## 4 Calculation of Damage to RC Arch
### Calculation of RC Arch Damage Evolution  
A distinct model was constructed within the ABAQUS software framework, utilizing the geometric dimensions of the arch rib depicted in Figure 6. The concrete material was represented using a three-dimensional hexahedral reduced integration element (C3D8R) with a unit size of 30 mm. Given the absence of consideration for adhesive slip, a two-node linear truss element (T3D2) was employed to simulate the steel, with a unit size of 40 mm. In order to ensure a connection between the concrete and steel, an embedded contact method was employed. Furthermore, a tie constraint was applied to constrain the concrete and loading pad. The loading pad was subjected to static loading through displacement coupling at its center. The concrete arch ring was modelled as a cantilever arch, with the end arch feet simulated as fixed supports (U1 = U2 = U3 = 0; UR1 = UR2 = UR3 = 0). The finite element model is depicted in Figure 8.  
In this model, the compressive and tensile strengths of the matrix concrete are set at \(f_{c}\) = 22.4 MPa and \(f_{t}\) = 1.86 MPa, respectively. The volume fraction of steel fibers is designated as \(v_{f}\) = 1%, with a length-to-diameter ratio of \(l_{f}/d_{f}\) = 60. The yield stress of the steel reinforcement bar is 335 MPa, the ultimate stress is 432 MPa, and the elastic modulus is 203 GPa. The loading pad is defined as an elastic rigid body. The concrete damage model employs the CDP model, wherein the values for inelastic strain and damage factor (\(d-\varepsilon_{t,c}^{in}\)) are calibrated according to the formula provided in Section 2 and are subsequently inputted. Table 3 presents the remaining plasticity parameters of the CDP model.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Model**} & \multicolumn{1}{c}{**Remarks**} \\ \hline
1 & RC arch & Ordinary reinforced concrete hingeless arch \\
2 & FRC arch & Uniform distribution of steel fibers throughout the arch section \\
3 & LSFRC arch & Layering of steel fibers in the damaged cross-section area \\ \hline \hline \end{tabular}
\end{table}
Table 2: Finite element model.  
Figure 8: Finite element model for static loading of RC arch vaults.  
Figure 7: Schematic of each model cross-section.  
Figure 9 shows the tensile damage region along the arch axis of the RC arch at the moment of ultimate load-carrying capacity. When the arch is subjected to a concentrated load, the RC arch first shows cracking damage at the lower edge of the arch crown, with a cracking load of 17.2 kN, which gradually expands to the upper edge with an increase in the load. As the load reached 32.1 kN, both sides of the arch spandrel appeared damage cracking; at the same time, the top plate position of the arch waist also appeared to be subjected to slight tensile damage. At a holding load of 39 kN, the upper edge of the arch waist appeared to be subjected to obvious tensile damage. From the damage evolution process of the whole RC arch, it can be seen that when the arch is subjected to a concentrated load, the lower edge of the arch spandrel, the upper edge of the arch waist, and the lower edge of the arch crown are the key cross-sections prone to damage, and the corresponding damage areas when it reaches the peak load are [0, L/8], [L/8, 3L/8], and [3L/8, 5L/8], with the trend of cracking damage evolution being shown in Figure 10.