## 7 Conclusions  
This paper proposes a new numerical finite strain procedure for a circular tunnel in MC strain-softening rock masses with a non-associated flow rule. First, the explicit form of the differential relation between the original coordinate and the deformed coordinate is established in the Lagrangian coordinate. Using the governing equations, yield criterion, evolutional law, the solutions in the residue region, and the softening region are derived as two sets of differential equations of radial stress and plastic shear strain and the solutions in the elastic region are expressed as the differential equations of radial stress and tangential stress. By directly solving the differential equations using the Runge-Kutta method, the  
Figure 11: Sensitivity index of tunnel deformation. Figure 12: Stochastic sampling results: (**a**) tunnel displacement and support pressure; (**b**) tunnel displacement and support stiffness. problem of solving for the finite strain solution transforms into the problem of searching for two adjacent boundaries of three regions (\(R_{\rm s}\) and \(R_{\rm P}\)) and the initial values on the excavation boundary (\(R_{0}\)). Then the bisection method is adopted to approximate the true value based on the characteristics that the dependent variables (radial stress, plastic shear strain and tangential stress) in each region increase or decrease monotonously with the deformed radius. Two numerical procedures of calculating \(p_{\rm P}^{*}\) and \(p_{pe}^{*}\) and three procedures for the solutions in the residue-softening-elastic state, the softening-elastic state, and the elastic state respectively are programmed and integrated into the finite strain numerical procedure. The proposed finite strain procedure is validated by comparing with recursive solutions and FLAC simulation results. By increasing \(k_{\infty}\) and \(n\), the calculation relative error \(|\alpha|\) decreases to about \(10^{-14}\). When \(k_{\infty}\) and \(n\) are greater than 1000 and 10 respectively for this numerical example, \(\mu_{0}/R_{0}\), \(R_{\rm P}/R_{0}\) and \(R_{\rm r}/R_{0}\) become rather stable. The proposed procedure directly solves three sets of differential equations and uses the bisection method to approximate the boundary conditions, realizing quantitative control for the calculation error. More meaningfully, the highly efficient computing performance of the proposed procedure creates conditions for its engineering application in the global sensitivity analysis and the reliability design of the Zhongyi tunnel, which requires large sample data. Moreover, this paper develops a numerical procedure for circular tunnels excavated in strain-softening rock masses based on the small deformation hypothesis still using the iteration method. The correctness of this iteration method was verified again by comparing with a self-similar solution. However, this solution underestimates the self-bearing capacity of the surrounding rock and is inappropriate to be used to analyze a large deformation tunnel. In the end, the finite strain procedure was adopted for the sensitivity analysis and the reliability design of Zhongyi tunnel. The sensitivity indices of tunnel deformation to support pressure and geostress are much higher than others. This study recommends 4.3 MPa/m as the designed support stiffness to guarantee a tunnel displacement lower than 0.5 m.  
Conceptualization, W.C. and D.Z. ; methodology, W.C. and Q.F. ; software, W.C.; validation, W.C.; formal analysis, W.C.; investigation, W.C., X.C. and T.X. ; writing--original draft preparation, W.C.; writing--review and editing, D.Z. and Q.F. ; visualization, W.C., X.C. and T.X. ; supervision, D.Z. ; project administration, D.Z. and Q.F. ; funding acquisition, D.Z. All authors have read and agreed to the published version of the manuscript. This research was supported financially by the National Key Research and Development Program of China (Grant No. 2021 YFB2300803 and Grant No. 2017 YFC0805401) and National Natural Science of China (Grant No. 51738002).