## 1 Introduction  
With the rapid construction of transportation infrastructure, more and more tunnels have been built in soft rock under high geostress. Large deformation occurs frequently in such tunnel engineering and even causes roof collapse, steel ribs' twisting and breaking, shotcrete cracking, lining cracking, the invasion of primary lining into the space of secondary lining, and other disasters [1]. For instance, the maximum deformation of the Muzhailing Tunnel located upon the Lanzhou-Haikou National Expressway exceeds 2000 mm, far exceeding the deformation allowance [2]. Similar problems are encountered in the Muzhailing Railway Tunnel [3], Zhegu Mountain Tunnel [4], Huangjiazhai Tunnel [5], Minxian Tunnel [6] and have also been predicted to appear during the construction of 43 soft rock tunnels with high geostress in the Sichuan-Tibet railway [7]. The large deformation hazard raises significant challenges for the design of the tunnel support system. In recent years, scholars have been conducting considerable research on the mechanical calculation of tunnels excavated in strain-softening rock masses. [PERSON] et al. [8] and [PERSON] [9] derived a similarity solution for the circle tunnel in strain-softening rock mass, which it still needed to be solved by a numerical method. [PERSON] et al. [10] compared a simplified theoretical method with a rigorous theoretical method for tunnel excavation in the rock masses exhibiting strain-softening behavior. [PERSON] et al. [11], [PERSON] et al. [12], [PERSON] et al. [13], [PERSON] et al. [14], [PERSON] et al. [15], [PERSON] et al. [16], [PERSON] et al. [17], [PERSON] et al. [18], [PERSON] et al. [19], [PERSON] et al. [20],and [PERSON] et al. [21] proposed a series of new semi-analytical solutions and numerical procedures for calculating the stress and displacement around a circular tunnel excavated in strain-softening Mohr-Coulomb, generalized Hoek-Brown, or three-dimensional nonlinear Hoek-Brown rock mass. Although these methods were applied well on the mechanical analysis and the tunnel design [16; 22; 23; 24; 25; 26; 27], they are based on the small deformation hypothesis and not applicable to analyze a tunnel with a high risk of large deformation. For the large strain problem, [PERSON] [28], [PERSON] and [PERSON] [29], and [PERSON] [30; 31] derived finite strain closed-form solutions of circular opening issues in the elastic and the elasto-perfectly-plastic continuous medium based on a geometric equation in Lagrangian coordinates [32]. [PERSON] [33] proposed a large strain similarity solution, expressed as first-order ordinary differential equations, for a spherical or circular opening in elasto-perfectly-plastic rock masses. [PERSON] et al. [34] described the confinement-convergence responses for deep tunnels in linearly elastic, non-associated MC, and brittle Hoek-Brown rock masses using large-strain solutions of unified spherical and cylindrical cavity contraction. [PERSON] et al. [35] derived a large strain differential equations for elasto-brittle-plastic rock masses and implemented a numerical procedure. For a tunnel in strain-softening rock masses, [PERSON] et al. [36] analyzed a highly deformed circular tunnel in the Lagrangian coordinate by dividing the plastic zone into \(n\) concentric annuli and using analytical solutions for the elastic-brittle-plastic rock mass within each annulus. [PERSON] et al. [37] derived the differential equations in the original coordinate by the stress equilibrium equation and deformation compatibility equation and proposed one numerical procedure which updated the variables of \((i+1)\)th annulus in the softening region by the fifth-order Runge-Kutta method according to the data of the \(i\)th annulus. Then [PERSON] et al. [38] converted differential equations into integral equations and implemented a recursive procedure named'softening annulus loop'. In a similar method, [PERSON] et al. [39] developed a numerical solution for strain-softening rock masses obeying the GZZ strength criterion.