Based on the influence of nonsingular term in Williams expansion of the stress field at a crack tip on crack propagation, a theoretical solution of stress was proposed with considering the friction effect of crack surface and T-stresses. The modified maximum tangential stress criterion was used to analyze the effect of T-stresses on crack initiation and propagation under compression loading conditions. Through the comparison of the test results and the theoretical solutions under uniaxial compression conditions, the rationality of the theoretical solution was verified by considering the three T-stress components (T_x, T_y and T_xy). The results show that for a certain crack inclination angle β, both the maximum tangential stress σ_θmax and the maximum absolute value of the shear stress τ_rθmax at the crack tip decrease gradually with the increase of the friction coefficient f of the crack surface. Both σ_θmaxand τ_rθmax decrease first and then increase with the increase of the lateral pressure coefficient k. The possibility of shear crack is higher with larger k and β. The results also reveal that the T-stresses can be controlled to keep the crack growth direction away from the dangerous direction and the overall structure fracture can be effectively stopped or delayed.