# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 3 Slope Stability Analysis
### Slope Failure Criterion Suitable for the PRE-IE Method  
A rock slope with an inclined weak structural plane is a classic example in slope stability analysis. The slope model is shown in Figure 5: a vertical concentrated load \(F=1500\) kN is applied to the midpoint of a strip foundation, the angle between the weak structural plane and the horizontal direction is \(45^{\circ}\), the gravity of slope is not considered, and there are displacement constraints on the right and bottom of the slope. The material properties of the structural plane are cohesion \(c=200\) Pa, friction angle \(\varphi=0^{\circ}\), elasticity modulus \(E=1.0\times 10^{5}\) Pa, and Poisson ratio \(\
u=0.495\). It can be obtained from the calculation that the theoretical safety factor \(K^{*}=1.333\).  
The mesh model is shown in Figure 6. The slope is divided into rigid elements I and II, which are connected by an interface element with nine contact node pairs. Under the initial condition, all node pairs are in the stick mode. When the node pair changes to debonding or slip mode, the node pair is regarded as a failure. Using the PRE-IE method andthe SRM described above, we can obtain the rigid displacements of element I centroid and the number of failed node pairs under different reduction coefficients. The results are shown in Table 3 and Figure 7.  
It can be seen from Table 3 and Figure 7 that when the reduction coefficient increases from 1.0 to 1.333, the rigid displacement of the element I centroid is almost unchanged, and the number of failure node pairs is gradually increasing. When the reduction coefficient exceeds 1.333, the rigid displacement of the element I centroid changes dramatically, and all node pairs on the interface debond or slip. It can be seen in Figure 7 that when the Reduction coefficient is 1.333, there is an obvious displacement mutation in the X direction, and all node pairs fail. Therefore, we can draw a conclusion: when the reduction coefficient gradually approaches the theoretical safety factor, the contact node pairs on the interface will gradually slip or debond, and finally, all the node pairs will not stick while the slope failure occurs. Figure 8 is the Mises stress contour plot of the interface element when the reduction coefficient is 1.334. From Figure 8a, we can see that all the node pairs are in the stick mode before the iteration, and the stress is concentrated in the middle part of the interface element. After the nonlinear contact iterative method described in Figure 4 is used, all the node pairs change to the slip mode. Therefore, it can be seen from Figure 8b that the stress is modified and redistributed according to Table 2, which satisfies the criterion  
Figure 5: Rock slope model with an inclined weak structural plane.  
Figure 6: PRE-IE mesh model with 9 contact node pairs.  
of plastic zone penetration. Accordingly, the failure of all node pairs on the interface indicates that the infinitely developing plastic deformation appears on the slip surface. At the same time, there is no need to set convergence criteria and select nodes to observe the displacement mutation, which prevents the subjective effect on the safety factor. To sum up, the failure of all the contact node pairs is very suitable to be used as the slope failure criterion of PFE-IE method.