## 5 Analysis of Response-Influencing Factors
### Effect of Seismic Loading Frequency on Dam Stability  
The frequency of natural seismic waves typically ranges from 5 to 15 Hz. This study analyzed the dynamic response characteristics of the dam model using seismic wave frequencies of 5, 10, 15, 20, and 25 Hz. The acceleration response increases with elevation, exhibiting a significant amplification effect that peaks at the top of the slope. As the frequency of the input seismic wave increases, the amplification effect of acceleration with elevation tends to decrease (refer to Figure 14). At an input frequency of 5 Hz, the peak ground acceleration ratio (PGAR) amplification coefficient is at its highest as the dominant frequency is lowest at this point, resulting in a pronounced amplification effect on the slope body. Conversely, higher input frequencies contain more high-frequency energy, which enhances the filtering effect of the geotechnical body, thereby reducing the PGAR amplification coefficient response. Figure 15 illustrates the time course curves of displacement in the x-direction at monitoring point C4 for different frequencies. The maximum displacement value decreases as the frequency increases due to the reduced acceleration response of the higher-frequency seismic waves. When the seismic load frequency exceeds 15 Hz, the displacement value of the model due to the seismic load remains nearly unchanged, indicating that high-frequency waves have a lesser impact on the dam body.  
The dynamic response of the model is characterized by its macroscopic deformation. According to the theory of elastic wave scattering, seismic stress waves split the wave field upon encountering a heterogeneous interface to maintain equilibrium. This phenomenon is primarily manifested through reflections from the free surface and both reflection and refraction at the slope discontinuity interface. Consequently, the seismic wave field near the interface becomes a complex combination of various wave types. As illustrated in Figure 16, the maximum permanent displacement in the horizontal direction of the tailings pond decreases with the increasing frequency of the input seismic wave. When the frequency exceeds 15 Hz, the displacement amplification ceases, likely due to the tailings' filtering effect and the dam's inherent inhibition.  
Figure 13: PGAR at different friction angles.  
Figures 17-20 illustrate the maximum displacement cloud of the dam body at various frequencies. At lower frequencies, the maximum displacement of the dam body model is concentrated at the foot of the slope, with the maximum permanent displacement values distributed stratified along a specific radial pattern. As the frequency increases, the maximum displacement of the dam body model gradually shifts towards the top of the slope. Notably, at 25 Hz, the distribution of the maximum permanent displacement differs significantly from that observed at other frequencies. Figure 21 depicts the deformation damage at frequencies of 5 Hz and 25 Hz, respectively. At lower frequencies, the deformation damage to the dam model is concentrated in the shallow surface area and at the foot of the slope, resulting in localized cracking. In contrast, at 25 Hz, deformation damage primarily occurs at the bottom of the dam body, characterized by the predominance of lateral crack development, while the slope surface exhibits minimal damage.  
Figure 14: PGAR of C4 at various frequencies.  
Figure 15: X-displacement time course of monitoring point C4 under different frequencies.  
Figure 16: Change rule of maximum permanent displacement in x-direction with elevation at different frequencies.  
Figure 17: The maximum permanent displacement at a frequency of 5 Hz.  
Figure 20: The maximum permanent displacement at a frequency of 25 Hz.  
Figure 18: The maximum permanent displacement at a frequency of 15 Hz.