## 4 Analysis of the Dynamic Response Law of Layered Slopes
### Dynamic Response Analysis of Slopes Characterized by Laminated Joints
#### 4.2.2 Influence of Rock Hardness and Rock Structure on the Amplification Effect of  
Choosing appropriate indicators is crucial for studying the amplification effect of jointed slopes. This study selected the ratio of PGA at a certain point on the slope to PGA at the foot of the slope (M\({}_{\text{PGA}}\)) as the analysis parameter. M\({}_{\text{PGA}}\) reflects the acceleration amplification factor at that specific point, i.e., the acceleration amplification effect. Figure 11a-d depict the variation of M\({}_{\text{PGA}}\) with elevation for jointed slopes in soft and hard rock formations.  
From Figure 11a,c, it can be observed that the M\({}_{\text{PGA}}\) on the surface of both soft and hard rock slopes generally increases in fluctuation with the elevation before decreasing. For soft rock slopes, the maximum M\({}_{\text{PGA}}\) values on the surface exhibit the following trend in descending order of joint inclination: 45\({}^{\circ}\)\(>\) 60\({}^{\circ}\)\(>\) 75\({}^{\circ}\)\(>\) 90\({}^{\circ}\)\(>\) 30\({}^{\circ}\)\(>\) 15\({}^{\circ}\)\(>\) 0\({}^{\circ}\) \(>\) without joints. The maximum M\({}_{\text{PGA}}\) value at a joint inclination of 45\({}^{\circ}\) is 2.04, while at 0\({}^{\circ}\), it is 1.48. Conversely, for hard rock slopes, the maximum M\({}_{\text{PGA}}\) values on the surface follow the following order of joint inclination: 60\({}^{\circ}\)\(>\) 45\({}^{\circ}\)\(>\) 30\({}^{\circ}\)\(>\) 75\({}^{\circ}\)\(>\) 90\({}^{\circ}\)\(>\) 15\({}^{\circ}\)\(>\) 0\({}^{\circ}\) \(>\) without joints. The maximum M\({}_{\text{PGA}}\) value at a joint inclination of 60\({}^{\circ}\) is 1.72, while at 90\({}^{\circ}\), it is 1.30.  
From this analysis, it is evident that the M\({}_{\text{PGA}}\) on the surface of both soft and hard rock slopes is positively correlated with the elevation within the relative elevation range of 0.2 to 0.6, increasing with the elevation. This indicates that under the influence of blast-induced vibration waves, both soft and hard rock slopes exhibit elevation amplification effects. There is a significant difference in the M\({}_{\text{PGA}}\) increment at the same elevation among slopes with different joint inclinations. Moreover, between relative elevations of 0.6 and 1.0, the fluctuation of M\({}_{\text{PGA}}\) decreases with an increasing elevation, suggesting that the elevation amplification effect occurs only under certain conditions and is influenced by factors such as lithology, topography, elevation, and blast-induced vibration waves. The decrease in the amplification effect observed here is attributed to the dissipation of energy from blast-induced vibration waves. Joint inclination has a complex nonlinear effect on the elevation amplification effect of slopes, with slopes inclined at 45\({}^{\circ}\) and 60\({}^{\circ}\) exhibiting more pronounced elevation amplification effects compared to those at 0\({}^{\circ}\) and slopes without points. Additionally, rock hardness significantly influences the elevation amplification effect of slopes, with soft rocks exhibiting a more pronounced effect than hard rocks.  
From Figure 11b,d, it is evident that both soft and hard rock slopes exhibit a minor degree of the elevation amplification effect near the lower point or middle part of the slope. However, overall, their M\({}_{\rm PGA}\) demonstrates a decrease in fluctuation with an increasing slope height, mainly indicating the characteristic of M\({}_{\rm PGA}\) attenuation with an increasing distance from the blast center. By comprehensively comparing the variations in M\({}_{\rm PGA}\) within the slope and on the surface, it can be concluded that under the influence of blast-induced vibration waves, both soft and hard rock slopes exhibit amplification effects in terms of surface elevation and jointing, with the maximum amplification effect occurring at the middle part of the slope.