Shared loading on composite beam section

[StrEng007]

Do you guys agree with the following equation?

Scenario: You have a composite section of two different materials making a beam. The top half (above the geometric centroid) is material 1 & the bottom half (below the geometric centroid) is material 2. Considering equal deflections in both materials, an equation is written to proportion the distributed load based on the relative stiffness of each part.

From W_total, the loading on material 1 (w_matl1) = W_total / [1 + (E2*I2)/(E1*I1)]
and thus, the loading on material 2 (w_matl2) = W_total / [1 + (E1*I1)/(E2*I2)]

QUESTION: I've seen a text reference the equation above then proceed to use each distributed load to calculate the maximum bending stress in each material as though it was a single section beam (Mc/I relative to only the top or bottom section, independently). There was no modular transformation performed (n=E1/E2) or transformed moment of inertia calculated, which provides a very different answer from the typical transformed section method.

Also, what is your approach to finding the overall deflection of this composite section? The problematic equations above are based on equivalent deflections... what is the direct method to finding that overall deflection?

Do any of you know a reference that backs up this load distribution theory?

Thanks!

 

[wannabeSE]

The method described appears to be for a non-composite section. To find the deflection of these non-composite members, reference the 2nd premise in the scenario. Once the load to each member is determined, find the deflection of one member using it's portion of the total load. It can be checked by calculating the deflection of the other member and the result should be the same as the first.

 

[KootK]

I agree with WannabeSE. All signs point to your beam section being non-composite. Let us know if that`s not the case as the discussion is quite a bit more interesting with a truly composite shape.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[StrEng007]

I'm pulling this example from calculations provided for a beam. The engineer analyzes it as a "composite" section by stating that the dissimilar materials act together as one beam. This is why it bothers me that there is no modular transformation being done. The only benefit I see in doing the example this way is that the bending stresses for each beam are extremely low (which I do not agree with). However, the deflection is almost twice as large compared with a section that was truly composite.

Do you know of any references that expand further on this proportioning of loads through modulus and deflection?

 

[Agent666]

How are the two materials joined? If they are capable of transferring the longitudinal shear between then it will act in a composite manner?

 

[StrEng007]

There is no shear flow taken into account here. Ironically, the interface where these two members connect is right at the point of maximum shear of the cross section (ie in the middle where the N.A.) would be. My issue is whether or not its allowable to just assume that these two members act independently and take portions of the load (while not acting together as a composite section). I've just haven't seen examples where this was illustrated.

 

[StrEng007]

Simply put, can anyone point me to a reference where a beam is made of two separate parts, is non composite and the bending stresses of each individual cross section are calculated, while maintaining the theory that both members are deflecting the same amount?

Thanks!

 

[Agent666]

I would have thought so if the deflections are constrained and there is no connection between them to allow them to act composite. It's sort of like having two different depth steel beams beside each other sharing a common load. Load will be shared based on relative stiffness.

If they are connected, and modulus of elasticity differs the geometric centroid won't be the neutral axis of the combined section. Just pointing this out as you are making this assumption. For them to act independently you will get slip at the interface.

 

[Jerehmy]

if you have analysis software, model two beams like 2 ft apart(one above the other) with the appropriate cross section and material strength. Add in a link between the two beams every 0.5ft or so. Make sure the link is hinged both sides so it can only transmit axial load. Load up the top beam. Should give a pretty good analysis of your situation.

if you don't have analysis software, you can make some conservative estimations with the inverse of the deflection formulas. Otherwise you'll have to do some beam stiffness matrix analysis.