Context. The structure characteristics of the critically-rotating accretor in binaries are investigated in this paper, on the basis of the potential function including rotational and tidal distortions. 

Aims. Our aim is to investigate the structure of the accretor when the accreting star reaches the critical velocity. 

Methods. In this paper, we have implemented the prescription described by Kippenhahn & Thomas (1970, Proc. IAU Colloq., 4, 20) and Landin et al. (2009, A&A, 494, 209). 

Results. The traditional model merely included the hydrodynamical effect of rotation. When comparing this model with ours, we find that it is very necessary for the rapidly rotating accreting star to include the gravitational potentials from tides Psi(tide), and the distortions of the star resulting from rotation Psi(dis,rot).Furthermore, we find that the mean effective gravitational acceleration can be decreased in the distort model, and the star shifts towards low temperature and low luminosity. Rotation and tides can extend the convection zone below the surface, and reduce the convective core in the center of stars due to the Solberg-Hoiland criterion. Rotational distortions derived from Psi(dis,rot) can intensify the critical velocity whereas the tide force derived from Ilitide tends to reduce the critical velocity. Rapid rotation induced by mass transfer also causes the central temperature to decrease, and triggers efficient mixing which can significantly modify the H-profile.