## 4 Improved Expression of Longitudinal Deformation Profile (LDP) for Soft Rock Tunnel  
To demonstrate the application of the improved numerical simulation method, an improved expression of the longitudinal deformation profile (LDP) for soft rock tunnels considering confining pressure influence is proposed according to many numerical simulation results for soft rock tunnels obtained by the improved numerical simulation method. Therefore, this section first obtains many longitudinal deformation profile (LDP) results for soft rock tunnels with different initial pressure conditions by an improved numerical simulation method. Furthermore, an improved expression of the LDP is proposed by a mathematical fitting method based on many numerical simulation results of the LDP under different pressure conditions.  
Longitudinal deformation profiles for soft rock tunnels under 1-20 MPa initial pressure conditions (interval of condition is 1 MPa) are obtained by an improved numerical simulation method. The tunnel radius is 3 m, the supporting force is 0 kPa, the excavation footage is 0.5 m and the constitutive model of the surrounding rock element is the Mohr-Coulomb model. Variable models for the mechanical parameters of surrounding rock are listed in Table 8.  
Because tunnels are circular and axisymmetric structures, a quarter numerical finite element model is established as shown in Figure 16a. The left and bottom boundaries of the model are fixed boundaries, and the outer surface of the model is the stress boundary with the value of the initial pressure. Meanwhile, the longitudinal deformation profile (LDP) of the surrounding rock is represented by the normalized release coefficient of displacement and the normalized distance from the tunnel face [27; 28], as shown in Figure 16b.  
Figure 16: (**a**) Finite element model and (**b**) normalized longitudinal deformation profile of surrounding rock.  
Figure 17 shows the numerical simulation results of the LDP under 1-20 MPa initial pressure conditions. For a better display, the complete LDP is divided into two parts, with the tunnel face as the discontinuity point: (a) LDP in front of the tunnel face and (b) LDP behind the tunnel face.  
According to the influence analysis of multiple factors on the LDP from previous scholars [28; 29], the initial pressure, tunnel radius and mechanical parameters of surrounding rock all have a significant influence on the LDP; however, the mechanical parameters of surrounding rock for soft rock tunnels are variable, resulting in the mechanical parameters of surrounding rock not acting as independent variables to fit the expression of the LDP. Therefore, \(R^{*}\) is introduced into the improved expression of the LDP and \(R^{*}=R_{\max}/r_{0}\) (\(R_{\max}\) is the maximum plastic zone radius of surrounding rock without support, and \(r_{0}\) is the tunnel radius). \(R^{*}\) can implicitly reflect the comprehensive effect of the initial pressure, mechanical parameters of surrounding rock and tunnel radius [28]. The Levenberg-Marquardt and universal global optimization methods are applied in the fitting process for the expression of the LDP for soft rock tunnels. The improved expressions of the LDP for soft rock tunnels are listed in Table 9.  
Where \(u_{0}^{*}=u(x=0)/u_{max}\), \(R^{*}=R_{max}/r_{0}\), \(x^{*}=x/r_{0}\), \(x\) is the longitudinal distance from the tunnel face, \(r_{0}\) is the tunnel radius, \(u\) is the displacement around the tunnel at \(x\) distance from the tunnel face and \(R_{\max}\) is the maximum plastic zone radius of the surrounding rock without support.