# Investigating the Impact of Random Field Element Size on Soil Slope Reliability Analysis
## 3 Results and Discussion  
In this study, an approximate relative error (RE) is used to study the impact of RFE size on the _PF_. Considering the slope discretized into square elements, each with a width  
Figure 1: Flow chart for computing _PF_. denoted as \(D_{e}=w_{1}\), \(w_{2}\), \(w_{3}\ldots w_{n}\), where \(w_{1}\) represents the minimum value among \(w_{1}\), \(w_{2}\), \(w_{3}\ldots w_{n}\) and is deemed sufficiently small. Thus, the RE pertaining to \(PF\) can be formulated as follows:  
\[\text{RE}=\frac{PF^{i}-PF^{1}}{PF^{1}} \tag{6}\]  
where \(PF^{1}\) is the \(PF\) with \(D_{\text{e}}=w_{1}\), and \(PF^{i}\) is the \(PF\) with \(D_{\text{e}}=w_{i}\). The critical dimension of RFE corresponds to the minimum size at which any further decrement in \(D_{\text{e}}\) fails to elicit a notable reduction in the absolute RE of \(PF\), thereby indicating the optimal balance between accuracy and computational efficiency. This study utilizes an analysis program that was developed by the author using the MATLAB programming language (R2020a, 9.8.0). Example 1:  
Example 1 proposed by [PERSON] [14] is typical slope with single layer undrained cohesive. Table 1 displays the soil parameters for Example 1, including the mean (\(\mu_{c}\)) and covariance (COV\({}_{c}\)) of \(c\) and the soil weight (\(\gamma\)), mean (\(\mu_{\varphi}\)), and covariance (COV\({}_{\varphi}\)) of \(\varphi\). This study considered the scenarios with constant \(\delta_{h}\) (\(\delta_{h}=40\) m) and different \(\delta_{v}\) (\(\delta_{v}=3\) m, \(5\) m, \(7\) m). In this example, we modeled the shear strength \(c\) as a random field using the K-L expansion method with different RFE sizes; \(D_{e}=10\) m, \(5.0\) m, \(2.5\) m, \(1.0\) m, \(0.5\) m, \(0.25\) m, \(0.2\) m, \(0.15\) m, \(0.1\) m, \(0.09\) m, and \(0.08\) m. And the corresponding ratios of slope height and RFE size (\(H/D_{e}\)) are \(0.5\), \(1\), \(2\), \(5\), \(10\), \(20\), \(25\), \(33\), \(50\), \(55\), and \(62.5\), respectively. When employing the MCS for slope reliability analysis, the accuracy of the results improves with the increasing number of simulations. Typically, a relatively accurate result can be obtained after 10,000 simulations. To ensure the precision of the computational outcomes in this study, we generated 50,000 random field realizations using MCS. Figure 2 shows the typical random field realization by the RFE size of \(5\) m, \(1\) m, \(0.25\) m, and \(0.1\) m, respectively. In this section, the effects of the REF size on \(F_{s}\) is analyzed. Figure 3 illustrates the distribution of \(F_{s}\) for various \(H/D_{e}\) of Example 1. The \(F_{s}\) corresponding the maximum probability density function value is around \(1.25\). It can be seen that when \(H/D_{e}\) is smaller than \(5\), the probability density of \(F_{s}\) becomes higher, and the distribution of \(F_{s}\) becomes more and more concentrated at \(1.25\) with the increase in \(H/D_{e}\). Nevertheless, when the element has a relatively smaller size like \(H/D_{e}>10\), the distribution of \(F_{s}\) tends to be stable without much change. The effect of the RFE size on the \(PF\) of slopes was analyzed in this section. Figures 4 and 5 show the \(PF\) values for various \(H/D_{e}\). As shown in Figure 4, when the RFE size \(D_{e}\) is large and \(H/D_{e}\) is small, the value of \(PF\) was obviously overestimated, and as \(D_{e}\) decreases and \(H/D_{e}\) increases, it shows a downward trend and gradually becomes a steady state. This is because when the RFE size is relatively large, the number of elements into which the slope is divided is smaller, making the slope appear relatively homogeneous. This means that the \(c\) values at different locations are very similar. Therefore, the average \(c\) at the slip surface has a relatively high probability to take a very low value. But when the \(D_{e}\) value is large, this phenomenon will be reversed due to the influence of spatial averaging.