## 4 Numerical Simulation
### Internal Force Analysis of Initial Support  
The initial support axial force and bending moment diagrams of the tunnel are shown in Figures 20 and 21, respectively (taking the collapse treatment as an example). The axial forces on the vault, arch waist, and sidewall were all pressure forces, while the middle part of the inverted arch experienced tension forces. After treatment, the maximum axial compression force of the initial support occurred at the arch waist, with a value of \(-\)5253.45 kN/m. The thickness of the concrete slab element in the model was 0.32 m, so the maximum axial compressive force at the arch waist was \(-\)1681.1 kN. The maximum axial tensile force occurring at the inverted arch was 552.08 kN. The maximum bending moment occurred at the right corner, with a value of \(-\)163.73 kN\(\cdot\)m/m.  
According to the Code for the Design of Highway Tunnels [40], the strengths of the concrete eccentric compression members were checked according to the failure stage. The formula was as follows:  
\[K=\frac{N_{limit}}{N}\geq K_{regulations} \tag{6}\]  
where \(K\) is the safety factor of the tunnel lining structure. When the concrete reaches the ultimate compressive or shear strength, the safety factor should be \(\geq\)2.0. When the concrete reaches the ultimate tensile strength, the safety factor should be \(\geq\)2.4.  
When controlled by compressive strength, that is, \(e=\frac{M}{N}\leq 0.2h\), the ultimate bearing capacity is calculated as follows:  
\[N_{limit}=\varphi\alpha R_{a}bh \tag{7}\]  
where \(e\) is the eccentricity, \(M\) is the bending moment, \(N\) is the axial force, \(\varphi\) is the longitudinal coefficient of the member, taken as \(\varphi=1\), \(R_{a}\) is the ultimate compressive strength of concrete, \(b\) is the section width and \(\alpha\) is the eccentricity influence coefficient of the axial force. The value \(\alpha\) is determined according to the following empirical formula:  
\[\alpha=1-1.5\frac{e}{h} \tag{8}\]  
where \(h\) is the section thickness.  
When controlled by tensile strength, that is, \(e=\frac{M}{N}\leq 0.2h\), the ultimate bearing capacity is calculated as follows:  
\[N_{limit}=\varphi\frac{1.75R_{1}bh}{\frac{e\epsilon}{h}-1} \tag{9}\]  
where \(R_{1}\) is the ultimate tensile strength of concrete.  
According to the above formula, the safety factors of each position in the original design scheme (Table 4) and reinforcement treatment (Table 5) were calculated. As shown in Table 4, when using the original design scheme, the safety factors of the arch crown and the left and right corners did not meet the required value of 2.4 specified in the code [26]. Other locations were resistant to compression damage. Although the safety factor met the required value of 2.0, the safety factor was generally low. After the collapse treatment, the safety factors at each position of the initial support met the requirements. The minimum safety factors at the left and right wall feet were 2.7 and 2.6, respectively, but they were also greater than the tensile strength safety factor of the reinforced concrete structure of 2.4. Thus, the collapse treatment measures proposed in this paper meet the engineering requirements and can serve as references for similar projects.  
Figure 21: Bending moment diagram of each node of the initial support after treatment.