## 3 Hybrid Analysis
### Strategy for Symmetric Rigid-Body Displacements  
Hybrid analysis of a full-scale test of the symmetrically assembled tunnel ring, shown in Figure 1, subjected to symmetric external loadings, was carried out in [34]. Both external loading and measured discontinuities at the segmental joints were employed as input. It was found that the rigid-body displacements of the segments are responsible for 95% of the convergences, while the deformations of the segments resulting from the internal forces contribute to the convergences only by approximately 5%.  
The radial displacement of the axis of the tunnel lining at an arbitrary value of the angular coordinate, \(\varphi\), denoted as \(u(\varphi)\), was calculated as [34]  
\[u(\varphi)=u^{L}(\varphi)=\sum_{j=1}^{6}R\Delta\theta_{j}\sin(\varphi-\varphi_ {j})H(\varphi-\varphi_{j})\,. \tag{21}\]  
\(u(\varphi)\) results from the rigid-body displacements caused by the relative rotations of the joints, \(\Delta\theta_{j}\), located at \(\varphi=\varphi_{j}\) with \(j=1,2,\dots,6\). With respect to symmetric rigid-body displacements, i.e., \(\Delta\theta_{j}=\Delta\theta_{7-j}\) with \(j=4,5,6\), the vertical and horizontal convergences, denoted as \(C_{\triangledown}\) and \(C_{h}\), respectively, are obtained as  
\[C_{\triangledown} = u(\pi)=\sum_{j=1}^{3}R\Delta\theta_{j}\sin(\pi-\varphi_{j})\,, \tag{22}\] \[C_{h} = 2u\Big{(}\frac{\pi}{2}\Big{)}=2\sum_{j=1}^{2}R\Delta\theta_{j} \sin\Big{(}\frac{\pi}{2}-\varphi_{j}\Big{)}\,. \tag{23}\]  
The input variables, \(\Delta\theta_{j}\), are quantified and improved with the help of the strategy for symmetric rigid-body displacements. The Bernoulli-Euler hypothesis (index \(BEh\)) was employed for the first estimates of the relative rotations as the difference of the circumferential displacements, measured at the inner and outer surfaces, \(\Delta v_{j}^{\theta}-\Delta v_{j}^{i}\), divided by the sensor distance, denoted as \(l_{s}\), i.e.,  
\[\Delta\theta_{j}^{BEh} = \frac{\Delta v_{j}^{\theta}-\Delta v_{j}^{i}}{l_{s}}\,,\quad j=1,2,\dots,6\,. \tag{24}\]  
Then, \(\Delta\theta_{j=1,2,\dots,6}^{BEh}\) were symmetrized with respect to the vertical axis of the tunnel ring as  
\[\Delta\theta_{j}^{sym} = \frac{\Delta\theta_{j}^{BEh}+\Delta\theta_{7-j}^{BEh}}{2}\,,\quad j =1,2,3\,. \tag{25}\]This results in the reduction in the number of variables from six to three. Increments \(\Delta\theta_{j}^{corr}\) are added to \(\Delta\theta_{j}^{sym}\) to obtain relative rotations referring to rigid-body displacements (index \(rbd\)) as  
\[\Delta\theta_{j}^{rbd} = \Delta\theta_{j}^{sym}+\Delta\theta_{j}^{corr}\,,\quad j=1,2,3\,. \tag{26}\]  
In order to calculate \(\Delta\theta_{j}^{corr}\), the following conditions must be fulfilled: (1) the segmental rings must remain closed rings (continuity conditions), i.e., the displacements at both \(\varphi=0\) and \(\varphi=2\pi\) must be the same; (2) the discontinuities \(\Delta\theta_{j}^{rbd}\) refer to pure rigid-body displacements that do not generate internal forces in the ring; and (3) the quantities \(\Delta\theta_{j}^{corr}\) are defined as the minimum of the sum of their squares. Thus, they refer to the smallest improved increments. The solution for \(\Delta\theta_{j}^{corr}\) reads as [35]  
\[\Delta\theta_{1}^{corr} = -0.7651\Delta\theta_{1}^{sym}-0.3933\Delta\theta_{2}^{sym}+0.158 3\Delta\theta_{3}^{sym}\,, \tag{27}\] \[\Delta\theta_{2}^{corr} = -0.3933\Delta\theta_{1}^{sym}-0.3417\Delta\theta_{2}^{sym}-0.2651 \Delta\theta_{3}^{sym}\,,\] (28) \[\Delta\theta_{3}^{corr} = +0.1583\Delta\theta_{1}^{sym}-0.2651\Delta\theta_{2}^{sym}-0.893 3\Delta\theta_{3}^{sym}\,. \tag{29}\]