## 5 Effect of Slope Inclination
\begin{array}{c}-\\ \mathbf{u}_{x}\text{\bf m}\end{array} \) & \( \begin{array}{c}-\\ \mathbf{u}_{y}\text{\bf m}\end{array} \) & \( \begin{array}{c}-\\ \mathbf{u}_{y}\text{\bf m}\end{array} \) & \( \begin{array}{c}-\\ \mathbf{u}_{y}\text{\bf m}\end{array} \) \\ \hline A & 2:1 & 2.58 & 0.39 & 0.35 & 0.53 \\ \hline B & 4:3 & 1.71 & 0.41 & 0.47 & 0.63 \\ \hline C & 1:1 Figure 11: Probability density of the coefficient \(f_{y}=u_{y}/u_{y}\) of vertical permanent displacement for three slopes (\(u_{y}\) = vertical permanent displacement, \(u_{y}\) = vertical permanent displacement of homogeneous slope). Figure 12: Probability density of the coefficient \(f\) = \(u/u\) of total permanent displacement for three slopes (\(u\) = total permanent displacement, \(u\) = total permanent displacement of homogeneous slope). To examine the impact of slope inclination on permanent displacements, it is of interest to utilize the derived statistical distribution of the ratio \(f_{x}=u_{x}/\overline{u}_{x}\) (or \(f_{y}=u_{y}/\overline{u}_{y}\)) and the results of permanent displacements for homogeneous slopes \(\overline{u}_{x}\) (or \(\overline{u}_{y}\)) given in Table 8, combined with the statistical distribution of normalized displacements expressing the temporal and frequency variability of the seismic excitation. For each slope inclination investigated, the combined effects of the spatial variability of soil properties and earthquake characteristics are expressed by the relations:  
\[u_{x} = \overline{u}_{x}\ \mathbf{RV}\ \big{(}\text{Normal Distribution}\ \big{[}\mu_{f_{x}},\sigma_{f_{x}}\big{]}\big{)}\ \times\ \mathbf{RV}\ \big{(}\text{Uniform Distribution}\ [r_{\text{min}},r_{\text{max}}]\big{)} \tag{2a}\] \[u_{y} = \overline{u}_{y}\ \mathbf{RV}\ \big{(}\text{Normal Distribution}\ \big{[}\mu_{f_{y}},\sigma_{f_{y}}\big{]}\big{)}\ \times\ \mathbf{RV}\ \big{(}\text{Uniform Distribution}\ [r_{\text{min}},r_{\text{max}}]\big{)} \tag{2b}\]  
where  
\(u_{x}\), \(u_{y}\) = horizontal and vertical permanent displacements of heterogeneous slope \(\overline{u}_{x}\), \(\overline{u}_{y}\) = horizontal and vertical permanent displacements of homogeneous slope \(\mathbf{RV}\) = random variate  
\(\mu_{f_{x}},\mu_{f_{y}}\) = mean values of the coefficients \(f_{x},f_{y}\)  
\(\sigma_{f_{x}},\sigma_{f_{y}}\) = standard variations of the coefficients \(f_{x},f_{y}\)  
\(r_{\text{min}},r_{\text{max}}\) = bounds of uniform distribution  
For the slope with inclination 2:1, Figure 14a plots a histogram with the distribution of the computed horizontal permanent displacement derived from a sample of 500 values of \(u_{x}\) using Equation (2a). The distribution of \(u_{x}\) is fitted with the Weibull, Extreme Value, and Gamma distributions. Moreover, Figure 14b,c, plot the computed values of the vertical permanent displacements \(u_{y}\) and total permanent displacements \(u\). Figure 13: Probability of exceedance of a given value \(f^{*}\) of the coefficient \(f=u/a\) of total permanent displacement for three slopes (\(u=\) total permanent displacement, \(a=\) total permanent displacement of homogeneous slope). Similar results on the distribution of permanent displacements for slope inclination 4:3 are presented in Figure 15, and for slope inclination 1:1 in Figure 16. A comparison of the best fit test parameters for the three statistical distributions in all results presented in Figures 14-16 shows that the Weibull distribution clearly provides the best fit for all cases. Figure 14: Computed permanent displacements for slope inclination 2:1: (**a**) horizontal displacement \(u_{x}\), (**b**) vertical displacement \(u_{y}\), and (**c**) total displacement \(u\) fitted by Weibull, Extreme Value, and Gamma distributions. Figure 15: Computed permanent displacements for slope inclination 4:3: (**a**) horizontal displacement \(u_{x}\), (**b**) vertical displacement \(u_{y}\), and (**c**) total displacement \(u\) fitted by Weibull, Extreme Value, and Gamma distributions.