## 1 Introduction  
**Rock bolts have been used in underground engineering to stabilize rock masses for more than 100 years** **[**1**]****. At present, many kinds of rock bolts have been developed, among which, fully grouted passive bolts are the most popular type and have been widely employed due to their economical and convenient nature. The analytical solution for a deep circular tunnel reinforced with passive bolts in a hydrostatic field is a classical problem in underground engineering. Although the practicability of the analytical solutions is limited by their assumptions, they can be employed to quickly assess the mechanical behaviors of the rock mass and reinforcement and can be used for preliminary design analyses or the pre-dimensioning of the reinforcement** **[**2, 3**]****. **By assuming that the rock mass and the passive bolts have the same strain--that is, no relative displacement occurs at the bolt-rock interface--a series of analytical and semi-analytical solutions were formulated by many authors [2; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] in terms of the elastic, viscoelastic, and elastoplastic rock mass, starting in the 1980s. However, according to the in situ test results reported by [PERSON] [14], the relative displacement between the bolts and rock mass is produced with the advance of the tunnel face, which results in shear stresses at the bolt-rock interface and induces a tension force in the bolts. In addition, [PERSON] et al. [15] reported that this assumption causes an overestimation of the tensile force in the bolts and leads to an overly conservative design. Therefore, it is necessary to reasonably consider the interaction between the bolts and the rock mass. To simplify this problem, an equivalent material method was developed by [PERSON] and [PERSON] [16; 17] to consider a proper interaction mechanism, and the analytical solution for a reinforced deep circular tunnel in a Mohr-Coulomb rock mass is proposed. [PERSON] and [PERSON] [18; 19] and [PERSON] and [PERSON] [20] improved this solution by incorporating the Hoek-Brown failure criterion to represent the rock mass behavior using a more rigorous assessment of the strain compatibility. However, these solutions determine the position of the neutral point of the passive bolts using an empirical equation [21] instead of a rigorous theoretical analysis. The empirical equation assumes that the position of the neutral point is only related to the tunnel radius and the bolt length, which is inconsistent with the results reported by [PERSON] et al. [22; 23] and [PERSON] and [PERSON] [24], which state that the position of the neutral point is related to the properties of the bolts and rock mass. The aforementioned studies show that it is quite difficult to derive an analytical solution to reasonably assess the interaction between passive bolts and the rock mass due to the complexity of this problem. [PERSON] et al. [25], [PERSON] [26], and [PERSON] [27; 28] all proposed a semi-analytical solution of great interest; however, [PERSON] et al. [25] mechanically defined the position of the neutral point and neglected the effect of the end plate. [PERSON] [26] modeled the shear distribution at the bolt-rock interface with a predefined interaction curve for the shear stress. [PERSON] et al. 's and [PERSON]'s solutions are limited to the Mohr-Coulomb failure criterion. Moreover, in both [PERSON] et al. 's and [PERSON]'s solutions, the advance of the tunnel face is simulated using a gradually decreased support pressure from the initial stage to the final stage. In each intermediate stage, an iterative process is required to evaluate the mechanical behaviors of the passive bolts and rock mass; therefore, numerous stages are needed to ensure the accuracy of the analysis, which is cumbersome and time-consuming. Recently, [PERSON] et al.