## 3 Analysis of Energy Response of Multi-Stage Loess Slope under the Action of Earthquake
### Critical Acceleration of Potential Sliding Soil
#### 3.2.2 Negative Downward Critical Acceleration  
When the seismic acceleration changes from positive to negative, the velocity of the potential sliding soil relative to the lower soil gradually decreases until the seismic acceleration reaches the negative critical acceleration of the potential sliding soil. Then, the potentially sliding soil will be in limit equilibrium state about to slide upward under  
Figure 6: Calculation diagram of positive critical acceleration.  
seismic excitation, and the angle between its velocity \(v^{\prime}\) and the potential sliding surface is \(\beta\), and the direction is upward. The calculation diagram is shown in Figure 7.  
At this time, the power made by the external force \(\dot{w}_{0}\) and the dissipation power of the internal force \(\dot{w}_{i}\) are:  
\[\dot{w}_{0}=-m(g+a_{y})v^{\prime}\sin(\theta+\beta)+m(\ddot{x}_{g}+\ddot{x})v^{ \prime}\cos(\theta+\beta) \tag{8}\]  
\[\dot{w}_{i}=clv^{\prime}\cos\beta+csv^{\prime}\cos\beta+mv^{\prime}\big{[}(g+ a_{y})\cos\beta+\big{(}\ddot{x}_{g}+\ddot{x}\big{)}\sin\beta\big{]}\tan\varphi\cos\beta \tag{9}\]  
From \(\dot{w}_{0}=\dot{w}_{i}\)[23], the negative critical acceleration \(a_{\infty}(t)\) can be obtained when the potential sliding soil is in the limit equilibrium state, and it should be noted that the negative critical acceleration is a negative value:  
\[\begin{array}{rl}a_{\infty}(t)=&-\frac{cl\cos\beta+\cos\beta+m(g+a_{y})\cos \beta\tan\varphi\cos\beta}{m\cos(\theta+\beta)\cos\alpha-m\sin\theta\tan\varphi \cos\beta}\\ &-\frac{m(g+a_{y})\sin(\theta+\beta)}{m\cos(\theta+\beta)\cos\alpha-m\sin \theta\tan\varphi\cos\beta}\end{array} \tag{10}\]  
Influence of Inclination Angle of Slip Surface \(\theta\) and Soil Parameters on Positive and Negative Critical Acceleration  
Since the vertical acceleration of potential sliding soil is no longer a fixed value, it is a more complex value with the change in inclination angle and soil parameters when the inclination angle \(\theta\) and soil parameters (internal friction angle \(\varphi\) and cohesive force \(c\)) are changed. Therefore, the vertical acceleration \(a_{y}\) is not considered in the analysis of this section, which makes it equal to 0. The effects of the inclination angle of the potential slip surface \(\theta\) and the friction angle of the soil \(\varphi\) and cohesive force \(c\) of the soil on the critical acceleration of the potential sliding soil \(a_{c}\) are shown in Figures 8-10. In order to simplify the calculation process, the width of the potential sliding soil is 1 m. The geometric, physical and mechanical parameters of the slope are shown in Table 3.  
The Newmark slider displacement analysis method does not consider the negative critical acceleration, so it does not consider the reverse displacement.  
Figure 8: Influence of inclination angle of potential slip surface \(\theta\) on critical acceleration \(a_{c}\).  
Figure 9: Influence of internal friction angle of soil \(\varphi\) on critical acceleration \(a_{c}\).  
It can be seen from Figures 9 and 10 that the friction angle and cohesive force of soil have much less influence on the critical acceleration relative to the inclination angle of the potential sliding surface. It is proved that the energy dissipated by damping and soil deformation is much smaller than the energy dissipated by the motion of the potential sliding soil.