## 4 Random Analysis  
Random analysis is performed based on an actual engineering case to analyze the effect of the spatial variability of soil elastic modulus on the responses of the existing tunnel to the new tunnel undercrossing. [PERSON] et al. [19] reported the responses of an overlying metro tunnel to the excavation of a new tunnel in Shanghai, China. Both the new tunnel and the existing tunnel were constructed by the shield method and have an outer diameter of 6.2 m. The cross-sectional view of the new and existing tunnel is shown in Figure 6. The buried depths of the existing and new tunnels are 9.1 m and 20.1 m, respectively, and the skew angle between the existing and new tunnel is 90\({}^{\circ}\). The vertical pillar depth \(P\) is 4.8 m. The parameters of segments and bolts for typical metro shield tunnels in Shanghai are listed in Table 1 referring to [PERSON] et al. [35]. Joint parameters \(\lambda\) and \(\xi\) are assumed to be 0.4725 and 10, respectively [36; 42]. Therefore, the equivalent longitudinal flexural and shearing stiffness of the existing tunnel are 1.36 \(\times\) 10\({}^{5}\) MN\(\cdot\)m\({}^{2}\) and 4.2 \(\times\) 10\({}^{4}\) MN, respectively, which  
Figure 5: Calculation procedure of proposed RTSAM. are close to the parameter values used in the study of [PERSON] et al. [19]. The volume loss caused by the new tunneling is set to 0.6%, and the Poisson's ratio of the soil is 0.3. The mean value \(\mu_{E}\), the coefficient of variation COV\({}_{E}\), and the SOF \(\theta_{E}\) of the soil elastic modulus are assumed to be 16.49 MPa and 0.5 and 18.6 m (3\(D\)), respectively [19]. Figure 7 illustrates the longitudinal settlement, bending moment, and shear force of the existing tunnel calculated from 500 random field realizations. The results from the deterministic analysis with a constant soil elastic modulus of 16.49 MPa are also included. As shown in Figure 7, the deterministic results are contained in the distribution ranges of the longitudinal tunnel responses on the basis of the random field. It can be seen that the random results at the extreme points of the response curves exhibit larger variability due to the spatial variability of the soil elastic modulus. The higher uncertainties in these extreme values may result in higher possibilities of maximum tunnel responses exceeding allowable limits. Furthermore, though the greenfield soil settlement due to new tunneling is symmetric with respect to the centerline of the new tunnel, the curves of the longitudinal tunnel responses may show asymmetric profiles because of the asymmetric longitudinal distribution of the soil elastic modulus. The asymmetry is more apparent for the curves of the internal force and cannot be reflected when the longitudinal variation in the soil elastic modulus is ignored. The settlements of the existing tunnel obtained from the measurement data are also shown in Figure 7a. It can be found that most measurement data fall within the random results, which proves the applicability of the proposed RTSAM in practical engineering to a certain extent. The results indicate that the proposed RTSAM can offer the random responses of the existing tunnel under the random field of the soil elastic modulus \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline
**Segmental rings** & \\ Outer diameter (m) & 6.2 \\ Inner diameter (m) & 5.5 \\ Thickness (m) & 0.35 \\ Elastic modulus (MPa) & \(3.45\times 10^{4}\) \\ Length (m) & 1 \\ Poisson’s ratio & 0.2 \\
**Bolts** & \\ Longitudinal bolts number & 17 \\ Length (mm) & 400 \\ Diameter (mm) & 30 \\ Elastic modulus (MPa) & \(2.06\times 10^{5}\) \\ Poisson’s ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical structure parameters of shield tunnels used in the Shanghai metro line. Figure 6: Relative position for the existing tunnel and the new tunnel in random analysis. and provide a simplified methodology for the uncertainty analysis of the longitudinal tunnel responses to new tunneling underneath.