Abstract: This paper studies the bounding problems of the complete solution of limit analysis for a rigid-perfectly plastic medium, allowing for the discontinuity of plastic flow. A generalized variational principle involving conditions of the rigid-plastic interface and the discontinuous surface of a velocity field has been advanced for the mixed-boundary value problem. Based on this principle, a set of variational formulae of limit analysis is established. The safety factors obtained by these formulae lie between the upper and lower bounds obtained by the classical bounding theorems with the same kinematically and statically admissible field.Moreover, extended bounding theorems have been derived and proved, which hold a broader stress and velocity field than the statically and kinematically admissible field. The corollaries of these theorems indicate the relationship between the variational solution and the complete solution of limit analysis. Applications of these theorems show that a close approximation can be obtained by the proposed method with different admissible fields.