## 4 Extended Hybrid Method
### Presentation and Discussion of Results  
The improved strategy of symmetric rigid-body displacements was applied to Tests 1, 2, and 3 from the beginning of the loading process. The development of \(\alpha\), as the larger convergence \(C\) increases, is shown in Figure 8. For Test 1, \(\alpha\) is small, and it changes only mildly during the loading process. This underlines that realistic relative rotations of the joints are obtained by means of combining the measured circumferential displacement jumps at the inner and outer surfaces of the joints with the Bernoulli-Euler hypothesis, followed by symmetrization. The deformed structural configuration is reproduced with high accuracy. However, for Tests 2 and 3, \(\alpha\) increases strongly when the larger convergences exceed 75 mm (serviceability class C). This implies that, for Tests 2 and 3, extra rigid-body displacements are required in order to reproduce the measured convergences once the structure enters serviceability class C. The reason for this situation is that sudden failure, such as spalling of concrete, occurred in the vicinity of the segmental interfaces when the structures were close to SLS. This resulted in movements of the displacement sensors deployed on the surfaces of the interfaces. This leads to inaccuracies in the measured data inthe tests at the last load steps. Such failure was observed in the tests. As shown in Figure 9, the outer concrete spalled at joint 1 at load level 22 in Test 2 and at joint 6 at load level 31 in Test 3. Hence, in such situations, additional data, i.e., measured convergence, are employed to reproduce the accurate deformed structural configuration. Different from the asymmetric problem, where the asymmetric structural behavior develops in a progressive and subtle fashion [13], for the third class of problems, this behavior develops in a sudden fashion.  
Figure 8: \(\alpha\)—larger convergence \(C\) diagrams using the improved strategy of symmetric rigid-body displacements.  
Figure 9: Concrete spalling at the outer surface: (a) at segmental joint 1 at load level 22 in Test 2 and (b) at segmental joint 6 at load level 31 in Test 3.  
In practical applications, the outer surface is inaccessible. Thus, the cracking and spalling of the concrete at the outer surface remain unnoticed. If the tunnel is in an aggressive environment, with a particular risk of corrosion of the reinforcement of the segments, cracking at the outer surface is not acceptable. In such a case, it is better to reduce \(C_{ref}=0.4\%D_{o}\) to \(C_{ref}=0.3\%D_{o}\), used herein and in [13]. Otherwise, \(C_{ref}=0.4\%D_{o}\) is sufficient.  
To reproduce the deformed structural configurations, the following procedure is proposed. As shown in Figure 10, at first, the symmetry indicator \(\mathcal{I}_{as}\) is calculated with the help of the measured vertical and horizontal convergences to categorize the problem as symmetric or asymmetric. As for the latter, the strategy of general rigid-body displacements is to be employed. For symmetric problems, the strategy of symmetric rigid-body displacements is to be used. The accuracy of the simulation is checked by the error index \(\Delta C/t\). If this index is smaller than 0.01, the calculation is finished. However, if it is larger than 0.01, the here-proposed strategy of symmetric rigid-body displacements should be employed.