If a material has different young modulus in tension than compression the neutral axis of the beam will shift from the section center of area. However the usual integrating over the beam without knowing in advance which part is under tension and which part under compression looks like a challenge.
You should seek the neutral axis first. There is one principle that two forces on two areas cutted by the neutral axis are opposite. You can depend on the principle to make programmes to find the neutral axis. In some condition, the neutral axis is not a linear line,may be a curve.
Thanks Yanli, In the lack of axial force the neutral axis location should stay constant. Do you know any closed form solution for different section profiles (circular, etc)?
The problem is that I calculate it as a function of (E_t/E_c) but it does not look reasonable and I want to verify with some source.
Why don't transform the section into a common homogeneous material (with one of the elastic modulii) and then continue to use Euler-Bernoulli Beam theory as usually. This issue can be found in any textbook on mechanics of materials
I just tried the following relation: the resultant moment of the internal stresses at a point O' (with position x1 on the length L0 of the beam) on the interface (in the bent specimen) is equal to the moment of the applied forces on the complementary part of the specimen with length (L0-x1) . The vertical position of the neutral line emerges that depends on the elastic constants, geometric moments of inertia of specimen section (2 different types of geometric moment of inertia, one with x2*x2ds and the other with x2ds; ds surface element at position x2 along the vertical direction in the specimen), radius of curvature R of the neutral line ... I have considered the problem of a rectangular bar bent by terminal transverse load
This looks like just what you are searching for:
http://www.scientific.net/AMM.94-96.1115
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