## 3 Dynamic Response of the Slope  
### Analysis of the Amplification Effect  
The dynamic response of the slope is most intuitively represented by specifying the increase in acceleration experienced by each part of the slope compared to the initial input  
Figure 5: The nature of the input acceleration used in this work (Wolong waves of the Wenchuan earthquake), showing: (**a**) the acceleration history with a holding time of 5 s and E-W, U-P, and N–S PGA values of 9.28, 6.39, and \(-\)6.08 m/s\({}^{2}\), respectively, and (**b**) the corresponding Fourier spectra and earthquake dominant frequency range (1.5–6 Hz). Figure 6: Spatial locations of the acceleration (A) and displacement (D) monitoring points and the profiles upon which they are located in the slope model. dynamic acceleration, i.e., the amplification effect. For the sake of simplicity and clarity, the acceleration amplification factor (\(AAF\)) calculated using \(PGA\) values is usually used to evaluate the amplification effect, i.e., the PGA value measured at a particular point is compared (ratioed) with the PGA value of the initial dynamic input [23; 24]. In this paper, the \(AAF\)s in the \(X\)-, \(Y\)-, and \(Z\)-directions are denoted by \(AAF_{x}\), \(AAF_{y}\), and \(AAF_{z}\), respectively, and are calculated using the expressions:  
\[\left. \begin{array}{l}AAF_{x}=x PGA_{n}/x PGA_{0}\\ AAF_{y}=y PGA_{n}/y PGA_{0}\\ AAF_{z}=z PGA_{n}/y PGA_{0}\end{array} \right\} \tag{6}\]  
where \(xPGA_{n}\), \(yPGA_{n}\), and \(zPGA_{n}\) are the PGA values measured in the three directions at a given point in the model and \(xPGA_{0}\), \(yPGA_{0}\), and \(zPGA_{0}\) are the input \(PGA\) values in the three directions at the bottom of the model (point A1). The regularity of the spatial distribution of the dynamic amplification effect inside the model is considered first. Figure 7 shows the \(AAF\) results obtained using a slope model with \(\alpha=40^{\circ}\) and \(\beta=60^{\circ}\) when the slope is subjected to dynamic loading in the form of natural seismic waves and sinusoidal waves. Overall, the \(AAF\)s produced using sinusoidal waves are generally larger than those produced using natural seismic waves (and their spatial variation pattern is also more obvious). We can compare these results with those obtained in previous research where the amplification effect in anti-dip bedding rock slopes subjected to seismic action was derived using test models and numerical simulations [17; 24; 25]. According to Figure 7a,b, the slope does not exhibit a single 'elevation effect' under the action of three-directional dynamic loading. More specifically, in the plane problem, the \(AAF\)s increase monotonically as the elevation increases--here, however, the maximum values appear in the middle and upper parts of the slope (i.e., points A5 and A9). Similarly, it can be found that the'surface effect' that occurs in the plane problem (in which the larger \(AAF\) values are encountered closer to the surface of the slope) does not appear under three-directional loading conditions. Instead, according to Figure 7c, the largest \(AAF\) values are concentrated at the trailing edge (i.e., points A16-A18) and are further away from the slope surface. We also note that, on the whole, the three-directional accelerations along profiles 1, 2, and 3 do not increase monotonically. Instead, a pattern of increasing (decreasing) values at first followed by decreasing (increasing) values is found. According to elastic wave scattering theory, the seismic S-wave is split when it interacts with a structural plane and is decomposed into a reflected S-wave and refracted P-wave. When a wave propagates from the bottom of the slope to the top it passes through several structural planes and so a complex vibrational wave field is formed within the slope. As a result, a rhythmic pattern of variation is generated in the \(AAF\)s. According to Figure 7a,b, we find that \(AAF_{z}>AAF_{y}\) in the upper part of the slope and vice versa in the lower part.