## 1 Introduction  
Circular shield tunnels are widely employed around the world because they have the advantage of the arching effect. Circular segmental tunnel linings are sensitive to the reduction in the horizontal load and the increase of the vertical load. The damage of the tunnel lining of the Panchiao Line in Taipei resulted from an adjacent excavation [1]. Moreover, serviceability problems occur because of unfavorable environmental conditions, such as the cracking of concrete [2], water ingress through the tunnel lining [3], changes in the groundwater level [4], and the opening of segmental joints [5]. Convergences, i.e., changes in the diameter of a circular tunnel lining, are chosen as the indicator to define the serviceability limit states (SLSs) of shield tunnels. Large convergences have a negative influence on traffic operation because of the reduction in the cross-section. In the design of future tunnels, the convergence-related SLSs in soft clay were defined as 0.3-0.5% of the outer diameter of the circular tunnel lining, \(D_{o}\), in several design codes [6; 7; 8; 9] and in the literature [10; 11]. As shown in Table 1, four convergence-related serviceability classes were defined for the operation of the existing shield tunnel lings [12; 13], in which \(C\) denotes the maximum of the absolute values of the horizontal and vertical convergences, i.e., \(C_{h}\) and \(C_{v}\):  
\[C=\max\{|C_{h}|;|C_{v}|\}\,. \tag{1}\]  
A reference value, \(D_{ref}\), is defined as \(0.4\%D_{o}\). The serviceability of a segmental tunnel lining is excellentwhen \(C\) is smaller than or equal to \(2D_{ref}\), i.e., \(0.8\%D_{o}\). It is good when \(C\) is larger than \(0.8\%D_{o}\) and smaller than or equal to \(1.2\%D_{o}\), endangered when \(C\) is larger than \(1.2\%D_{o}\) and smaller than or equal to \(2.0\%D_{o}\), and violated when \(C\) is larger than \(2.0\%D_{o}\). If the serviceability is endangered, strengthening methods, such as bonded steel plate strengthening [14; 15; 16; 17], should be applied to the tunnel lining. There are many technologies to measure the convergences of tunnels, including tape extensometers [18; 19; 20; 21], laser profilers [22; 23; 24], and total station surveying [25]. Moreover, Light Detection and Ranging (LiDAR) is employed to acquire three-dimensional point cloud data quickly [26]. This technology is used in various geological environments, such as tunnels [27], landslides [28], volcanic environments [29], and rockfalls [30]. It is used to collect convergence data with high accuracy and efficiency, which facilitates the back analysis of the structural behavior of tunnels. However, specific machines and a great deal of space are required for the application of this measuring technology. Hence, tape extensometers are still widely used for the convergence measurements in the laboratory. The analysis of the structural behavior of segmental tunnel linings is complicated because of the complexity of the structure-soil-water interaction and the discontinuity of the segmental joint. A hybrid approach, which combines numerical studies and analytical solutions, was employed to develop a model considering the tunnel lining as a continuous ring with adjusted rigidities. This improves the calculation efficiency of the numerical simulations [31]. A machine learning algorithm, based on the results from the model and numerical tests, was used to analyze the effects of the parameters that affect the structure-soil-water interaction behavior [32]. Another hybrid model was established with which the internal forces of the tunnel lining were obtained by employing the in situ-monitored convergences as the input data of a numerical model [33]. The structural deformation state of a symmetrically assembled segmental ring, subjected to symmetric external loading, can be categorized as a symmetric or an asymmetric problem [13].