## 2 Equations Derivation of the Direct Calculation Method (DCM)
### Solution for the Support-Ground Interaction under the Plastic Condition  
When the stresses of the surrounding rock at the intrados of the tunnel are in the plastic region (Case II) [40]. Therefore, at the equilibrium state (point _E_), the mobilized support pressure (\(p_{s}\)) and the radial displacement (\(u_{R}^{s}\)) can be obtained as:  
\[P_{s}=k_{s}\left(\frac{u_{R}^{s}}{R}-\frac{u_{R}^{d}}{R}\right)=\sigma_{R}^{s} =\frac{2}{\left(K_{p}-1\right)}\left[\lambda_{e}\left(\frac{R}{R_{R}^{s}} \right)^{K_{p}-1}-N\right]\sigma_{v} \tag{5}\]  
\[\frac{2G}{\sigma_{o}}\frac{u_{R}^{s}}{R}=\lambda_{e}\Bigg{[}C_{1}+C_{2}\left( \frac{R}{R_{p}^{s}}\right)^{K_{p}-1}+C_{3}\left(\frac{R_{p}^{s}}{R}\right)^{K_ {p}+1}\Bigg{]} \tag{6}\]  
where \(K_{\psi}\) is the coefficient of the plastic flow, and \(C_{1}\), \(C_{2}\), and \(C_{3}\) are the coefficients of ground. In addition, the plastic zone radius at the moment of installing support (\(R_{p}^{d}\)) can be represented as:  
\[\frac{R_{p}^{d}}{R}=\Bigg{[}\frac{2\lambda_{e}}{\left(K_{p}+1\right)\lambda_{ e}-\left(K_{p}-1\right)\lambda}\Bigg{]}^{\frac{1}{K_{p}-1}} \tag{7}\]  
The mobilized pressure of the support system blocks the continuous expansion of the plastic zone radius and maintains a fixed value (\(R_{p}^{s}\)) at the equilibrium state (point _E_). For the expansion of the plastic zone, the calculation of the plastic zone radius due to the advancing excavation of the tunnel at different stress state (between point \(C\) and _E_) plays an important role in the analysis of DCM.  
As the stresses are in the plastic state, through the function of the DCM algorithmic process, a numerical method called [PERSON]'s recursive method is proposed, which can find the solution of the intersection between the two curves (SCC and GGC) in the equilibrium state (point \(E\)). Therefore, this function can be represented as:  
\[f\Big{(}\frac{R_{p}^{s}}{R}\Big{)}=\left[C_{2}-\frac{4G}{k_{s}(k_{p}-1)}\right] \left(\frac{R_{p}^{s}}{R}\right)^{1-K_{p}}+C_{3}\left(\frac{R_{p}^{s}}{R} \right)^{K_{p}+1}-\left[C_{2}\left(\frac{R}{R_{p}^{s}}\right)^{K_{p}-1}+C_{3} \left(\frac{R_{p}^{s}}{R}\right)^{K_{p}+1}\right]-\frac{4 GN}{3,k_{s}(k_{p}-1)} \tag{8}\]  
\[f^{\prime}\Big{(}\frac{R_{p}^{s}}{R}\Big{)}=\left[C_{2}-\frac{4G}{k_{s}(k_{p}-1 )}\right]\left(1-K_{p}\right)\left(\frac{R_{p}^{s}}{R}\right)^{-K_{p}}+C_{3} \big{(}K_{\phi}+1\big{)}\left(\frac{R_{p}^{s}}{R}\right)^{K_{\phi}} \tag{9}\]  
where the derivative of the function \(f(R_{p}^{s}/R)\) is represented by \(f^{\prime}(R_{p}^{s}/R)\). Therefore, the plastic zone radius (\(R_{p}^{s}\)), can be given as:  
\[\left(\frac{R_{p}^{s}}{R}\right)_{n+1}=\left(\frac{R_{p}^{s}}{R}\right)_{n}- \frac{f\Big{(}\left(\frac{R_{p}^{s}}{R}\right)_{n}\Big{)}}{f^{\prime}\Big{(} \left(\frac{R_{p}^{s}}{R}\right)_{n}\Big{)}} \tag{10}\]  
where the incremental step in the recursion is represented by \(n\). For substituting Equation (10) into Equations (5) and (6), the radial displacement and the mobilized support pressure at the equilibrium state can finally be obtained.  
Procedure of Numerical Analysis for the Direct Calculation Method (DCM) and Finite Element Method (FEM)