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When I encounter a single cantilever beam with a uniform load, I know what to do. When I know the uniform load, I can calculate the deflection of the end of the cantilever beam.

Now, I have a composite beam, consisting of two materials (the two materials are on top of eachother and the load is applied from the top). The materials have a different thicknes $d$ and a different Young's modulus, $E$. I am trying to find the deflection for this composite beam under a uniform load.

I do not have a very solid background in statics, so I am struggling a bit to find the deflection for this configuration. I've found some sources that seem relevant here and here, but I do not really manage to properly add the pieces of information I have now. I could go down the route with a transformation of the material via an equal axial stiffness, but this does not seem very convincing yet.

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  • $\begingroup$ The material transformation approach makes sense when you have a "matrix composite" like reinforced concrete, carbon fiber reinforced epoxy etc. Lots is elements of one or the other stiffness bonded together. If you have two dissimilar materials bonded together (layer A glued to layer B) that method is not really appropriate. Can you clarify how you use the word "composite" here? $\endgroup$ – Floris Sep 18 '17 at 11:49
  • $\begingroup$ @Floris It is really just two layers. I hope I did not misuse 'composite' $\endgroup$ – Bernhard Sep 18 '17 at 12:00
  • $\begingroup$ You didn't misuse it - but your second link relates to the "other kind" of composite so I was just clarifying. Do you have simple rectangular sections? $\endgroup$ – Floris Sep 18 '17 at 12:01
  • $\begingroup$ @Floris Yes, they are simple rectangular. $\endgroup$ – Bernhard Sep 18 '17 at 13:03
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There are some fairly useful lecture notes on the website of Missouri S&T. This shows how you could transform the problem you have into another simpler problem by changing the cross section of one of the materials. The key is that the analysis requires you to find the neutral axis, and that can be done easily and intuitively when the material is the same everywhere.

Assuming you start out with two rectangular sections A and B with Young's modulus $E_A$ and $E_B$, and with the same width $w$, you know that your equations (bending moment, strain) would be unchanged if you changed material A to material B while at the same time changing the width of A to $w' = \frac{E_B}{E_A}w$:

Cross section of beams

For a given strain, each slab of A' will produce the same stress as a slab of A did - so now the calculation has been simplified. We find the neutral axis by determining the center of the second moment of area, and after that the calculation can proceed as for any beam of non-uniform cross section.

I hope that is enough to get you going - the rest of the exercise should be straightforward, and will solidify your understanding. But ask for help if you still get stuck.

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  • $\begingroup$ Thanks, this is really useful. "Those portions of the cross section which were unaltered in the transformation process carry the same stresses on both the original and transformed sections". In other words, if I calculate the deflection (vertical direction in your sketch) for given uniform load, then this is the same in both situations. Did I understand correctly? $\endgroup$ – Bernhard Sep 18 '17 at 14:08
  • $\begingroup$ Yes, that's exactly what this method should allow you to do. $\endgroup$ – Floris Sep 18 '17 at 14:34

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