## 2 Pseudo-Dynamic Analysis Method of Bedding Slope Stability
### Determination of Forces of Slip Block
#### 2.3.2 Determination of Horizontal Seismic Inertia Force of Slip Block  
In the present study, the coordinate origin of the calculated model is located at the toe of the slope, as shown in Figure 1. It is important to note that the free surface at the top of the slope is defined as the coordinate origin in the pseudo-dynamic method, thus the value  
Figure 2: Analytical model of bedding slope under the seismic inertia forces. of \(z\) needs to be converted with the transformation of coordinate in Figure 2. To sum up, the total horizontal seismic forces of the whole slip body can be written as  
\[ \begin{split}\sum\limits_{i}^{n}E_{\text{hi}}&=\int_{H- h_{n}}^{H}a_{\text{h}}(z,t)\text{d}m(z)\\ &=\int_{H-h_{n}}^{H}\bigg{\{}\gamma[(z-b_{n})\cot\beta_{n}-z\cot \alpha]\Big{[}1+\frac{z}{H}(f-1)\Big{]}k_{\text{h}}\sin\left[\omega(t-\frac{z} {v_{\text{s}}})\right]\bigg{\}}\text{d}z\\ &=\gamma k_{\text{h}}\Bigg{\{}\begin{array}{c}m_{1}\sin\left[ \omega\bigg{(}t-\frac{H}{v_{\text{s}}}\bigg{)}\right]+m_{2}\sin\left[\omega \bigg{(}t-\frac{H-h_{n}}{v_{\text{s}}}\bigg{)}\right]\\ +m_{3}\cos\left[\omega\bigg{(}t-\frac{H-h_{n}}{v_{\text{s}}}\bigg{)}\right]+m _{4}\cos\left[\omega\bigg{(}t-\frac{H}{v_{\text{s}}}\bigg{)}\right]\end{array} \Bigg{\}}\end{split}  \tag{8}\]  
where the coefficients \(m_{1}\)-\(m_{4}\) can be expressed as follows:  
\[ \begin{split}\left\{\begin{array}{c}m_{1}=(\cot\beta_{n}-\cot \alpha)\frac{v_{\text{s}}^{2}}{\omega^{2}}+\frac{f-1}{H}[(2H-h_{n})\cot\beta_{n }-2H\cot\alpha]\frac{v_{\text{s}}^{2}}{\omega^{2}}\\ m_{2}=-(\cot\beta_{n}-\cot\alpha)\frac{v_{\text{s}}^{2}}{\omega^{2}}-\frac{ f-1}{H}\{[2(H-h_{n})-b_{n}]\cot\beta_{n}-2(H-h_{n})\cot\alpha\}\frac{v_{\text{s}}^{2}}{ \omega^{2}}\\ m_{3}=-(\cot\beta_{n}-\cot\alpha)(H-h_{n})\frac{v_{\text{s}}}{\omega}+b_{n} \cot\beta_{n}\frac{v_{\text{s}}}{\omega^{2}}+2\frac{f-1}{H}(\cot\beta_{n}- \cot\alpha)\frac{v_{\text{s}}^{2}}{\omega^{3}}\\ -\frac{f-1}{H}\Big{\{}\Big{[}(H-h_{n})^{2}-(H-h_{n})b_{n}\Big{]}\cot\beta_{n} -(H-h_{n})^{2}\cot\alpha\Big{\}}\frac{v_{\text{s}}}{\omega}\\ m_{4}=(\cot\beta_{n}-\cot\alpha)H\frac{v_{\text{s}}}{\omega}-b_{n}\cot\beta_{n }\frac{v_{\text{s}}}{\omega}+\frac{f-1}{H}\big{[}(H^{2}-Hb_{n})\cot\beta_{n}- H^{2}\cot\alpha\big{]}\frac{v_{\text{s}}}{\omega}\\ -2\frac{f-1}{H}(\cot\beta_{n}-\cot\alpha)\frac{v_{\text{s}}^{2}}{\omega^{3}} \end{array}\right.\end{split}  \tag{9}\]  
Similarly, the total vertical seismic forces of the whole slip body can be written as  
\[ \begin{split}\sum\limits_{i}^{n}E_{\text{vi}}&=\int_{H- h_{n}}^{H}a_{\text{v}}(z,t)\text{d}m(z)\\ &=\int_{H-h_{n}}^{H}\bigg{\{}\gamma[(z-b_{n})\cot\beta_{n}-z\cot \alpha]\Big{[}1+\frac{z}{H}(f-1)\Big{]}k_{\text{v}}\sin\left[\omega(t-\frac{z} {v_{\text{p}}})\right]\bigg{\}}\text{d}z\\ &=\gamma k_{\text{v}}\Bigg{\{}\begin{array}{c}m_{5}\sin\left[ \omega\bigg{(}t-\frac{H}{v_{\text{p}}}\bigg{)}\right]+m_{6}\sin\left[\omega \bigg{(}t-\frac{H-h_{n}}{v_{\text{p}}}\bigg{)}\right]\\ +m_{7}\cos\left[\omega\bigg{(}t-\frac{H-h_{n}}{v_{\text{p}}}\bigg{)}\right]+m _{8}\cos\left[\omega\bigg{(}t-\frac{H}{v_{\text{p}}}\bigg{)}\right]\end{array} \Bigg{\}}\end{split}  \tag{10}\]  
where the coefficients \(m_{5}\)-\(m_{8}\) can be expressed as follows:  
\[