Abstract: The development of vertical cracks in loess slopes often affects slope stability. Compared with the plane strain mechanism, the slope stability analysis under the three-dimensional (3D) failure mechanism is closer to the actual slope instability. Based on the upper bound method of plastic limit analysis, different failure mechanisms (face failure, toe failure and base failure) of 3D loess slope with pre-existing cracks are considered, the energy balance equation and its dimensionless critical height expression γH/c are established, and the upper bound solution of critical height is obtained by random search method. The effects of constraint width, slope angle, internal friction angle and crack depth on the critical height of 3D vertical cracked loess slopes are analyzed. The results indicate that for the toe failure mechanism, the critical height decreases with the increase of crack depth, and the increase in crack depth no longer affects the critical height after reducing to the critical crack depth (δ /H)_min. The critical crack depth increases with the increase of slope angle β and decreases with the increase of internal friction angle φ. When the constraint width B/H<0.8, most of the failure mechanism is of face failure. When the constraint width  B/H =0.8 , internal friction angle φ =10°, and the constraint width B/H=0.6, internal friction angle φ =15°, the failure mechanism of the slope gradually transits from the face failure mechanism to the toe failure mechanism. The loess slope with vertical cracks has a smaller critical height than the intact slope. The constraint width and internal friction angle can affect the failure mechanism of 3D loess slopes.

Key words: slope instability, vertical cracks, failure mechanism, upper bound method