## 3 Elastic Foundation Beam Model Used in the Small Radius TBM Interval
### Model Solution
Therefore, additional deflection caused by the concentrated moment \(M_{i}\) is as below [34],  
\[y(\Phi)_{M_{i}}=M_{i}F_{y3}\Big{(}\Phi-\Phi_{M_{i}}\Big{)}\Big{(}\Phi\geq\Phi_ {M_{i}}\Big{)}, \tag{12}\]  
The additional defection caused by the concentrated force \(P_{i}\) when \(\Phi\geq\Phi_{Pi}\) is shown below [34],  
\[y(\Phi)_{P_{i}}=-P_{i}F_{y4}\Big{(}\Phi-\Phi_{P_{i}}\Big{)}\Big{(}\Phi\geq\Phi _{P_{i}}\Big{)}, \tag{13}\]  
The additional defection caused by the radial distribution load \(q_{ri}\) when \(\Phi\geq\Phi_{ai}\) can be obtained [34],  
\[y(\Phi)_{q_{ti}}=-D\int_{\Phi_{ci}}^{\Phi}q_{t}(\delta)F_{y5}(\Phi-\delta)d \delta(\Phi\geq\Phi_{ci}), \tag{14}\]  
When \(\Phi\geq\Phi_{bi}\), the integral upper limit \(\Phi=\Phi_{bi}\) is selected. The additional defection caused by the tangential distribution load \(q_{ti}\) when \(\Phi\geq\Phi_{ci}\) can also be obtained [34],  
\[y(\Phi)_{q_{t}}=-D\int_{\Phi_{c}}^{\Phi}q_{t}(\delta)F_{y5}(\Phi-\delta)d \delta(\Phi\geq\Phi_{c}), \tag{15}\]  
When \(\Phi>\Phi_{di}\), the integral upper limit \(\Phi=\Phi_{di}\) is selected. With the initial conditions while \(\Phi=0\) and the above external load, the deflection of the curved beam at any points can be achieved [34],  
\[ \begin{array}{l}y(\Phi)=&y_{0}F_{y1}(\Phi)+\theta_{0}F_{y2}(\Phi)+M_{0}F_{y3 }(\Phi)+Q_{0}F_{y4}(\Phi)+N_{0}F_{y5}(\Phi)+\sum\limits_{i=1}^{n_{M}}M_{i}F_{ y3}\Big{(}\Phi-\Phi_{M_{i}}\Big{)}\\ &-\sum\limits_{i=1}^{n_{P}}P_{i}F_{y4}\Big{(}\[PERSON]-\[PERSON]_{P_{i}}\Big{)}-\sum \limits_{i=1}^{n_{M}}D\int_{\[PERSON]_{ai}}^{\[PERSON]}q_{ri}(\delta)F_{y4}(\[PERSON]-\delta )d\delta-\sum\limits_{i=1}^{n_{M}}D\int_{\[PERSON]_{ci}}^{\[PERSON]}q_{ti}(\delta)F_{y5} (\[PERSON]-\delta)d\delta\end{array}  \tag{16}\]  
With the same procedure, we can obtain the analytical expressions of rotation angle \(\theta\), bending moment \(M\), shear force \(Q\), and axial force \(N\) when some loads are acting on the beam span. The TBM tunnel segment experiences leaving the shield tail, pea gravel backfilling, and grouting in turn. There are differences in the stiffness of the medium near the segment and the deformation of segment cross-section with loading. Therefore, the bending of the limited length variable cross-section circular curved beam on the inhomogeneous Winkler foundation needs further discussion. Figure 6 shows the equal section circular curved beam subjected to the distributed load on the homogeneous Winkler foundation. The analytical expressions of the displacement and the internal force can be expressed in the following matrix form [34]. \[A(\Phi)_{5\times 1}=B(\Phi)_{5\times 5}A(0)_{5\times 1}+C(\Phi)_{5\times 1}+D( \Phi)_{5\times 1}, \tag{17}\]  
where,  
\[A(\Phi)_{5\times 1}=[y(\Phi),\theta(\Phi),M(\Phi),Q(\Phi),N(\Phi)]^{T};\]  
\[B(\Phi)_{5\times 5}=\left[ \begin{array}{cccc}F_{y1}(\Phi)&F_{y2}(\Phi)&F_{y3}( \Phi)&F_{y4}(\Phi)&F_{y5}(\Phi)\\ F_{\theta 1}(\Phi)&F_{\theta 2}(\Phi)&F_{\theta 3}(\Phi)&F_{\theta 4}(\Phi)&F_{ \theta 5}(\Phi)\\ F_{M1}(\Phi)&F_{M2}(\Phi)&F_{M3}(\Phi)&F_{M4}(\Phi)&F_{M5}(\Phi)\\ F_{Q1}(\Phi)&F_{Q2}(\Phi)&F_{Q3}(\Phi)&F_{Q4}(\Phi)&F_{Q5}(\Phi)\\ F_{N1}(\Phi)&F_{N2}(\Phi)&F_{N3}(\Phi)&F_{N4}(\Phi)&F_{N5}(\Phi)\\ \end{array} \right];\]  
\[A(0)_{5\times 1}=\left[y_{0},\theta_{0},M_{0},Q_{0},N_{0}\right]^{T};\]  
\[C(\Phi)_{5\times 1}=\left[C_{y}(\Phi),C_{o}(\Phi),C_{M}(\Phi),C_{Q}(\Phi),C _{N}(\Phi)\right]^{T};\]  
\[D(\Phi)_{5\times 1}=\left[D_{y}(\Phi),D_{\theta}(\Phi),D_{M}(\Phi),D_{Q}( \Phi),D_{N}(\Phi)\right]^{T};\]  
Figure 4: Uniform cross-section circular curved beam on the Winkler elastic foundation [35]. Figure 5: Diagram of force analysis on the microelement body [35]. The elements in the matrix \(C(\varPhi)_{5\times 1}\) are as follows [34],  
\[\left\{