## 4 Parametric Study
### Effect of Period of Lateral Shaking \(T\)  
Figure 13 shows the effect of a lateral shaking period \(T\) on the \(K_{\rm a}\), \(\alpha_{\rm a}\), and normalized dynamic active earth pressure distributions with \(f_{\rm a}\) = 1.4, \(\varphi\) = 30\({}^{\circ}\), \(\delta\) = \(\varphi\)/2, \(\theta\) = 20\({}^{\circ}\), \(i\) = 8\({}^{\circ}\), and \(k_{\rm v}\) = 0 in the cases of \(c/\gamma H\) = 0 and \(c/\gamma H\) = 0.05. The different \(H/\lambda\) (\(H/\lambda\) = 0.45, 0.30, and 0.15) values are shown in Figure 13 to represent the different \(T\) values, since \(T\) directly affects the value of \(H/\lambda\). If the period corresponding to \(H/\lambda\) = 0.30 is \(t_{0}\), then the periods corresponding to \(H/\lambda\) = 0.45 and 0.15 are 0.67 \(t_{0}\) and 2.00 \(t_{0}\), respectively. As \(V_{\rm P}/V_{\rm s}\) can be taken as 1.87 in most geological materials [31; 32], the value of \(\eta/\lambda\) is also taken as 1.87 in this paper. In Figure 13a, it can be seen that \(K_{\rm a}\) increases significantly as \(H/\lambda\) decreases from 0.45 to 0.15, increases significantly as \(T\) increases from 0.67 \(t_{0}\) to 2.00 \(t_{0}\). For low-period lateral shaking, portions of backfill above the failure surface behind the rigid retaining wall may be moving in opposite directions. The long wavelengths associated with higher-period motion cause the backfill above the failure surface behind the rigid retaining wall to move essentially in phase. This may be the reason why the active earth pressure increases as \(T\) increases. The existence of cohesion (\(c/\gamma H\) = 0.05) reduces \(K_{\rm a}\), as shown in Figure 13b. In Figure 13c, the failure wedge angle is decreased as \(H/\lambda\) decreases from 0.45 to 0.15. As \(k_{\rm h}\) increases, the difference in the failure wedge angle with different \(H/\lambda\) values tends to increase. When \(c/\gamma H\) changes from 0 to 0.05 and \(T\) remains the same, \(\alpha_{\rm a}\) tends to increase. Figure 13d shows the typical normalized dynamic active earth pressure distribution for different values of \(H/\lambda\) with \(k_{\rm h}=0.2\) and \(k_{\rm v}=0\). When \(H/\lambda\) is 0.45, 0.30, and 0.15 in the case of \(c/\gamma H=0.05\), the corresponding normalized critical depth \(z_{\rm c}/H\) is 0.147, 0.116, and 0.097, respectively. As can be seen from Figure 13d, the normalized earth pressure distribution obtained by the present study is nonlinear. When \(H/\lambda\) decreases from 0.45 to 0.15, the degree of nonlinearity of the distribution curves increases. It can be inferred that, as \(T\rightarrow\infty\) or \(H/\lambda\to 0\), the distribution curves will become straight lines. This is if the backfill is assumed to be rigid, which is the same assumption of the Mononobe-Okabe method. Effect of Horizontal and Vertical Seismic Acceleration Coefficients \(k_{\rm h}\) and \(k_{\rm v}\)  
It can be observed in Figures 8-13 that \(K_{\rm a}\) increases as \(k_{\rm h}\) increases. The failure wedge angle shows significant decreases with increases in \(k_{\rm h}\). In Figure 14a,b, \(K_{\rm a}\) decreases slightly for \(k_{\rm h}<0.2\) and \(c/\gamma H=0\) when \(k_{\rm v}=0\), but increases significantly for \(k_{\rm h}>0.2\). In Figure 14c, the failure wedge angle is decreased as \(k_{\rm v}\) decreases from 0 to \(k_{\rm h}\). The difference in the failure wedge angle with different \(k_{\rm v}\) tends to increase as \(k_{\rm h}\) increases. When \(c/\gamma H\) increases from 0 to 0.05 with \(k_{\rm v}\) held constant, \(\alpha_{\rm a}\) tends to increase, and this trend becomes more pronounced with higher \(k_{\rm h}\). Figure 14d shows the typical normalized dynamic active earth pressure distribution for different values of \(k_{\rm v}\) with \(k_{\rm h}\) = 0.2. When \(k_{\rm v}\) increases from 0 to 0.2 in the case \(c/\gamma H\) = 0, the dynamic active earth pressure at the bottom of the wall decreases from 0.844 to 0.767, which represents a decrease of about 9.1%.