# An Analytical Solution for Block Toppling Failure of Rock Slopes during an Earthquake
## 3 An Application Research on an Anti-Dip Rock Slope Model under Shake Table Test
Figure 6: The large scale shake table test on anti-dip layered slope model: (**a**) the layout of acceleration sensors on the surface and the failure status of the slope model (modified based on [26]); (**b**) the acceleration amplification coefficients increase with slope height on the slope surface (data from [26]). 3. As the inputting acceleration (horizontal direction) increased to 0.4 g, the toppling deformation was observed obviously and continued to develop noticeably around the slope crest. 4. As the inputting acceleration (horizontal direction) increased to 0.7 g, shear sliding deformation initiated in the slope, i.e., a curved-linear sliding surface was observed. 5. As the inputting acceleration (horizontal direction) increased to 0.8 g, the sliding surface propagated to nearly go through the crest area and some cracks parallel to the sliding surface were observed in the slope (Figure 6a). An equilibrium slope model was built based on the shake table test as shown in Figure 7. The adjacent area around the slope crest is determined for the equilibrium model according to the observed deformation blocks in the physical experiment (the dashed white line in Figure 6a as the overall block base). There are 26 blocks in total and the geometric parameters are shown in Table 2. It should be mentioned that the \(J_{c}\) of the base interface is estimated approximately as 0.51 to be equivalent to the joint cohesion (about 0.51 times of block cohesion). The original model before the experiment would be analyzed firstly following the method described in the previous section (Figure 5) and also by the following steps:  
Step 1. Calculating all the sizes needed using Equations (A1)-(A11); the results are shown in the 2 nd to 4 th columns in Table 3.  
Step 2. Determining the physical parameters of blocks and joints, which were described above, and the block weight \(W_{n}\) which can be obtained from the 5 th column in Table 3.  
Step 3. Calculating \(P_{n-1,s}\) and \(P_{n-1,t}\) from the top block downwards on the basis of Equations (4) and (11) with \(k_{1}=0\) and \(k_{2}=0\) for the static state. The results are shown in the 6 th and 7 th columns in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\Delta x\) (m) & 0.04 & \(\psi_{f}\) (\({}^{\circ}\)) & 40 \\ \(a_{1}\) (m) & 0.007 & \(\psi_{p}\) (\({}^{\circ}\)) & 30 \\ \(a_{2}\) (m) & 0.0135 & \(\psi_{b}\) (\({}^{\circ}\)) & 28 \\ \(b\) (m) & \(-0.0015\) & \(\psi_{s}\) (\({}^{\circ}\)) & 11 \\ \(J_{c}\) & 0.51 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The geometric parameters in the equilibrium slope model of shake table test (data based on [26]). Figure 7: The limit equilibrium model for physical model on the shake table test  
Similarly, the limit equilibrium analysis for the slope under various dynamic loads is also carried out according to the steps described above. It should be noted that the acceleration amplification coefficient is also considered, i.e., 1.5 for the horizontal direction and 1 for the vertical direction. Furthermore, \(k_{1}\) and \(k_{2}\) are converted from \(k_{x}\) and \(k_{y}\) according to slope angle. Some critical sheets with horizontal accelerations of 0.3 g, 0.4 g, 0.7 g and 0.8 g are shown in Table 4. The corresponding FOS and the number of shearing and tensile blocks under varied accelerations are shown in Figure 8. The progressive deformation process derived by the presented method is detailed as follows. 1. The toppling deformation initiates at the dynamic loads of \(k_{x}=0.3\) g and \(k_{y}=0\), i.e., there is one toppling block near the slope crest (Table 4a and Figure 8). 2. As the accelerations (both horizontal and vertical direction) increased to 0.3 g, two toppling blocks were estimated (Table 4b). 3. As the acceleration (horizontal direction) increased to 0.4 g, four toppling blocks was estimated around the slope crest (Table 4c and Figure 8). 4.