Modulus of compound beam member 1

[tmoe]

I have a built up beam and Im interested in calculating deflection under uniform loading conditions.
I have applied geometric principles to determine the compound moment of inertia, I, but I'm stumped on what modulus of elasticity, E, to use. I realize that the stiffer side members will absorb more of the load, but I'm unclear how to account for this with (as I call it) an equivalent modulus, E_eqv. Some sort of modular ratio perhaps?

your thoughts are appreciated.

 

[msquared48]

I would use E1I1 + E2I2 + E3I3 = EtIt, giving Et = 1.974

Mike McCann
MMC Engineering

 

[tmoe]

Thanks Mike, that reduces down to exactly the ratio I was looking for.

 

[BAretired]

I don't agree with the method suggested. The Parallel Axis Theorem can't be used in that way.

If three members are connected together adequately to act as a single unit, the transformed section must be determined. You can transform the middle piece to to the same material as the edge pieces. Then calculate the revised I_t based on the revised properties.

So I₁ = I₃ = 27
and I₂(transformed) = 1.6*I₂/2.8 = 0.57I₂.

I_t is found using Parallel Axis Theorem

Finally, EI(transformed) = I_t*E₁

Alternatively, the edge members may be transformed to the same material as the middle piece. Then I_t and E_t will be different but their product will be the same as found above.

If, on the other hand, the three members are nailed together but do not act as a composite section, then the Parallel Axis Theorem should not be used at all.
In that case EI(combined) = E₁I₁ + E₂I₂ +E₃I₃ and each member will carry load in proportion to its stiffness.

BA

 

[fegenbush]

BA got it this time. Very thorough answer.

 

[rb1957]

yep, "rule of mixtures" is what we call it ... transform the section to one E (make sure you're not changing the neutral axis when you do this. solve the beam; then transfrom the stresses by the ratio of the Es.

Quando Omni Flunkus Moritati

 

[tmoe]

Thanks all, appreciated.

 

[tmoe]

BA,

What type of connection would allow the members to behave as a true composite section rather than individually? You mentioned nailing would not. Why?

thanks

 

[paddingtongreen]

For a combined section, you need the strain rates to be uniform across the combined section. Nailing them at intervals will only make them deflect the same amount, but the strain rates will be different, thus they will not make a combined section.

Putting it differently, in the combined section, the compression stress in the top and bottom of the outer elements will not be equal. If they are simply made to deflect the same amount, they will be the same.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[BAretired]

BA,
What type of connection would allow the members to behave as a true composite section rather than individually? You mentioned nailing would not. Why?

I did not say nailing would not allow the members to behave as a composite section. But the usual number of nails used in wood assemblies is not enough to resist the horizontal shear stress at the interfaces.

It is conceivable that a combination of nailing or screwing and gluing could provide the necessary shear strength but the labor involved would be significant and I don't feel comfortable relying on field applied glue for structural purposes notwithstanding the fact that the quality of glue has improved considerably in recent years.

BA

 

[IDS]

The simplest way to calculate (assuming the connected elements will bend as a single piece) is to select a reference E value, then factor the width dimensions of any elements with a different E value by the modular ratio. The depth dimensions remain unchanged for all elements. The resulting I value can be used with the reference E to give EI.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

it's a fair question to ask how to practical make a section that the "rule of mixtures" applies to.

in practice it's an ideal analysis, it assumes perfectly rigid connection between the different components. in practice they are connected with a finite stiffness so there is some "slip" at the boundaries. ideally all elements are bending the same, in practice they bear against one another and transfer normal loads. i was trying to picture a situation where there'd be significant gapping (stiff member above a flexible one, flexible above stiff), it seems to me the most unideal setup is to have different elements beside one another.

at the end of the day, all the elements should deflect much the same. of course if they're all acting as independent members (as opposed to one composite assembly) then this deflection will be higher.

maybe the "better" analysis approach is to assume they're all individual elements; the first difficulty with this is calculating the bearing loads between the elements. then you'll be able to determine the amount of relative displacements at the boundaries, and then impose whatever stiffness is reasonably there. and iterate several times!

personally, i'll stick to the rule of mixtures.

Quando Omni Flunkus Moritati

 

[tmoe]

Great input everyone, it has really helped to clear things up for me.

Ultimately Im creating a spread spreadsheet for work, that will run this scenario.

I live in San Francisco and see a lot of remodels and additions to the old Victorian structures.

We are often sistering members onto existing beams that are or will be overloaded.

As of now I have run both scenarios in the spreadsheet ( rule of mixtures, and relative stiffness) and the results seem to make since, with the later scenario showing a bit more deflection.

Thanks again.