## 1 Introduction  
Slope stability is always an important topic of debate, not only in geo-mechanics, but also in engineering practice. Many studies have adopted various failure mechanisms to analyze the slope stability issues in recent years in some theoretical methods. For example, [PERSON] et al. investigated the soil slope displacement induced by the earthquake force based on the log spiral failure model proposed with considering tensile strength cut-off in upper bound theory [1, 2]. [PERSON] and [PERSON] [3] studied the stability of soil slopes incorporating the horizontal and vertical seismic loads based on the cylindrical failure model proposed by [PERSON] [4] in the limit equilibrium method. It should be noted that these failure models are all plan-strain failure mechanisms. However, recent researches show that the practical slope collapse in a critical state is a three-dimensional failure, and the plane-strain failure mechanisms will lead to conservative estimations for three-dimensional slope stability issues.  
Many 3D horn failure models were proposed to investigate the slope stability, e.g., 3D one-block collapse failure model, 3D multi-block mechanism, and 3D horn failure model [5, 6, 7]. The failure zone and stability number obtained using these analytical models were more consistent than the plane-strain failure mechanisms when compared with the experiments and numerical solutions [8]. In particular, the 3D horn failure model proposed by [PERSON] and [PERSON] was the most prevalent failure model used to investigate slope stability [7]. The 3D horn failure model was composed of a rotation mechanism and a plane strain insert mechanism. In recent years, based on the 3D horn failure model, the effects of seepage forces, inhomogenous soils, and pile reinforcement on slope stability have been addressed [8; 9; 10]. Note that these researches are all limited to calculating soil slope stability numbers with the Mohr-Coulomb failure criterion. However, according to the test results [11; 12; 13], the strength envelops of almost all rock materials have the feature of nonlinearity. At the same time, few pieces of research incorporated the nonlinear failure criterion on the slope stability issues based on the 3D horn failure model. Moreover, even though some studies have involved the effect of seismic loads on the soil or rock slopes stability [4; 9], rare researches conducted the estimation of the critical seismic acceleration of rock slopes with the modified Hoek-Brown (MHB) failure criterion. Given this, this study will fill the gaps of these researches.  
In this work, the MHB failure criterion upper limit theory will be introduced to study rock slopes' critical seismic acceleration coefficient based on the 3D rotational failure mechanism. The nonlinear Hoek-Brown failure criterion can be transformed into a linear criterion by using the generalized tangent technique. The feasibility and validity of the critical seismic acceleration coefficients obtained from this research will be verified by limit analysis based on the finite element method. Finally, a design chart will be provided for the particular situation of actual use in rock engineering.