## 3 Dynamic Response Analysis Method  
As many factors could influence the dynamic response of the slope, a homogeneous and elastic rock slope with horizontal layered-structure, slope angle of \(30^{\circ}\), and height of \(30\) m is specified to focus on the study of the parameters in which we are most interested (normalized fracture stiffness and dimensionless fracture spacing). As Figure 7 shows, the number of the joint in the slope model (\(N\)) is \(5\); the equal space of the joint (\(d\)) is \(10\) m. The joint model adopts the Coulomb slip model, which is commonly used in FLAC. High values are assigned to the cohesion and tension of the joint to make sure that the joint is linear and elastic. As in Section 2.1, cohesion is \(10\) GPa, the friction angle is \(30^{\circ}\), and the tension strength is \(10^{5}\) GPa.  
Figure 6: Relations between transmission coefficient \(T_{1P}\) and incident wave angle \(\theta\) for obliquely incident P wave transmission across a single joint.  
Figure 7: Schematic illustration of FLAC model for the numerical analysis.  
Through using the FISH language embedded in FLAC, the monitoring points are laid out at equal intervals in the upper slope in which we are most interested. The duration of Sv wave applied under the bottom of the slope was 1.0 s, allowing the incident wave to transmit between the bottom and top boundaries of the slope several times.  
For a slope with multiple horizontally layered structures, analytical solution of wave transmission through the horizontal layered-structure rock slope is impossible to obtain and experiments as well as in situ tests are difficult to conduct. Therefore, numerical simulation is the most appropriate approach to study the complexity of this problem.  
A FLAC model with five horizontally equal interval joints (\(d\) = 10 m), slope angle of 30\({}^{\circ}\), and height of 30 m is shown in Figure 7. The viscous boundaries are placed at the left side, right side, and bottom of the model. The incident harmonic Sv wave is applied at the bottom boundary of the slope model. The monitoring points are evenly distributed between the joints in the slope. The origin of the X-Z coordinates is located at the toe of the slope.  
It is known that the horizontal acceleration is greatly amplified at the crest of slopes [12]. In this section, the horizontal acceleration (i.e., X-acceleration) of monitoring points in the slope shown in Figure 7 is the concerned factor. For convenience, the horizontal acceleration is described as the acceleration, and the acceleration amplification coefficient of monitoring point is defined as \(\eta\), which can be expressed as:  
\[\eta=A_{0}/A_{n} \tag{6}\]  
where \(A_{0}\) is the acceleration amplitude of the reference point near the toe of the slope and \(A_{n}\) is the acceleration amplitude of any monitoring point in the slope shown in Figure 7.  
In order to study the dynamic response regularities of the slope, two parameters are considered. They are the normalized fracture stiffness (\(K\)) and the dimensionless fracture spacing (\(\xi\)), respectively. Through the monitoring points placed in the slope profile and Equation (6), the acceleration amplification coefficients of monitoring points in the whole slope can be obtained. It should be noted that the mechanical properties, spatial properties, and incident wave properties of the actual slope are very complicated. In this article, with the rock slope model specified, many factors are fixed. Therefore, the main study can focus on the most important parameters.