## 3 Analysis of Energy Response of Multi-Stage Loess Slope under the Action of Earthquake
### Energy Equation of Sliding Soil System  
From the analysis of Figure 6, it can be concluded that under the combined action of horizontal and vertical seismic excitation, sliding soil will move along the potential sliding surface after the acceleration of the sliding soil exceeds the yield acceleration, and the differential equation of motion is:  
\[-m(\ddot{x}_{g}+\ddot{x})\cos\theta-cl-\mu m\big{[}(g+a_{y})\cos\theta+(\ddot{ x}_{g}+\ddot{x})\sin\theta\big{]}+m(g+a_{y})\sin\theta=0 \tag{11}\]  
where \(m\) is the friction coefficient between the sliding soil mass and the lower soil mass, \(\mu\)= tan\(\varphi\). Equation (11) can be rearranged:  
\[(1+\mu\tan\theta)m(\ddot{x}_{g}+\ddot{x})+\frac{cl}{\cos\theta}+\mu m(g+a_{y}) -m(g+a_{y})\tan\theta=0 \tag{12}\]  
During the whole process of the earthquake, the soil of the slope has been in the energy field under the earthquake excitation, and the input, transformation and consumption of energy have been carried out and balanced. On the one hand, the soil is constantly transmitting energy; on the other hand, it continues to consume energy because of the displacement caused by motion. The differential equation of motion of the potential sliding  
Figure 11: Horizontal acceleration and positive critical acceleration time history. soil is established only in the limit equilibrium state, which can only reflect the energy state of one moment, and cannot represent the process of motion change process of potential sliding soil at other times in the duration range of earthquake. By quantifying the process of energy change in sliding soil in the energy field, the energy response equation of sliding soil under earthquake can be obtained [28; 29]. The absolute displacement of potential sliding soil \(x_{g}+x\) at both ends of Equation (12) is integrated in the earthquake duration range [0, \(t\)], and the absolute energy response equation of potential sliding soil is obtained:  
\[ \begin{array}{l}\int_{0}^{t}\left(1+\mu\tan\theta\right)m(\ddot{x}_{g}+\ddot {x})(\dot{x}_{g}+\dot{x})dt+\int_{0}^{t}\frac{cl}{\cos\theta}(\dot{x}_{g}+ \dot{x})dt+\\ \int_{0}^{t}\mu m(g+a_{y})(\dot{x}_{g}+\dot{x})dt-\int_{0}^{t}m(g+a_{y})\tan \theta(\dot{x}_{g}+\dot{x})dt=0\end{array}  \tag{13}\]  
Simplification can be obtained:  
\[ \begin{array}{l}\frac{1+\mu\tan\theta}{2}m(\dot{x}_{g}+\dot{x})^{2}+\Big{[} \frac{cl}{\cos\theta}+\mu m_{2}(g+a_{y})\Big{]}x-m(g+a_{y})x\tan\theta=\\ (1+\mu\tan\theta)\cdot\int_{0}^{t}m(\ddot{x}_{g}+\ddot{x})\dot{x}_{g}dt\end{array}  \tag{14}\]  
where \(\dot{x}_{g}\) is the velocity of the lower soil, \(x\) is the displacement of the potential sliding soil relative to the lower soil, that is, the permanent displacement of the potential sliding soil, \(\dot{x}_{g}+\dot{x}\) is the absolute velocity of the potential sliding soil, and \(\ddot{x}_{g}+\ddot{x}\) is the absolute acceleration of the potential sliding soil. Equation (14) is suitable for any time within the duration range of the earthquake; that is, the values of energy input and output of the whole sliding soil system are always balanced in the earthquake duration range. Figure 12 shows the acceleration of potential sliding soil and lower soil during a certain period of time when the vertical acceleration of the lower soil is not taken into account. During the time period of [0, \(t_{1}\)], the acceleration of the lower soil mass (the seismic acceleration) does not exceed the positive and negative critical acceleration of the potential sliding soil, and the potential sliding soil remains relatively static with the lower soil. The magnitude of the static friction force at the potential slip surface is equal to the seismic force at the moment, and the direction changes continuously with the direction of the seismic force to counteract the effect of the seismic force.