## 2 Materials and Methods
### Methods
#### 2.2.1 The Permanent Displacement Analysis  
The traditional seismic stability analysis of rock slope adopts the quasi-static method, which assumes that the seismic force is a constant force continuously applied in one direction, so only the peak ground acceleration (PGA) is considered. However, the seismic wave features such as type, frequency and duration are ignored, and the permanent displacement of the slope during an earthquake is not revealed. [PERSON] (1965) proposed the displacement analysis method based on the limited equilibrium theory of infinite slopes, which concluded that the permanent displacement of a sliding mass was caused by the displacement accumulation after instantaneous failure along the most dangerous sliding surface under seismic loading. The obtainment of the permanent displacement is divided into two steps: (1) The pseudo-static method is used to determine the critical acceleration of the potential sliding mass. When the acceleration at the most dangerous sliding surface exceeds the critical acceleration, the block slides along the failure surface. (2) The difference between the earthquake acceleration and the critical acceleration is quadratically integrated over time to obtain the permanent displacement (Figure 3). The calculation formula is as follows:  
\[a_{c}=(F_{s}-1)g\sin\delta \tag{1}\]  
\[D=\iint\limits_{0}^{t}\Bigl{[}a_{(t)}-a_{c}g\Bigr{]}dt^{2} \tag{2}\]  
where \(a_{c}\) is the critical acceleration, \(D\) is the permanent displacement, \(F_{s}\) is the pre-earthquake stability factor, g is the gravity acceleration (taken as 9.8 m/s\({}^{2}\)), \(\delta\) is the inclination of sliding surface, \(t\) is the time and \(a_{(t)}\) is the earthquake acceleration. The basic assumptions of the traditional Newmark method include the following: 1 the sliding block is rigid, such as granite, dolomites, limestone and sandstone. They will have a cataclastic texture under the forces [35; 36]. 2 The sliding block is displaced only when the seismic acceleration exceeds the critical acceleration. 3 The static and dynamic strength of the sliding surface is the same and fixed (without considering the deterioration). 4 Only the horizontal seismic force is considered (without considering the vertical seismic force). 5 The inertia force is not considered. Figure 3: Schematic diagram of the Newmark method, (**a**) acceleration versus time, (**b**) velocity versus time and (**c**) displacement versus time [34]. The sliding block is subjected to fixed sliding force and resistance force on the sliding surface when no earthquake occurs and remains in static equilibrium (Figure 4). The safety factor (\(F_{s}\)) of the sliding block is defined as follows:  
\[F_{s}=\left(\mu_{s}mg\cos\delta+cA\right)/\left(mg\sin\delta\right) \tag{3}\]  
where \(m\) is the mass of the sliding block, \(g\) is the gravity acceleration (taken as 9.8 m/s2), \(\delta\) is the inclination of the sliding surface, \(\mu_{s}\) is the static friction coefficient, \(c\) is the cohesion and \(A\) is the sliding surface area. During an earthquake, the critical equilibrium conditions of the sliding block are changed by ground vibration. The stability of the slider is controlled by the weight, the friction of the sliding surface and the acceleration in different directions. The seismic force acting on the slider can be decomposed perpendicular to the sliding surface (\(a_{n}\)), parallel to the sliding surface along the inclination (\(a_{d}\)) or parallel to the sliding surface along the direction (\(a_{s}\)). They can be obtained by recording the east-west acceleration (\(a_{E}\)), north-south acceleration (\(a_{N}\)) and vertical acceleration (\(a_{V}\)). The horizontal accelerations \(a_{E}\) and \(a_{N}\) are synthesized as acceleration \(a_{s}\) along the sliding direction, and then, \(a_{s}\) and \(a_{V}\) are synthesized as accelerations along the sliding direction \(a_{d}\) and vertical sliding direction \(a_{n}\).