + E2.d2.t2)

So for properties about the neutral axis:

EIna = E1 . (d1^3.t1/12 + d1.t1.((d1+t2)/2-Yx1)^2) + E2 . ([t2^3.d2/12 +]## d2.t2.Yx1^2)

Yt = d1+t2/2 - Yx1 yb = Yx1 + t2/2


## the [] term is often ommitted as a first approximation.


For applied Mx (about any horizontal axis)

ft = E1.Mx.Yt/EIna

fb = E2.Mx.Yb/EIna

(straint = Mx.Yt/EIna, strainb = Mx.Yb/EIna) ###


For deflection as in simply supported beam under central point load P

delta = 1/48.P.L^3/E/I = 1/48.P.L^3/EIna ###


### Sort the signs out yourself.
[/tt]

Just remove all the E1 and E2 and you have the formulas for a section with one material.

The use of stiffness with section properties is completely generalisable to all the familiar formulas for sections, provided E tension = E compression.

If Et <> Ec then it's a non-linear problem. For straightforward sections where the NA is in line with one of the section's axes of symmetry this can usually be solved by trial and error.

NB: I have made only simple checks of this, but errors aside I hope the principle is reasonably clear.