# 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  
A lot of research has been conducted on the dynamic response of rock slopes since the Wenchuan earthquake on 12 May 2008, e.g., the centrifuge test [24; 25] and shake table test [26]. The latter one will be adopted to put forward a comparative study with the presented analytical solution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Calculating Results** & \(P_{n-1}\) & **Deformation Status** \\ \hline \(P_{n-1,t}<0\) & \(P_{n-1,s}<0\) & - & 0 & stable \\ \(P_{n-1,t}>0\) & \(P_{n-1,s}<0\) & - & \(P_{n-1,t}\) & toppling \\ \(P_{n-1,t}>0\) & \(P_{n-1,s}>0\) & \(P_{n-1,t}>P_{n-1,s}\) & \(P_{n-1,t}\) & toppling \\ \(P_{n-1,t}<0\) & \(P_{n-1,s}>0\) & - & \(P_{n-1,s}\) & sliding \\ \(P_{n-1,t}>0\) & \(P_{n-1,s}>0\) & \(P_{n-1,s}>P_{n-1,t}\) & \(P_{n-1,s}\) & sliding \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Determination of the \(P_{n-1}\) and deformation status of the \(n\)th block. Figure 5: The flow chart to use the presented limit equilibrium method in sheet. The experiment was conducted using the shake table in the Institute of Engineering Mechanics China Earthquake Administration, which is 5 m \(\times\) 5 m; maximum horizontal and vertical accelerations are 1.0 g and 0.8 g respectively. The size of the model box is 3.6 m (length) \(\times\) 1.6 m (width) \(\times\) 1.8 m (height). The height of the physical slope model is 1.6 m, the slope angle is 40\({}^{\circ}\), and the anti-dip angle of the layer is 60\({}^{\circ}\). The model is built from lots of small blocks of 20 cm (length) \(\times\) 5 cm (width) \(\times\) 4 cm (height) (Figure 6a). The density of the blocks is 2.51 g/cm\({}^{3}\), the tensile strength \(\sigma_{bt}\) is 0.094 MPa, elastic modulus is 138.74 MPa, cohesion \(c_{b}\) is 0.711 MPa and internal friction angle \(\theta_{b}\) is 36.2\({}^{\circ}\). The friction angle of block side joints is 30\({}^{\circ}\). The friction angle and cohesion of the block base joints are 30\({}^{\circ}\) and 0.35 MPa respectively [26]. All these parameters have an influence on the analytical results in the presented limit equilibrium method, so they are strictly following the real case to verify the practicability of the analytical method. Sine waves with various frequency (5, 10, 15, 20 Hz), acceleration amplitude (0.1 g, 0.2 g,..., 0.8 g) and duration (10, 20, 30, 40 s) were input to study the dynamic response of the anti-dip slope. Rubber materials are used to eliminate the boundary effects. Acceleration sensors were laid out in horizontal and vertical lines to record the acceleration during the experiment in different positions, some of which are on the surface as shown in Figure 6a. A high speed camera was also used to record the progressive failure process. The results of acceleration data show that the acceleration enlarges with elevation in the vertical direction and also enlarges near slope surface in the horizontal direction. This phenomenon has been observed by extensive researches (e.g., [27; 28; 29]) and known as amplification effects. In the shake table tests, it indicates that under low acceleration (<0.2 g), that the horizontal acceleration on the slope surface increased over 1.5 times above 3/4 slope height compared with that in the slope toe, while the vertical acceleration did not increase obviously in the same zone except at the top position (Figure 6b). The progressive failure process with the increasing incident wave was recorded and described as follows [26]:  
1. As the inputting acceleration (horizontal direction) increased to 0.3 g, some flexible toppling deformation initiated around the slope crest. 2. As the inputting accelerations (both horizontal and vertical direction) increased to 0.3 g, the number of toppling blocks increased a little and several blocks fell down.