## 1 Introduction  
Tunnel excavations generate displacements/convergence owing to the redistribution of stresses at the excavation periphery. Such convergence/deformation in tunnels is an important factor for evaluating tunnel behavior. Tunnel deformations depend on the rock mass properties and excavation-induced stress environments. Analyzing tunnel convergence is considered important for implementing appropriate excavation and support measures. The longitudinal deformation profile (LDP) is a graph that longitudinally represents the radial displacement of the ground around the tunnel during excavation; it is used to determine the unsupported tunnel distance from the excavation face and considered important for determining the support installation distance and timing in the convergence-confinement method (CCM) [1]. This method (CCM) is widely used in road, hydropower, mining and nuclear waste repository projects that normally involve tunneling and underground excavations. The CCM can be used to estimate the load acting on a support installed behind the tunnel face and to control the displacement at the tunnel excavation face to ensure tunnel stability. The principle of the CCM has been discussed by several previous researchers [1; 2; 3]. The CCM involves three curves: (1) the LDP, (2) the ground reaction curve (GRC), and (3) the support characteristic curve (SCC); these are illustrated in Figure 1.  
Figure 1: Schematic of the longitudinal deformation profile (LDP), ground reaction curve (GRC), and support characteristic curve (SCC) [1]. A brief description of each point shown in Figure 1 is as follows and a list of nomenclature for symbols is given in Appendix A:  
1. Longitudinal deformation profile * Point F: Convergence in the tunnel face * Point I: Convergence at a location separated by distance L from the back of the tunnel face, \(u_{r}^{0}\). 2. Ground reaction curve * Point O: Initial state, \(p_{i}=\sigma_{0},\ u_{r}=0\) * Point M: Final point of reach if there are no support members, \(p_{i}=0,\ u_{r}=u_{r}^{m}\) * Point E: When a plastic area is created around the tunnel excavation surface, \(p_{i}=p_{r}^{cr}\) * Point N: Virtual ground pressure when installing the support members. 3. Support characteristic curve * Point K: The time of installing the support members, \(p_{s}=0,\ u_{r}=u_{r}^{0}\) * Point D: Status of ground pressure exerted by the additional excavation after installing the support members, \(p_{s}=p_{s}^{D}\) * Point R: Yield of support members, \(p_{s}=p_{s}^{max}\). The complexities of tunnel behavior are dependent on the geological and ground conditions, excavation method, and installed support. Many researchers have used numerical analysis methods to evaluate tunnel behaviors, because these methods can ideally model the tunnel excavation process under various conditions and assess the LDP efficiently. [PERSON] [4] analyzed the convergence during tunnel excavation using axisymmetric finite element analyses under elastic, plastic, and perfect elastoplasticity failure conditions and proposed an equation for convergence. [PERSON] [5] reported that the deformation due to tunnel excavation first occurs at the front of the excavation surface, resulting in approximately one-third of the total displacement at the excavation surface; this was found to vary according to the ratio of the rock mass strength to the in situ stress. [PERSON] and [PERSON] [3] performed a numerical analysis on various ground conditions based on the Hoek-Brown model and proposed a robust formula to estimate the longitudinal deformation profile (LDP) using plastic radius as a parameter. [PERSON] et al. [6] presented a new approach to calculate the simplified approximation formula of the plastic area and the LDP according to the \(GSI\) under strain-softening rock mass conditions. [PERSON] et al. [7] presented a new type of LDP using numerical simulations and regression analyses; this profile was a function of the \(GSI\) and tunnel cover depth.