## 3 Kinematical Analysis Model of 3D Soil Slopes with Two Steps
### Internal Energy Dissipation
In general, the overall internal energy dissipation of the failure mechanism relatively easily deduced from the Equation (29) can be classified as four portions along the profile of the two-step slope: \(D_{\text{AB}}\), \(D_{\text{BC}}\), \(D_{\text{CD}}\), and \(D_{\text{DE}}\) are the corresponding energy dissipation rates on the planes represented by AB, BC, CD, and DE, respectively, and their 3D portions can be shown as  
\[D_{\text{AB-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{0}}^{\theta_{0}} \int_{0}^{x_{1}^{*}}r_{0}^{2}\ \sin^{2}\ \theta_{0}\frac{\cos\theta}{\sin^{3}\theta}\text{d}x \text{d}\theta \tag{30}\]  
\[D_{\text{BC-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{0}}^{\theta_{0}} \int_{0}^{x_{2}^{*}}r_{0}^{2}\ \sin^{2}\ (\theta_{\text{B}}+\beta_{1})\frac{\sin^{2}\theta_{0}}{\sin^{2}\theta_{0}} \frac{\cos(\theta+\beta_{1})}{\sin^{3}(\theta+\beta_{1})}\text{d}x\text{d}\theta \tag{31}\]  
\[D_{\text{CD-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{0}}^{\theta_{0}} \int_{0}^{x_{3}^{*}}(r_{0}\sin\theta_{0}+\xi_{1}H)^{2}\frac{\cos\theta}{\sin^ {3}\theta}\text{d}x\text{d}\theta \tag{32}\]  
\[D_{\text{DE-3D}}=-2\omega c_{t}\cot\varphi_{t}\int_{\theta_{0}}^{\theta_{0}} \int_{0}^{x_{4}^{*}}r_{0}^{2}e^{2(\theta_{\text{B}}-\theta_{0})\tan\varphi_{t }}\sin^{2}(\theta_{\text{h}}+\beta_{2})\frac{\cos(\theta+\beta_{2})}{\sin^{3} (\theta+\beta_{2})}\text{d}  
In accordance with the geometric features, the angles \(\theta_{B}\), \(\theta_{C}\), and \(\theta_{D}\) in Figure 2 can be deduced as  
\[\theta_{\rm B}=\arccos\frac{\cos\theta_{0}-\eta_{2}}{\sqrt{\sin^{2}\theta_{0}+ \left(\cos\theta_{0}-\eta_{2}\right)^{2}}} \tag{42}\]  
\[\theta_{\rm C}=\arccos\frac{\cos\theta_{0}-\eta_{2}-\xi_{1}\cot\beta_{1}\eta_{ 1}}{\sqrt{\left(\sin\theta_{0}+\xi_{1}\eta_{1}\right)^{2}+\left(\cos\theta_{0} -\eta_{2}-\xi_{1}\cot\beta_{1}\eta_{1}\right)^{2}}} \tag{43}\]  
\[\theta_{\rm D}=\arccos\frac{e^{\left(\theta_{h}-\theta_{0}\right)\tan\varphi_{ t}}\cos\theta_{h}+\xi_{2}\cot\beta_{2}\eta_{1}}{\sqrt{\left(e^{\left(\theta_{h}- \theta_{0}\right)\tan\varphi_{t}}\sin\theta_{h}-\xi_{2}\eta_{1}\right)^{2}+ \left(e^{\left(\theta_{h}-\theta_{0}\right)\tan\varphi_{t}}\cos\theta_{h}+\xi _{2}\cot\beta_{2}\eta_{1}\right)^{2}}} \tag{44}\]  
in which \(\eta_{1}=H/r_{0}\) and \(\eta_{2}=L/r_{0}\) are two dimensionless expressions, shown as  
\[\eta_{1}=e^{\left(\theta_{h}-\theta_{0}\right)\tan\varphi_{t}}\sin\theta_{h}- \sin\theta_{0} \tag{45}\]  
\[\eta_{2}=\cos\theta_{0}-e^{\left(\theta_{h}-\theta_{0}\right)\tan\varphi_{t}} \cos\theta_{h}-(\xi_{1}\cot\beta_{1}+\xi_{2}\cot\beta_{2}+\lambda_{a})\eta_{1} \tag{46}\]  
\(L\) is the horizontal line AB's length at the top of the slope, as illustrated in Figure 2. Consequently, the overall internal dissipation for two-step slopes after adding the plane insert section can be expressed in the following form  
\[ \begin{array}{ll}D&=D_{\rm AB-3D}+D_{\rm BC-3D}+D_{\rm CD-3D}+D_{\rm DE-3D}+D _{\rm insert}\\ &\qquad\qquad\qquad\qquad\qquad=\omega c_{t}\cot\varphi_{t}r_{0}^{3}(g_{1}+g_ {2})\end{array}  \tag{47}\]  
where \(g_{1}\) and \(g_{2}\) are dimensionless expressions for the internal dissipation of 3D section and 2D plane-strain section, respectively, given as  
\[ \begin{array}{ll}g_{1}=&-2\sin^{2}\theta_{0}\int_{\theta_{0}}^{\theta_{0}} \frac{\cos\theta}{\sin^{2}\theta}\sqrt{f_{2}^{2}-f_{3}^{2}}{\rm d}\theta\\ &-2\sin^{2}(\theta_{\rm B}+\beta_{1})\sin^{2}\theta_{\rm B}\int_{\theta_{0}}^{ \theta_{\rm C}}\frac{\cos(\theta+\beta_{1})}{\sin^{2}(\theta+\beta_{1})}\sqrt {f_{2}^{2}-f_{4}^{2}}{\rm d}\theta\\ &\qquad\qquad\qquad-2(\sin\theta_{0}+\xi_{1}\eta_{1})^{2}\int_{\rm cc}^{ \theta_{\rm D}}\frac{\cos^{3}\theta}{\sin^{3}\theta}\sqrt{f_{2}^{2}-f_{5}^{2} }{\rm d}\theta\\ &-2e^{2(\theta_{h}-\theta_{0})\tan\varphi_{t}}\sin^{2}(\theta_{\rm h}+\beta_{ 2})\int_{\theta_{0}}^{\theta_{\rm h}}\frac{\cos(\theta+\beta_{2})}{\sin^{3}( \theta+\beta_{2})}\sqrt{f_{2}^{2}-f_{6}^{2}}{\rm d}\theta\end{array}  \tag{48}\]  
\[