# Deflection of a composite beam of two materials

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.

• 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? – Floris Sep 18 '17 at 11:49
• @Floris It is really just two layers. I hope I did not misuse 'composite' – Bernhard Sep 18 '17 at 12:00
• 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? – Floris Sep 18 '17 at 12:01
• @Floris Yes, they are simple rectangular. – Bernhard Sep 18 '17 at 13:03

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 $w' = \frac{E_B}{E_A}w$: