###### Abstract  
In engineering practice, the initial support system is commonly installed in the vicinity of the tunnel face after excavation, whereas the self-capacity of rock mass will fail to be utilized and the cost of the initial support system will be expensive. In this study, a methodology is proposed to determine the appropriate timing of initial support installation to find out the balance of tunnel safety and construction cost. Firstly, the global safety factor is introduced as the critical indicator to evaluate tunnel stability. Then, the comprehensive graphic relationship between the global safety factor and the distance to the tunnel face is established. Once the global safety factor decreases to an admissible value, the stability of the surrounding rock is in a critical state and the corresponding distance is the recommended location for installing the initial support. In these procedures, the installation time of the initial support at the typical tunnel section can be quickly designed and fed back by a direct indicator during construction. Meanwhile, several cases with different conditions have been carried out to discuss the regularity of the method. t 1999 Namely, a part of the load after redistribution around the excavation boundary will be carried by the face itself [8]. Therefore, this method emphasizes that the deformation of rock mass should be restricted for the stability, while the moderate deformation should be allowed to form the bearing rock arch to take full advantage of the rock capacity [9]. In this way, the design of the initial support is more economical and the unsupported span before support can be estimated. On the basis of NATM, the analytical method, namely the convergence-confinement method, is developed in 1960s and 1970s by [PERSON] [10] to illustrate the evolution of the rock-support system by three characteristic curves in Figure 1: (a) the Longitudinal Displacement Profile (LDP); (b) the Ground Reaction Curve (GRC); (c) the Support Characteristic Curve (SCC). Additionally, the rock-support system achieves equilibrium at the intersection of the GRC and SCC in Figure 1 [8]. This method provides a theoretical approach to describe the interaction between the rock mass and support structures during tunneling [11], and it can be further developed by combining with the time-dependent behavior of rock mass [12; 13]. Moreover, to obtain the in-situ observational data of rock and support structures, geotechnical monitoring, namely the observational method, develops as an essential part in the underground constructions by immediately evaluating the monitoring data and feeding back in design [14]. Then, due to the complexity of underground engineering, numerical analyses, such as the finite element method (FEM) [15], the finite difference method (FDM) [16; 17], the distinct element method (DEM) [18], the boundary element method (BEM) [19], and so on [20], have prevailed in tunnel design. What distinguished the numerical method from the conventional method is that this method could model the complicated geotechnical construction in the visual and convenient way. In engineering applications, the numerical method is commonly used to validate the reliability of the design [21]. Besides, the back analysis, combining the numerical method with the observational method, is also widely used [22]. In summary, the majority of the initial support design methods are aimed to choose appropriate support types and parameters, while the installation time of the initial support is commonly designed by semi-empirical methods. According to the modern support design theory, the early installation of support structures will lead to excessive load on the support structures and fail to make full use of the rock capacity, which is uneconomical. Contrarily, the delayed installation of support structures will lead to excessive deformation and threaten the safety of the tunnel. Consequently,  
Figure 1: Schematic representation of the convergence-confinement method [8].