## 2 Influence of the Comprehensive Slope Rate on Permanent Displacement
### When the Slope Rate of Each Grade of Multi-Stage Slope Is Changed  
In order to explore the influence of the change in the slope rate of each grade on the permanent displacement of potential sliding soil of multi-stage loess slope under the action of earthquake, a three-stage loess slope model is established by using GEO-Studio geotechnical finite element software. The total height of the slope is \(H=\)30 m, the height of each slope is \(h_{1}=h_{2}=h_{3}=10\) m, the width of the platform is 3 m, the friction angle of soil is \(\varphi=15^{\circ}\), the unit weight of soil is \(\gamma=16.6\) kN/m\({}^{3}\), the slope rate of each grade is 1:0.75-1:1.25, and the slope as a whole shows a steep trend at the upper part and gentle at the lower part. Additionally, a total of 12 working conditions are set for simulation calculation when the comprehensive slope rate is 1:1.2; the specific working conditions are shown in Table 1.  
The A-1 analysis model is shown in Figure 2. The model soil parameters are loess-like silt, the linear elastic model is adopted, the yield criterion is the Mohr-Coulomb yield criterion, the model Poisson's ratio is \(v=0.3\), and the initial dynamic shear modulus is \(G_{max}=220\) Mpa. First, the model is analyzed by the initial static analysis. The boundary condition of the model is that the horizontal and vertical directions at the bottom are fixed, the horizontal directions on both sides are fixed, and the vertical directions on both sides are free. Then, the model is calculated and analyzed by equivalent linearity calculation and analysis. The boundary condition of the model is that the horizontal and vertical directions at the bottom are fixed, the vertical directions on both sides are fixed, and the horizontal directions on both sides are free.  
In the dynamic calculation of the model, the EI-Centrol seismic wave with a duration of 30 s and a peak acceleration of 0.2 g is selected for each working condition, which is input  
Figure 1: Result of seismic stability analysis of slope under A-1 working condition.  
Figure 2: Grid diagram of dynamic analysis and calculation model of multi-stage loess slope under A-1 working condition.  
at the bottom of the model. The boundary condition of the model is that the horizontal and vertical directions at the bottom of the model are fixed, the vertical directions on both sides of the model are fixed, and the horizontal directions on both sides of the model are free. Because it is defined that the permanent displacement is the horizontal displacement of the slope after experiencing the whole duration range of the earthquake when the safety factor is less than 1.2, the maximum horizontal displacement after the calculation of the model under each working condition is selected, which is the permanent displacement. The calculated permanent displacement of the slope under each working condition in Table 1 is shown in Figure 3.  
As can be seen from Figure 3, when the slope rate of each grade slope changes but the comprehensive slope rate remains unchanged, the minimum horizontal permanent displacement of multi-stage loess slope under the action of earthquake is 0.15784 m under A-1 working condition, and the maximum value is 0.17633 m under A-10 working condition. Additionally, the difference between the maximum permanent displacement and the minimum permanent displacement is 0.01849 m, which is only 10.486% of the maximum permanent displacement. According to the above analysis, it can be shown that it is feasible to use the comprehensive slope rate to calculate the permanent displacement under the condition that the slope rate of each grade changes, but the comprehensive slope rate remains unchanged. It can also be seen from Figure 3 that the maximum horizontal displacement is most directly related to the slope rate of the first-stage slope. When the first-stage slope rate remains unchanged, the difference in the maximum horizontal displacement is very small, such as the working condition of A-10-A-12.