## 4 Random Analysis
### Effect of SOF \(\theta_{E}\)  
Figure 9 shows the changes in the mean values of the normalized maximum tunnel responses (\(w_{\text{m}}\), \(M_{\text{m}}\) and \(Q_{\text{m}}\)) under different SOF \(\theta_{E}\) values. It can be seen that the variation in \(\theta_{E}\) has a slight influence on the mean values of the normalized responses. With the increase in \(\theta_{E}\) from 0.5\(D\) to 5\(D\), the mean values decrease slightly at a decreasing rate. The mean values of the normalized responses almost remain stable when \(\theta_{E}\) is larger than 5\(D\). The variations in the COVs of the normalized maximum tunnel responses with increasing \(\theta_{E}\) are presented in Figure 10. As shown in Figure 10, the COVs of the normalized internal forces (\(M_{\text{m}}\) and \(Q_{\text{m}}\)) are always larger than those of the normalized tunnel settlement (\(w_{\text{m}}\)), indicating that the variability of the normalized internal forces is more apparent under the random field of the soil elastic modulus. The larger \(\theta_{E}\) results in larger COVs of the normalized maximum tunnel responses. This phenomenon can be explained according to the logic of the study of [PERSON] and [PERSON] [37], which investigated the stability of the tunnel face in spatially random soils. When \(\theta_{E}\) becomes smaller, the correlation between the soil elastic modulus of two arbitrary points becomes weaker, which makes the value of soil elastic modulus at different points under a single realization of the random field vary more greatly. The average soil elastic modulus within the influence range of the new tunneling changes less apparently among different random field realizations because of the decrease in \(\theta_{E}\), and the normalized maximum tunnel responses obtained through different realizations become closer to each other. When \(\theta_{E}\) is relatively large, the average soil elastic modulus under different realizations varies significantly, which leads to an increase in the variability of the  
Figure 8: The convergence of statistical results of normalized maximum tunnel responses: (**a**) Mean value, (**b**) Coefficient of variation.  
normalized maximum tunnel responses. The COVs of the normalized maximum tunnel responses increase remarkably when \(\theta_{E}\) varies in the range from \(0.5D\) to \(10D\). However, the variation of \(\theta_{E}\) has a limited effect on the COVs of the normalized responses when \(\theta_{E}\) is larger than \(10D\).  
The empirical cumulative distribution functions (CDFs) of \(w_{\text{m}}\), \(M_{\text{m}}\), and \(Q_{\text{m}}\) under different \(\theta_{E}\) (\(\theta_{E}=1D\), \(3D\), \(10D\)) are shown in Figure 11. The distribution ranges of \(w_{\text{m}}\), \(M_{\text{m}}\), and \(Q_{\text{m}}\) under \(\theta_{E}=1D\) are the narrowest, while the distribution ranges under \(\theta_{E}=10D\) are the widest. Therefore, the increasing \(\theta_{E}\) leads to higher probabilities of the occurrence of large \(w_{\text{m}}\), \(M_{\text{m}}\), and \(Q_{\text{m}}\). For example, with the increase in \(\theta_{E}\) from \(1D\) to \(10D\), the probability of \(M_{\text{m}}>0.17\) increases from \(0.03\) to \(0.1\). The findings presented here align with the results shown previously, i.e., the COVs of \(w_{\text{m}}\), \(M_{\text{m}}\), and \(Q_{\text{m}}\) under \(\theta_{E}=10D\) are the largest. Moreover, it is also seen that the CDF curves rotate clockwise about their intersection point as \(\theta_{E}\) increases.  
Figure 9: Mean values of normalized maximum tunnel responses under different \(\theta_{E}\) values: (**a**) Normalized maximum tunnel settlement, (**b**) Normalized maximum bending moment; (**c**) Normalized maximum shear force.  
Figure 11: CDFs of normalized maximum tunnel responses under different \(\theta_{E}\) values: (**a**) Normalized maximum tunnel settlement, (**b**) Normalized maximum bending moment; (**c**) Normalized maximum shear force.  
Figure 10: COVs of normalized maximum tunnel responses under different \(\theta_{E}\) values.