Nonlinear finite element models of functionally graded beams with power-law variation of material, accounting for the von-Karman geometric nonlinearity and temperature dependent material properties as well as microstructure dependent length scale have been developed using the Euler-Bernoulli as well as the first-order and third- order beam theories. To capture the size effect, a modified couple stress theory with one length scale parameter is used. Such theories play crucial role in predicting accurate deflections of micro- and nano-beam structures. A general third order beam theory for microstructure dependent beam has been developed for functionally graded beams for the first time using a modified couple stress theory with the von Karman nonlinear strain. Finite element models of the three beam theories have been developed. The thermo-mechanical coupling as well as the bending-stretching coupling play significant role in the deflection response. Numerical results are presented to show the effect of nonlinearity, power-law index, microstructural length scale, and boundary conditions on the bending response of beams under thermo-mechanical loads. In general, the effect of microstructural parameter is to stiffen the beam, while shear deformation has the effect of modeling more realistically as a flexible beam.