## 5 Conclusions  
The effect of spatial variability of soil elastic modulus on shield tunnel responses to new tunneling underneath is investigated in this paper using the random two-stage analysis method. A Timoshenko-Winkler-based deterministic analysis method considering the variation in the subgrade reaction coefficient along the longitudinal direction is first established. The RTSAM is then developed by combining the deterministic analysis method and the discretized random field of the soil elastic modulus. Random analysis is performed to investigate the influence of calculation parameters on the responses of the existing tunnel under the random field. The following conclusions can be summarized:  
1. The spatial variability of the soil elastic modulus induces apparent variabilities of the longitudinal tunnel responses caused by new tunneling and may cause asymmetric tunnel responses. With the increase in \(\theta_{E}\), the mean values of the normalized maximum tunnel responses (\(w_{\rm m}\), \(M_{\rm m}\) and \(Q_{\rm m}\)) slightly decrease, but the COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\) increase significantly. The variations in \(\theta_{E}\) within \(10D\) lead to a strong effect  
Figure 21: Effect of \(\xi\) on COVs of normalized maximum tunnel responses: (**a**) Normalized maximum tunnel settlement, (**b**) Normalized maximum bending moment; (**c**) Normalized maximum shear force. in the variation in the COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\). However, the effect of variations in \(\theta_{E}\) becomes limited when \(\theta_{E}\) is larger than \(10D\). The increasing \(\theta_{E}\) causes higher probabilities of the occurrence of large \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\). 2. The increase in COV\({}_{E}\) barely affects the mean values of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\), but it significantly increases the COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\). The larger the \(\theta_{E}\), the more obvious the effect of variations in COV\({}_{E}\) on the COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\). Furthermore, the reductions in the pillar depth and the mean value of the soil elastic modulus and the increase in the skew angle lead to larger COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\), i.e., they amplify the effect of spatial variability of the soil elastic modulus on tunnel responses. 3. There is a reduction in the mean values of \(w_{\rm m}\) and \(Q_{\rm m}\) and an increase in the mean values of \(M_{\rm m}\) due to the increase in the equivalent shear stiffness of the existing tunnel. Higher variabilities in \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\) are observed when the equivalent shearing stiffness increases, and the variation in COVs of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\) can be fitted well by the hyperbolic equation. The Euler-Bernoulli beam model overestimates the variabilities of the tunnel responses caused by spatial variability of the soil elastic modulus. This study provides theoretical references for the uncertainty analysis of the longitudinal responses of the existing tunnel affected by the new tunneling underneath and the spatial variability of the soil elastic modulus. However, it should be noted that the nonlinear deformations of the soil and the tunnel structure are not considered in this study and need to be investigated in further research. Conceptualization, [PERSON] ([PERSON]) and N.L. ; methodology, [PERSON] ([PERSON]) and N.L. ; software, [PERSON] ([PERSON]); validation, [PERSON] ([PERSON]); formal analysis, [PERSON] ([PERSON]) and N.L. ; investigation, [PERSON] ([PERSON]); resources, [PERSON] ([PERSON]); data curation, [PERSON] ([PERSON]); writing--original draft preparation, [PERSON] ([PERSON]); writing--review and editing, A.B. and N.L. ; visualization, [PERSON] ([PERSON]); supervision, [PERSON] ([PERSON]); project administration, N.L. ; funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript.