# Safety Analysis of Secondary Lining of Yulinzi Tunnel Based on Field Monitoring
## 4 Analysis of Test Results
### Calculation of Internal Force and Safety Factor
#### 4.2.1 Calculation Idea of Section Internal Force  
The idea of calculating the internal force of the section is first based on the four assumptions of the calculation of the bearing capacity of the normal section of the eccentrically compressed member of the rectangular section, which are as follows:  
1. Deformation coordination introduces the assumption of plane section. The assumption of the plane section refers to the assumption that the average strain distribution of concrete and longitudinal reinforcing bars within the height range of a concrete structural member along the positive section is linear after the member is subjected to forces. This assumption is applicable to eccentrically compressed members where each section satisfies this assumption during the load-bearing process;
2. Assume synchronous deformation of steel bar and surrounding concrete when deformation is coordinated. Ribbed steel bars are used in reinforced concrete, and there is significant friction between the steel bar surface and the concrete. During the structural loading process, the relative displacement between the steel bars and the concrete is very small. Therefore, it can be approximately assumed that the compressive strain of the steel bars is equal to the strain value of the corresponding position in the concrete;
3. Neglect the tensile strength of the concrete in the tension zone when the internal force is balanced. The tensile stress in concrete is very small, and once the concrete cracks, the tensile zone of the concrete no longer bears the load. Only the concrete near the neutral axis still bears partial tensile stress, but due to its small lever arm, the internal moment it bears can be neglected in the calculation process. Assuming the stress-strain relationship of the concrete in the compression zone as a rectangle. 4. Introduce the ideal elastic-plastic assumption of steel bars when the internal force is balanced. The reinforcement stress obeys [PERSON]'s law before reaching the yield stress; After reaching the yield stress, the stress remains unchanged. Based on the assumption of the normal section in the above assumptions, the strain distribution of the concrete section can be obtained. The strain distribution diagram of the section is shown in Figure 11.  
Based on the figure, the expression for the height of the compressed area of the section can be derived, as shown in Equation (2):  
\[x_{c}=\frac{\varepsilon_{t}a_{s}+he_{c}-a_{s}^{\prime}\varepsilon_{s}}{ \varepsilon_{t}+\varepsilon_{s}} \tag{2}\]  
The calculation Equation for the bearing capacity of the positive section of the eccentrically compressed member with a rectangular section is shown in Figure 12. When calculating the bending moment and axial force of the section, the bearing capacity can be calculated through the balance of force and torque. Figure 11: Strain distribution of the concrete section. Eccentrically loaded members can be divided into two types of stress states: large eccentrically loaded members or small eccentrically loaded members. The main difference lies in the range of the concrete compression zone and the stress state of the reinforcement in the tension zone. The stress state of the large eccentric compression section is characterized by tension on one side of the reinforcement in the tension zone, and the small eccentric compression section state is that the reinforcement on the tension side may be tensioned or compressed. Therefore, when calculating the internal force, as long as the direction is changed according to the compression or tension state, a unified calculation equation for large and small eccentric compression can be established. Taking the sum of the internal and external forces in the longitudinal direction of the member as zero, Equation (3) can be obtained. \[N=\sigma_{s}^{\prime}A_{s}^{\prime}+\sigma_{c}bx-\sigma_{s}A_{s} \tag{3}\]  
By summing up all the moments of forces on the section about the center of the section, we can obtain Equation (4).