## 4 Calculation of Damage to RC Arch
### Calculation of LSFRC Arch Damage Evolution  
Consequently, the specific location of the damage to the critical cross-section of the RC arch previously mentioned can be employed as a foundation for localized steel fiber doping, with the objective of enhancing the stress performance of the vulnerable cross-section of the RC arch through localized reinforcement. Therefore, the finite element model of steel fiber doping for the LSFRC arch is depicted in Figure 11.  
Figure 10: Damage evolution of critical sections of RC arches.  
Figure 9: Regional distribution of RC arch damage.  
As outlined in Section 4.2, the damage locations of the foot, arch waist, and vault sections of the RC arch can be utilized as a foundation for local steel fiber incorporation. Local reinforcement can be implemented in the areas of [0, L/8], [L/8, 3L/8], and [3L/8, 5L/8], with the objective of enhancing the stress performance of the damage-prone sections of the RC arch. Consequently, the schematic diagrams of steel fiber incorporation in LSFRC arches and the established FEM (Model 3) are presented in Figure 11.  
Figure 12 illustrates the results of a comparison of load-displacement curves for an RC arch, an LSFRC arch, and an FRC arch. Among these, the cracking loads of the LSFRC arch and the FRC arch are 18.8 kN and 19.4 kN, respectively, which are 9.3% and 12.7% higher than the cracking load of the RC arch, which is 17.2 kN. Furthermore, their peak loads are 190 kN and 224.2 kN, representing 6.6% and 25.7% increases compared to 178.3 kN for the RC arch, respectively. The analysis indicates that the incorporation of steel fibers into the vulnerable cross-section area of the RC arch does not significantly enhance the cracking load, but it does markedly improve the load-carrying capacity. Upon reaching the peak load, the FRC arch curve does not exhibit a significant downward segment. This phenomenon can be attributed to the uniform distribution of steel fibers over the entire cross-section of the FRC arch, which results in a higher toughness of the structure at this stage as well as a higher deformation capacity after the peak load. In contrast, in the LSFRC arch, the vulnerable cross-section forces are shared by both the FRC and RC layers. Consequently, the falling section of the load-displacement curve exhibits a trend similar to that of the RC arch, yet it is more ductile after being damaged and requires a higher amount of energy dissipation.  
Figure 13 illustrates the deflection curves of the RC arch, LSFRC arch, and FRC arch along the span direction under a load of 178.3 kN. As illustrated in Figure 13, the deflection of the RC arch, LSFRC arch, and FRC arch at the vault section (l/2) is 4.97 mm, 4.14 mm, and 2.56 mm, respectively, while the deflection at the waist section (l/4) is 1.56 mm, 1.1 mm, and 0.65 mm, respectively.  
Figure 11: LSFRC arch calculation model: (**a**) a diagrammatic representation of the location of the incorporation section for steel fiber; (**b**) ABAQUS finite element model.  
Figure 14 and 15 illustrate the evolution of damage in the RC, LSFRC, and FRC arches following deformation. In comparison to the RC arch, the crack damage area of the LSFRC and FRC arches is significantly reduced. The analysis of deflection difference and the damage evolution trend indicates that the incorporation of steel fiber can enhance the ultimate bearing capacity of RC arches whilst also significantly reducing the deflection at each section, effectively reducing the degree of damage and the number of cracks. While the full-section incorporation of fibers (FRC arch) demonstrates the greatest enhancement, its economic benefit falls short of that of the LSFRC arch.  
Figure 14: The region of arch extrados damage evolution.  
Figure 12: Comparison of load–displacement curves of RC arch with steel fiber in different sections.  
Figure 13: Deflection curves of RC, LSFRC, and FRC arches along the span direction.