Composite Structure: Area Transformation Method

I've been able to use the "Area Transformation Method" as shown in MERM page 51-20 to solve various composite structure problems. However, I don't really understand the origins of the method - I can get started by assuming that the strain in the two materials, but I don't understand the part where the "transformed" area is used for the stress calculations.

Can anyone describe a derivation of this method?

The method aims to transform the entire composite structure into a single material one (the weakest). In order to do this, you transform the area of all the other materials into equivalent areas of the weakest material. For example, let's say the modulus of elasticity of steel is 1.5 times that of copper. This means that for the same strain (since the materials are perfectly bounded, the strain on each material can be assumed to be equal) 1 square foot of steel carries 1.5 the stress of 1 square foot of copper. Or 1 square foot of steel is equivalent to 1.5 square feet of copper which is called the transformed or the expanded area. The derivation comes straight from the stress/strain formula by equating the strain.

How did you deduce that the ratio of stresses is equal to the ratio of areas? i.e. "sigma-st over sigma-c equals A-c over A-st"