## 3 Hybrid Analysis  
### Transfer Relations  
Transfer relations, obtained by means of the first-order theory of slender circular arches, are employed for the structural analysis. They read as [34]  
\[\left[ \begin{array}{c}u(\varphi)\\ v(\varphi)\\ \theta(\varphi)\\ M(\varphi)\\ N(\varphi)\\ V(\varphi)\\ 1\end{array} \right]=\left[ \begin{array}{cccccccc}\cos\varphi&\sin\varphi&T_{1 3}(\varphi)&T_{14}(\varphi)&T_{15}(\varphi)&T_{16}(\varphi)&\overset{\prime} {\underset{\}{\sum}}u^{L}(\varphi)\\ -\sin\varphi&\cos\varphi&T_{23}(\varphi)&T_{24}(\varphi)&T_{25}(\varphi)&T_{26} (\varphi)&\overset{\prime}{\underset{\}{\sum}}v^{L}(\varphi)\\ 0&0&1&T_{34}(\varphi)&T_{35}(\varphi)&T_{36}(\varphi)&\overset{\prime}{\underset {\}{\sum}}\delta^{L}(\varphi)\\ 0&0&0&1&T_{45}(\varphi)&T_{46}(\varphi)&\overset{\prime}{\underset{\}{ \sum}}M^{L}(\varphi)\\ 0&0&0&0&\cos\varphi&-\sin\varphi&\overset{\prime}{\underset{\}{\sum}}N^{L}( \varphi)\\ 0&0&0&0&\sin\varphi&\cos\varphi&\overset{\prime}{\underset{\}{\sum}}V^{L}( \varphi)\\ 1\end{array} \right]\left[ \begin{array}{c}u_{i}\\ v_{i}\\ \theta_{i}\\ M_{i}\\ N_{i}\\ V_{i}\\ 1\end{array} \right], \tag{2}\]  
Figure 4: Positions of the displacement sensors measuring the inner and outer circumferential displacements of the joints [35]; the sensor distances, \(\ell_{s}\), amounted to 0.40 m in Test 1 and to 0.35 m in Tests 2 and 3. \begin{table}
\begin{tabular}{l c c c c c} \hline class & A & B & C & D \\ serviceability & excellent & good & endangered & violated \\ criterion & \(C\leq 50\) mm & \(50\) mm \(<C\leq 75\) mm & \(75\) mm \(<C\leq 125\) mm & \(C>125\) mm \\ \hline \end{tabular}
\end{table} 
Table 3: Convergence-related serviceability of the tested segmental tunnel linings resulting from use of \(D_{0}\) in Table 1.  
Figure 3: Experimental data of Test 3: (**a**) intensities of the point loads; (**b**) measured convergences; (**c**,**d**) refer to measured inner and outer circumferential displacements of the joints, respectively. where [34]  
\[ \begin{array}{l}T_{13}(\varphi)=R\ \sin\varphi\,,\qquad\quad T_{14}(\varphi)= \frac{R^{2}}{EI}(\cos\varphi-1)\,,\\ T_{15}(\varphi)=\frac{R}{EI}\frac{1}{2}\varphi\sin\varphi+\frac{R^{3}}{EI} \Big{(}\frac{1}{2}\varphi\sin\varphi+\cos\varphi-1\Big{)}\,,\\ T_{16}(\varphi)=\frac{R}{EI}\Big{(}\frac{1}{2}\varphi\cos\varphi-\frac{1}{2} \sin\varphi\Big{)}+\frac{R^{3}}{EI}\Big{(}\frac{1}{2}\varphi\cos\varphi-\frac{1 }{2}\sin\varphi\Big{)}\,,\\ T_{23}(\varphi)=R\ (\cos\varphi-1)\,,\quad T_{24}(\varphi)=\frac{R^{2}}{EI}( \varphi-\sin\varphi)\,,\\ T_{25}(\varphi)=\frac{R}{EI}\Big{(}\frac{1}{2}\varphi\cos\varphi+\frac{1}{2} \sin\varphi\Big{)}+\frac{R^{3}}{EI}\Big{(}\varphi-\frac{3}{2}\sin\varphi+ \frac{1}{2}\varphi\cos\varphi\Big{)}\,,\\ T_{26}(\varphi)=\frac{R}{EI}\Big{(}-\frac{1}{2}\varphi\sin\varphi\Big{)}+\frac {R^{3}}{EI}\Big{(}1-\cos\varphi-\frac{1}{2}\varphi\sin\varphi\Big{)}\,,\\ T_{34}(\varphi)=-\frac{R}{EI}\varphi\,,\qquad\quad T_{35}(\varphi)=\frac{R^{2 }}{EI}(\sin\varphi-\varphi)\,,\quad T_{36}(\varphi)=\frac{R^{2}}{EI}(\cos \varphi-1)\,,\\ T_{45}(\varphi)=R\ (1-\cos\varphi)\,,\quad T_{46}(\varphi)=R\ \sin\varphi\,.\end{array}  \tag{3}\]  
In Equations (2) and (3), \(u\) and \(v\) denote the radial and circumferential displacements of the axis of the arch, \(\theta\) denotes the rotation of the cross-section, and \(\varphi\) denotes the circumferential coordinate. Moreover, \(R\), \(EI\), and \(EA\) stand for the radius of the axis of the arch, the bending stiffness, and the extensional stiffness, respectively. Further, \(M\), \(N\), and \(V\) stand for the bending moment, the axial force, and the shear force, respectively. The vector on the left-hand side of Equation (2) refers to the radial and circumferential displacements, the cross-sectional rotation, and the inner forces of the cross-section at an arbitrary value of the angular coordinate \(\varphi\).