## 4 Extended Hybrid Method  
### Application of the Hybrid Method Using the Strategy of Symmetric Rigid-Body Displacements  
As described in Section 3.2, for symmetric problems, the covergences are linear functions of the radial displacements. This enables subdividing the calculation into two load cases. Load case I refers to point loads, while the relative rotations are set equal to zero. It was found that \(95\%\) of the convergences result from load case II [35]. As for load case II, the input variables, \(\Delta\theta_{j}\), are quantified and improved by the strategy of symmetric rigid-body displacements. The Bernoulli-Euler hypothesis (index \(BEh\)) was employed for computation of first estimates of the relative rotations as the difference in the circumferential displacement jumps, measured at the inner and the outer surfaces, \(\Delta v_{j}^{o}-\Delta v_{j}^{i}\), divided by the sensor distance \(l_{s}\); see Equation (24). Then, the quantities \(\Delta\theta_{j=1,2,\ldots,6}^{BEh}\) are symmetrized with respect to the vertical axis of the ring; see Equation (25). The increments \(\Delta\theta_{j}^{corr}\) are added to \(\Delta\theta_{j}^{sym}\) to obtain relative rotations referring to rigid-body displacements (index \(rbd\)); see Equation (26). The quantities \(\Delta\theta_{j}^{corr}\) are calculated by inserting Equation (26) into the conditions (30)-(32). With the help of the additional condition  
Figure 6: Categorization of real-scale tests of segmental tunnel linings as symmetric and asymmetric problems, respectively, based on the index \(\mathcal{I}_{\text{as}}\); the abscissa is limited to convergences that are so large that the serviceability is endangered; see Table 3.  
\[\frac{d\big{[}(\Delta\theta_{1}^{corr})^{2}+(\Delta\theta_{2}^{corr})^{2}+( \Delta\theta_{3}^{corr})^{2}\big{]}}{d\Delta\theta_{1}^{corr}} = 0\,, \tag{46}\]  
involving the smallest improved increments, \(\Delta\theta_{j}^{corr}\) is computed. The overall displacements are calculated by superposition of the displacements obtained from load case I and load case II. In load case I, the convergences are calculated with the help of the transfer relations (2). Point load intensities, recorded during testing, are inserted into Equations (4)-(9) as the load integrals of the external loading. The load integrals of the relative rotations are set equal to zero. \(u_{i}\), \(v_{i}\), and \(\theta_{i}\) follow from Equation (14). \(M_{i}\), \(N_{i}\), and \(V_{i}\) are obtained from Equations (18)-(20). In load case II, the convergences are calculated with the help of the transfer relations (2), where the load integrals of the external loading are set equal to zero. The load integrals of the relative rotations are obtained by inserting the relative rotations into Equations (11)-(13). \(u_{i}\), \(v_{i}\), \(\theta_{i}\), \(M_{i}\), \(N_{i}\), and \(V_{i}\) are set equal to zero. The above-described method was applied to the three tests. The error of the simulation is defined as  
\[\Delta C = \sqrt{\frac{\left[\Delta C_{p}^{2}+\Delta C_{h}^{2}\right]}{2}}\,, \tag{47}\]  
where  
\[\Delta C_{p} = C_{p}^{exp}-C_{v}^{mod}\,, \tag{48}\] \[\Delta C_{h} = C_{h}^{exp}-C_{h}^{mod}\,. \tag{49}\]  
\(C_{v}^{mod}\) and \(C_{h}^{mod}\) denote the vertical and horizontal convergences, respectively, obtained from the model simulation. Dividing \(\Delta C\) by the thickness of the segments, \(t\), a dimensionless index is obtained as  
\[\frac{\Delta C}{t} = \frac{\sqrt{\frac{1}{2}\left[\Delta C_{p}^{2}+\Delta C_{h}^{2} \right]}{t}}{\,. \tag{50}\]  
\(\Delta C\) and \(\Delta C/t\) are shown in Figure 7 as functions of the larger convergence, \(C\). The following categorization is proposed. Provided that the error index \(\Delta C/t\) is smaller than or equal to 0.01, the accuracy is acceptable. If the error index \(\Delta C/t\) is larger than 0.01, the accuracy is not acceptable. It is shown that the structural displacements in Test 1 are reproduced with high accuracy during the whole loading process.