Thedevelopment of vertical cracks in loess slopes often affects slope stability.Compared with the plane strain mechanism

the slope stability analysis underthe three-dimensional (3D) failure mechanism is closer to the actual slopeinstability. Based on the upper bound method of plastic limit analysis

different failure mechanisms (face failure

toe failure and base failure) of 3Dloess slope with pre-existing cracks are considered

the energy balanceequation and its dimensionless critical height expression/ are established

andthe upper bound solution of critical height is obtained by random searchmethod. The effects of constraint width

slope angle

internal friction angleand crack depth on the critical height of 3D vertical cracked loess slopes areanalyzed. The results indicate that for the toe failure mechanism

the criticalheight decreases with the increase of crack depth

and the increase in crackdepth no longer affects the critical height after reducing to the criticalcrack depth (/). The critical crack depth increases with the increaseof slope angle and decreases withthe increase of internal friction angle .When the constraint width /<0.8

most of the failure mechanism is of face failure.When the constraint width  /=0.8

internal friction angle =10°

and the constraint width /=0.6

internal friction angle =15°

the failure mechanism of the slope gradually transits fromthe face failure mechanism to the toe failure mechanism. The loess slope withvertical cracks has a smaller critical height than the intact slope. Theconstraint width and internal friction angle can affect the failure mechanismof 3D loess slopes.