# An Analytical Solution for Block Toppling Failure of Rock Slopes during an Earthquake
## 1 Introduction  
Toppling failures always occur in rock masses with a dominant discontinuity set (usually bedding or foliation) which strike parallel to the slope and dip inwards. This failure mode is one of the most common seen in natural or excavated rock slopes, e.g., rock slopes induced by construction including of hydropower stations [1; 2; 3], open-pit mines [4; 5], railway [5] and other engineering works [6]. In recent years, quite a number of toppling failures occurred during earthquakes and caused huge losses to lives as well as infrastructures. For example, Xiaojianping landslide and Guantan landslide [7] were both triggered by the Wenchuan earthquake and characterized by an anti-dipping structure (Figure 1). This type of event always occurs abruptly and results in catastrophic loss, e.g., a dammed lake with deposits of about 6,000,000 m\({}^{3}\) caused by Xiaojianping landslide. Toppling failure of rock slopes can be roughly differentiated into three modes: block toppling, flexural toppling and combined block and flexural toppling [8]. Numerical approaches, physical models and theoretical analysis are three main tools to study the toppling rock slopes. This study concentrates on the theoretical analysis on the basis of limit equilibrium method. The quantitative limit equilibrium method to evaluate the stability of block toppling rock slopes was originally proposed by [8], after which this method has been largely further developed. [PERSON] (1983) [9] gave a design chart for rock slopes susceptible to toppling based on the original analytical solution. [PERSON] (1999) [10] presented an analytical method for the toppling failure considering the toppling mass as continuous medium rather than a discrete assemblage of blocks. [PERSON] et al. (2001) [11] derived an analytical solution also considering the infinitesimal thickness of blocks and concluded that this method is suitable for slopes higher than 20-30 times the average block thickness. [PERSON] et al. (2005) [1] made an extension of the original solution considering the connectivity of block base. [PERSON] et al. (2008, 2010) [12; 13] extended the continuum solution developed by [10; 11] and presented an analytical approach for analyzing block toppling of those slopes that can be considered as continua. Considering rock layers as cantilever beams, [PERSON] and [PERSON] (1992) [14] proposed a limit equilibrium stability analysis method for slopes and underground openings under various loading conditions against the flexural toppling. [PERSON] et al. (1997) [15] adopted this limit equilibrium approach to analyze a centrifuge test data and found this method can accurately predict the failure load for the tests. [PERSON] and [PERSON] (2011) [16] conducted studies on flexural toppling considering the length of cracks. They presented an equation for length of critical crack, and stated that if the length of natural crack is smaller than that of critical crack, the solid mechanics should be adopted, while otherwise, the fracture mechanics are more suitable. There are few reports about limit equilibrium on the combined block and flexural toppling mode. It is worth mentioning that seismic loads were taken into account in some new developed analytical solutions in recent years. For example, [PERSON] and [PERSON] (2013) [17] presented a 3D failure mode chart for toppling and sliding failure of slope, in which the seismic load is considered as a pseudo-static horizontal force acting at the centroid. However, just one block is considered in this analysis. [PERSON] et al. (2014) [18] proposed a solution for toppling failure of rock slopes subjected to earthquake, with the slenderness ratio of the block over 20. However, the joint persistence of the  
Figure 1: XiaoJianping landslide induced by Wenchuan earthquake on 12 May 2008.  
block base is not considered in their solution. [PERSON] et al. (2015) [19] investigated the rocking and overturning response of slender rock blocks on a stiff base submitted to simple cyclic excitations and to natural earthquakes.