## 5 Modeling Results  
### Reinforcement Performance of Different Materials at Variable Cross-Section  
This study primarily investigates the reinforcing effects of two distinct materials--steel fiber concrete and reinforced concrete--on varying degrees of lining crack damage at the variable sections of the arch. The stress concentration areas of these materials at the variable cross-sections are numerically analyzed, and the stress values at five characteristic points (A, B, C, D, and E) are determined; each characteristic point is depicted in Figure 11. Due to space constraints, only the load-stress curves for component 4, reinforced with two different materials, are presented.  
Under varying load conditions, the load-stress correlation curves at Points A and E are illustrated in Figure 12. The diagrams reveal that at Points A and E, the load and stress for both materials--reinforced concrete and steel fiber concrete--are approximately proportional. Under identical loading conditions, the stress in steel fiber concrete exceeds that in reinforced concrete, though the difference is marginal.  
Under varying load conditions, the load-stress correlation curves at Points B and C are illustrated in Figure 13. The figure illustrates that, in the initial loading stages, the concrete on both sides of the crack predominantly undergoes compression. Upon reaching a specific load threshold, the compressive stress diminishes, and the arch, along with the original lining, begins to share the load. However, as the load increases, the concrete transitions to a tensile state, with tensile stress escalating primarily in the arch. Furthermore, the stress in the reinforced concrete exceeds that in the steel fiber-reinforced concrete.  
Figure 11: Numerical model.  
Figure 12: Relationship curve between load and stress.  
Under varying load conditions, the load-stress correlation curve at Point D is depicted. As depicted in Figure 14, during the initial loading stages, the concrete at the composite surface of the arch and lining primarily undergoes compression. Upon reaching a specific load, the compressive stress diminishes. Concurrently, the arch and the original lining share the load, resulting in cracks at the composite surface. As the load increases, cracks at the composite surface continue to widen, and the concrete predominantly experiences tension. Under identical design loads, the stress in reinforced concrete at concentrated stress regions (A, B, C, D, and E) exceeds that in steel fiber-reinforced concrete by 16.15%, 6.01%, 12.68%, 36.62%, and 4.82%, respectively, the reinforcement performance of steel fiber-reinforced concrete is superior to that of reinforced concrete. Additionally, this difference is more pronounced in the mid-span region.  
The reinforcement effects of two materials on various damaged linings were compared and analyzed. During the initial loading stages, the overall stiffness of the reinforced concrete damaged lining exceeds that of the steel fiber-reinforced concrete; however, as the load increases, the overall stiffness of the steel fiber-reinforced concrete damaged lining surpasses that of the reinforced concrete. The steel fiber-reinforced concrete damaged lining undergoes plastic failure under the design conditions without component destruction, whereas the reinforced concrete damaged lining incurs damage under the same conditions. In the stress concentration areas (A, B, C, D, and E), the stress in the steel fiber-reinforced concrete damaged lining is less than that in the reinforced concrete. Based on the analysis, the reinforcement performance of steel fiber-reinforced concrete is superior to that of reinforced concrete, with both materials meeting the structural stress requirements.  
Figure 14: Relationship curve between load and stress at Point D.  
Figure 13: Relationship curve between load and stress.