## 1 Introduction  
Flexure toppling failure is widely found in anti-dip bedding rock slopes. According to [PERSON], the toppling failure modes of anti-dip bedding rock slopes can be divided into three categories: block toppling, flexure toppling, and block-flexure toppling, depending on the nature of the structural planes present [1]. To date, a large number of studies have been conducted on these three failure modes in anti-dip bedding rock slopes subject to static (self-weight) conditions. These include theoretical analyses [2, 3, 4], tests using models and, for example, centrifuges [5, 6, 7], and numerical analyses [4, 6, 7]. Therefore, our understanding of the stability (and mechanism responsible for the failure) of antichinal rocky slopes under static conditions is relatively mature. However, the dynamic response and stability of anti-dip bedding rock slopes under dynamic effects (e.g., seismic and blasting disturbance) are much more complicated compared to those under static conditions. The 8.0 \(M_{s}\) Wenchuan earthquake that occurred in 2008 induced more than 60,000 landslide hazards and caused more than 20,000 casualties [8]. The recently-initiated Sichuan-Tibet Railway is a major engineering project that includes 12 potential problem areas along its corridor, wherein the basic seismic intensity can reach up to 0.3 g or higher [9]. Therefore, the need to undergo engineering and construction tasks in areas with high seismic intensities, and also the need to generally prevent disasters occurring due to secondary geological hazards such as landslides, have led to new and more stringent requirements being placed on studies focused on the stability of rock slopes subjected to seismic action. Compared to bedding rock slopes, anti-dip bedding rock slopes are generally more stable under seismic action. However, once the seismic intensity reaches a certain critical value, insidious processes are set in motion that can lead to disaster and the scale of the instability may become enormous (deep failure). Therefore, it is of great engineering significance to study the failure mechanisms and dynamic responses of anti-dip bedding rock slopes under seismic action. At present, the stability of anti-dip bedding rock slopes subjected to dynamic action is mainly studied using pseudo-static methods, centrifugal model tests, and numerical calculation. The pseudo-static methods usually simplify the problem by separating the dynamic seismic action into inertia forces acting in two directions: the horizontal direction (with the slope facing outwards) and the vertically upward direction. The limit equilibrium and other methods are then used to make refinements compared to static conditions, thus enabling the stability of the slope under the seismic action to be evaluated. This approach thereby converts the original dynamic problem into a static equilibrium problem. Pseudo-static methods are easy to implement and so they are widely used in engineering practice [10]. However, they cannot be used to study the dynamic response of the slope under dynamic action and so they cannot accurately describe the failure mechanism involved. As a result, researchers usually apply pseudo-static methods in combination with other methods to more comprehensively evaluate the stability of slopes under seismic action [11; 12; 13]. [PERSON] and [PERSON] [11], for example, used a pseudo-static method to analyze the ultimate equilibrium state of a single block in a slope and then deduced the relationship between the failure mode of the rock mass and the slope angle, the width-to-height ratio, and the friction angle. [PERSON] et al. [12] derived an analytical solution for the stability of anti-dip bedding rock slopes subject to dynamic action using limit equilibrium theory that can be readily applied to shaking-table tests. Their method predicts that the slumping or sliding of the blocks from the top of the slope to the foot gradually increases as the input acceleration is increased, while the safety factor gradually decreases.