## 4 Mixed Formulation of Lower Bound Limit Analysis
### Equilibrium Constraint on the Interface between Block Elements and Finite Elements  
In this study, a mixed numerical discrete method is employed to simulate the mechanical behavior of a jointed rock slope while a fracture of the rock bridges is considered. The rigid block element and the triangular finite element are respectively used to discretize the rock blocks and rock bridges. According to the lower bound theorem, a statically admissible discontinuity permits the tangential stress to be discontinuous at the interfaces between the rigid block elements and finite elements, but this requires that the continuity of the corresponding normal and shear components be preserved. Figure 5 describes the force of interaction on an interface \(L_{j}\) between a rigid block element and several triangular finite elements. The interface should meet the following equilibrium constraint:  
\[N_{j}=\int_{L_{j}}\sigma_{n}^{j}dL,\ V_{j}=\int_{L_{j}}\tau_{s}^{j}dL \tag{18}\]  
where \(N_{j}\) is the normal force applied on interface \(L_{j}\) for the rigid block element \(B_{j}\); \(V_{j}\) is the shear force applied on the interface \(L_{j}\) for the rigid block element \(B_{j}\); \(\sigma_{n}^{j}\) is the normal stress in the \(n_{j}\) direction on the interface for finite elements; \(\tau_{s}^{j}\) is the shear stress in the \(s_{j}\) direction on the interface for finite elements; \(L_{j}\) is the length of the interface between the rigid block element \(B_{j}\) and finite elements. Figure 5 illustrates one block element and a few finite elements sharing an edge \(L_{j}\). The edge is divided into \(n_{j}^{e}\) triangular finite elements on the side of the rock bridge, and there are \(2n_{j}^{e}\) nodes in contact with the block element \(B_{j}\). Because the node stresses vary linearly in each finite element, Equation (18) can be written as:  
\[N_{j}=\sum_{i=1}^{n_{j}^{e}}\frac{\sigma_{n1}^{j/i}+\sigma_{n2}^{j/i}}{2}l_{i },\ V_{j}=\sum_{i=1}^{n_{j}^{e}}\frac{\tau_{s1}^{j/i}+\tau_{s2}^{j/i}}{2}l_{i} \tag{19}\]  
where \(j=(1,\cdots,n_{j})\), and \(n_{j}\) is the interface number between the rock blocks and finite elements on the entire rock slope; \(n_{j}^{e}\) is the number of triangular finite elements on interface \(L_{j}\); \(l_{i}\) equals the length of the interface between finite element \(E_{i}\) and the rigid block element \(B_{j}\); \((\sigma_{n1}^{j/i},\sigma_{n2}^{j/i})\) is the normal stress in the \(n_{j}\) direction of node 1 and node 2 of the finite element \(E_{i}\) on the interface, respectively; \((\tau_{n1}^{j/i}\), \(\tau_{n2}^{j/i})\) is the shear stress in the \(s_{j}\) direction of node 1 and node 2 of the finite element \(E_{i}\) on the interface, respectively. The equilibrium constraints of the interface can be expressed in matrix form:  
\[\text{A}^{j}\sigma^{j}=\text{B}^{j} \tag{20}\]  
with  
\[\text{A}^{j}=\left[\text{A}_{1}^{j}\quad\cdots\quad\text{A}_{i}^{j}\quad\cdots \quad\text{A}_{n_{j}^{j}}^{j}\right];\]  
\[\text{A}_{i}^{j}=\frac{1}{2}l_{i}\big{[}\text{T}^{j}\quad\text{T}^{j}\big{]};\]  
\[\sigma^{j}=\left[\sigma_{1}^{j\,\text{T}}\quad\cdots\quad\sigma_{1}^{j\,\text {T}}\quad\cdots\quad\sigma_{n_{j}^{j\,\text{T}}}^{j}\right]^{\text{T}};\]  
\[\sigma_{i}^{j}=\left[\sigma_{x1}^{j/i}\quad\sigma_{y1}^{j/i}\quad\sigma_{x \text{y}1}^{j/i}\quad\sigma_{x2}^{j/i}\quad\sigma_{y2}^{j/i}\quad\tau_{xy2}^{ j/i}\right]^{\text{T}};\]  
\[\text{B}^{j}=\left[N_{j}\quad V_{j}\right]^{\text{T}};\,\text{T}^{j}=\left[ \begin{matrix}\sin^{2}\theta_{j}&\cos^{2}\theta_{j}&-\sin 2\theta_{j}\\ -\frac{1}{2}\sin 2\theta_{j}&\frac{1}{2}\sin 2\theta_{j}&\cos 2\theta_{j} \end{matrix}