Mean value of Second moment of area

[MarkusLAndersson]

Hi

I would like to know if someone knows hove to calculate the mean value of Second moment of area (I) for serial connected steel beams.

I my case I have one case with 4 beams, and one with 11, with different I and different length placed on top of each other and would like to calculate I for the entire beam.

If some one can help me I would be really glad

 

[rb1957]

parallel axis theorem ...
you can take each beam as a lumped area at their centroids with its MoI about its centroid,
then the neutral axis of the assembled beams is ...
ybar = sum(Ay)/sum(A)
and MoI = sum(I)+sum(Ay^2)-sum(A)*ybar^2

good luck

 

[Drej]

Not sure I understand your "mean" SMOA. Not sure what an "average" value for a section actually means (if that is what you mean). If you want to calculate I for the entire section, you can do so from first principals. It should be easy for this type of section, just use either one of the parallel/perpendicular axis theorems and treat the individual sections as plane areas.

I_PAXIS = I_CM + Ah² (parallel axis)

where:

I_PAXIS = SMOA about the parallel axis
I_CM = SMOA of the section about its centre of mass
A = area
h = distance to the centre of mass

The sum of the above for the entire section will give you the SMOA for the section.


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[IFRs]

If they are just "placed" on top of each other then just add their section properties together. If and only if they are connected together such that they will truly act as one member (welded or bolted enough) then use the methods above. If they are different lengths, you have varying cross sections and strengths. If the short beams are spliced end-to-end, the splices have to be able to take all loads.

 

[MarkusLAndersson]

I am not shore if this is what I want. In my case I have several beams standing on top of each other bolted together and acting as one. This one beam is then connected in one end to the ground (bolted).

For every section I have one Second moment of area, I1 I2 I3 and I4, and the height H1 H2 H3 and H4. These sections is made out of equal angle steel members

Do I approached this problem the wrong way or is there any other method to calculate the total I of the beam

 

[gwolf]

The total "I" for the beam has no analytical use that I know of - what were you planning to do with this value anyway?

 

[rb1957]

oh! ... you have a column, with varying I, and I guess you want the column allowable ?

 

[MarkusLAndersson]

Yes I want to calculate the total deflecton of the column it cant be more then H/200. For this I nead the total "I"

 

[rb1957]

it may be reasonable to average the I's over their lengths (sum(IL)/sum(L) ... but probably you thought of this already.

i think a hand calc will be very messy very quickly

an FE model should give you the answer. the euler load for a column is the first eigenvalue solution for the stiffness matrix, the following link describes this ...

http://www.cesup.ufrgs.br/ansys/guide_55/g-str/GSTR7.htm

 

[gwolf]

I have done these sorts of calcs for a step-tapered carbon model wing spar. I used a spreadsheet and Roarke to calculate cumulative deflection and angle at each station out to the end - quite easy realy.

You could also just try applying bending and deflection theory by hand for the four sections which is not too much hassle. Don't forget to consider the way the deflection angles add up along the length.

From memory, castigliano's theorem is useful for this sort of problem - but I haven't used it in years.

gwolf.

 

[rb1957]

if you've got a copy of bruhn, section C2.6 has a methodology

 

[UcfSE]

Are the columns of varying length arranged in a particular order, such as in order of length? If they are not, such as a very long one next to a very short one, then the long one will take more of the load than simply an "average" of the beams. Because the cross section varies, you need to evaluate the deflection at each change in section. If the beams do act as one unit and the depth of the beam is large compared to the length, you will need to account for the shear deformation as well. Try going back to your mechanics of materials, strength of materials or solid mechanics books and take a look at the procedures for dealing with non-prismatic members.