# An Analytical Solution for Block Toppling Failure of Rock Slopes during an Earthquake
## 2 Mechanical Model and Analytical Solution to Toppling Slope Failure  
Similar to [1; 8; 12], the geometry of a toppling slope was simplified as shown in Figure 2. The variables and the corresponding description are addressed in the Abbreviations section at the end of this paper. The mechanical parameters including cohesions, friction angles of joints or bedding planes and tensile strength of blocks can be obtained based on lab or field experiments. Some of the basic geometric parameters can be obtained through field work, such as \(\psi_{b}\), \(\psi_{p}\), \(\psi_{s}\), \(\psi_{d}\), \(\psi_{f}\) and \(\Delta x\) can be obtained based on field work. The equations to reach other geometric parameters including \(a_{1}\), \(a_{2}\), \(b\) and \(y_{n}\) can be addressed in Appendix A. It is shown that the interface between deformation blocks and the base is stepped, and its overall dip \(\psi_{b}\) can be valued in the range shown in Equation (1) [8; 14; 23]. Figure 2: The geometric model of a toppling slope (modified based on [22] after [8]). \[\psi_{b}\approx\left(\psi_{p}+10^{\circ}\right)\ to\ \left(\psi_{p}+30^{\circ}\right) \tag{1}\]  
The model includes stable and unstable blocks, the latter of which can be divided into toppling blocks and sliding blocks according to the deformation mechanism. The mechanical conditions of the \(n\)th toppling block are interpreted in Figure 3. It can be seen that there are normal and shear forces (\(P_{n}\), \(Q_{n}\)) from the upper block on the right side, (\(P_{n-1}\), \(Q_{n-1}\)) from the lower block on the left side, and (\(R_{n}\), \(S_{n}\)) from the matrix on the base. The approach to reach the geometric parameters \(M_{n}\), \(L_{n}\) above, at and below the crest is addressed in Appendix A. The seismic load acting on the block can be broken down into two components, i.e., \(k_{1}\) and \(k_{2}\). The equilibrium equation for sliding of the \(n\)th block can be given as (Figure 3b)  
\[R_{n}=W_{n}\cos\psi_{p}+(P_{n}-P_{n-1})\tan\phi_{js}-k_{2}W_{n}/g \tag{2}\]  
\[S_{n}=J_{c}R_{n}\tan\phi_{jb}+(1-J_{c})(R_{n}\tan\phi_{b}+c_{b}\Delta x) \tag{3}\] \[=W_{n}\sin\psi_{p}+(P_{n}-P_{n-1})+k_{1}W_{n}/g\]  
So the solution of the force needed from the lower block to keep the \(n\)th block from sliding can be obtained as follows:  
\[P_{n-1,s}=P_{n}-\frac{\left[\sin\psi_{p}+k_{1}/g-\left(J_{c}\tan\phi_{jb}+ \xi\tan\phi_{b}\right)\left(\cos\psi_{p}-k_{2}/g\right)\right]W_{n}-\xi c_{b} \Delta x}{\left(J_{c}\tan\phi_{jb}+\xi\tan\phi_{b}\right)\tan\phi_{js}-1} \tag{4}\]  
The equilibrium equation for toppling of the \(n\)th block can be given as (Figure 3c):  
Figure 3: The mechanical condition for the \(n\)th block with non-persistent joint on the base under earthquake (modified based on [22] after [8]). (**a**) External forces from adjacent blocks acting on the \(n\)th block; (**b**) The limit equilibrium condition of the \(n\)th toppling block; (**c**) The limit equilibrium condition of the \(n\)th sliding block. \[ \begin{array}{l}P_{n-1}L_{n}+W_{n}\big{(}\cos\psi_{p}-k_{2}/g\big{)}\Delta x/2+P_{n }\tan\phi_{js}\Delta x+M_{b}\\ =P_{n}M_{n}+W_{n}\big{(}\sin\psi_{p}+k_{1}/g\big{)}y_{n}/2+R_{n}\varepsilon \Delta x/2\end{array}  \tag{5}\]  
The bases of toppling blocks are acted on with normal forces \(R_{n}\), shear forces \(S_{n}\) and moment \(M_{b}\) (Figure 3b). It should be noted that the effects of other forces have been incorporated in the three values. The block base can be seen as an eccentric compression bar (Figure 4), so the stress in the base section induced by \(M_{b}\) as  
\[\sigma_{M}=\frac{M_{b}x}{I_{y}} \tag{6}\]  
in which \(x\) denotes distance to a neutral point, and \(I_{y}\) denotes the product of inertia of the cross-section to the \(y\) axis, \(I_{y}=\frac{1}{\Omega}\big{(}\xi\Delta x\big{)}^{3}\), \(\xi=1-I_{c}\).