Propagation model of hard X-ray beams from partially coherent synchrotron source through different optical elements is proposed based on the coherent mode decomposition of Gaussian-Schell model and wave-front propagation. The fractional Talbot effect of phase gratings illuminated by focused partially coherent synchrotron radiation is simulated. Both the focusing beam′s intensity distribution and the coherence properties are obtained. And the self-imaging of gratings illuminated by defocused beam is analyzed. The influences of different factors on the shape of Talbot images at different fractional Talbot distances are analyzed through the comparison of the self-imaging fringes in the diffraction pattern of gratings illuminated by collimated beam. The diffraction images of rectangular phase gratings are submitted to a Fourier transform procedure, yielding the Fourier coefficients of different orders as a function of propagation distance, and the information on the lateral coherence of the beam is obtained.