It is clear that both Euler-Bernoulli and Timoshenko beam theories estimate the same vertical displacement as the height of the beam tends to 0.
However, depending on the boundary conditions (supports of the beams), both methods may estimate different or the same shear forces V(x) and bending moments M(x); this is because the values of the integration constants C1 and C2 depend on those boundaries.
In this sense, under which conditions both methods estimate the same shear forces V(x) and bending moments M(x)?
The picture below shows the fundamental equations of Timoshenko beam theory.
I don't know if I have understood your question properly. Let me try and answer based on my understanding of this question.
I believe that for a cantilever beam subjected to a pure bending couple at any point on the beam, the Euler-Bernoulli beam and the Timoshenko beam would give the same displacements, shear force ( zero) and same moment.
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