## 4 Analysis of the Dynamic Response Law of Layered Slopes
### Dynamic Response Analysis of Slopes Characterized by Laminated Joints
#### 4.2.4 Dynamic Stability Analysis of Different Joint Dip Slopes Under Blasting Loads  
The stability analysis of slopes under a blasting vibration can be conducted by determining the safety factor at different time intervals of the vibration wave, describing the dynamic stability of the slope. Specifically, based on the dynamic wave input time of 1.2 s, the safety factor can be calculated for the slope under dynamic loading at intervals of 0.2 s. This can be achieved through custom FISH scripts embedded within FLAC30 for solving the time steps.  
Figure 13 depicts the variation of safety factors over time for slopes with different joint dip angles under a blasting vibration. It can be observed from the graph that at 0.2 s, the safety factors of all jointed slopes reach their minimum values, except for the 0\({}^{\circ}\) joint dip angle slope, where the safety factor remains above zero. This indicates a tendency towards instability in the jointed slopes at this moment, attributed to the peak damage caused by the blasting vibration waveform between 0.1 s and 0.3 s. As time progresses, the vibration waveform attenuates, leading to an increase in safety factors for all jointed slopes, with the safety factors reaching their maximum values again at the end of the blasting wave. However, overall, at 1.2 s, the safety factors are still lower than those under static conditions, suggesting irreversible rock damage caused by the blasting vibration, compromising the overall stability of the slopes. Furthermore, the safety factor of the 0\({}^{\circ}\) joint dip angle slope is generally higher than those of the 60\({}^{\circ}\) and 90\({}^{\circ}\) joint dip angle slopes, indicating its lower susceptibility to damage, consistent with the jointing dip angle patterns described in previous sections.  
To assess the influence of blasting vibration on the slope safety factor, we employed Equation (13), as follows:  
\[\rm K=\left|\frac{k_{1}-k_{2}}{k_{2}}\right| \tag{13}\]  
where K is the rate of reduction of the factor of safety under the blasting vibration, k\({}_{1}\) is the factor of safety of the slope under the static load, and k\({}_{2}\) is the factor of safety of the slope under the breaking load.  
Figure 12: Maximum M\({}_{\rm PGA}\) values of slopes with different joints.  
The variation trend of the reduction rate of the safety factor for each joint inclination angle slope is shown in Figure 14. As depicted, the reduction rate of the safety factor exhibits an overall inverted U-shaped distribution with changes in the slope angle. Specifically, the slope with a 60\({}^{\circ}\) joint inclination angle experiences the highest reduction rate, reaching 16.5%, indicating its susceptibility to the influence of blasting loads and potential failure. Conversely, the slope with a 0\({}^{\circ}\) jointing dip angle shows the lowest reduction rate at only 6%, suggesting its resilience to the impact of blasting loads. Additionally, the slope with a 90\({}^{\circ}\) joint inclination angle is prone to tension failure, resulting in a relatively high reduction rate of the safety factor, reaching 14.8%.  
Figure 14: Change pattern of safety factor reduction rate of jointed slopes.  
Figure 13: Variation pattern of dynamic safety coefficient of jointed slopes.