## 4 Random Analysis
### Effect of Modified Factor \(\xi\)  
The effect of the modified factor of equivalent shearing stiffness \(\xi\) on the mean values of the normalized maximum tunnel responses under \(\theta_{E}=3D\) is shown in Figure 20. For comparison, Figure 20 also includes the mean values calculated by the Euler-Bernoulli beam model (\(\xi=\infty\)). As \(\xi\) increases from 0.5 to 10, the mean value of \(w_{\rm m}\) decreases by 8%, and the mean value of \(Q_{\rm m}\) decreases by 93%. On the contrary, the mean value of \(M_{\rm m}\) increases by 42% because of the increase in \(\xi\). The results imply that the mean values of the normalized maximum tunnel responses vary at a decreasing rate and gradually come closer to the calculation results according to the Euler-Bernoulli beam model for larger values of \(\xi\).  
Figure 19: Effect of \(a\) on COVs 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.  
The COVs of the normalized maximum tunnel responses for different \(\xi\) values under \(\theta_{E}\) = 3\(D\) are presented in Figure 21. It can be found that the COV of \(w_{\rm m}\), \(M_{\rm m}\), and \(Q_{\rm m}\) increases by 13%, 31%, and 35%, respectively with the increase in \(\xi\) from 0.5 to 10, implying that the increasing equivalent shearing stiffness of the existing tunnel results in higher variabilities of the normalized tunnel responses. It can also be deduced that the existing tunnel tends to displace following the greenfield soil settlement and is barely affected by the spatial variability of the soil elastic modulus when the tunnel is infinitely flexible. The comparison of the calculation results from the Timoshenko beam model and the Euler-Bernoulli beam model indicates that the variabilities in \(w_{\rm m}\) and \(M_{\rm m}\) may be significantly overestimated by the Euler-Bernoulli beam model. Furthermore, the calculation results show that the COV of the normalized maximum tunnel responses can be fitted with the following hyperbolic equation:  
\[\rm{COV_{n}}=\frac{a_{h}}{1+b_{h}/\xi} \tag{38}\]  
where \(\rm{COV_{n}}\) is the COV of the normalized maximum tunnel responses and \(a_{\rm h}\) and \(b_{\rm h}\) are two coefficients of the hyperbolic equation. The hyperbolic fitting curve shows good agreement with the calculated results and can be used to establish the mathematical relationship between \(\rm{COV_{n}}\) and \(\xi\). For \(w_{\rm m}\) and \(M_{\rm m}\), the fitting coefficient ah corresponds to  
Figure 20: Effect of \(\xi\) on mean values of normalized maximum tunnel responses: (**a**) Normalized maximum tunnel settlement, (**b**) Normalized maximum bending moment; (**c**) Normalized maximum shear force.  
the value of COVn under \(\xi=\infty\) (0.0697 for \(w_{\rm m}\) and 0.1489 for \(M_{\rm m}\)) and is basically identical to the COVn calculated by the Euler-Bernoulli beam model (0.0703 for \(w_{\rm m}\) and 0.1481 for \(M_{\rm m}\)). These findings further verify the effectiveness of the fitting curve and provide new understanding of the differences between the Timoshenko beam and the Euler-Bernoulli beam under the random field of the soil elastic modulus.