## 2 Refined Numerical Modelling Approach
### Performance Degradation Simulation Approach  
#### 2.5.1 Box Girder Crack  
The cracking of the box girder results in a local stiffness loss, which causes an increase in deflection and stress redistribution near the crack area. Among the existing simulation techniques, there are two main methods to simulate the post-crack behavior of concrete, called the smeared crack model and the discrete crack model, respectively. The former method simulates cracks equivalently through inelastic cracking strains. Essentially, crack development is predicted by adjusting the material softening constitutive relationship with the fracture energy conservation criterion. Since the smeared crack model needs to consider the softening behavior of the concrete, the iterative step is usually required to be very small, leading to a slower solution speed. The discrete crack model describes the discontinuous displacement field formed by the crack by setting interface elements at the crack position. The advantage of this model is that the crack area can be preset and the stiffness reduction and stress concentration effects caused by cracking can be considered, and these are widely used to analyze the mechanics issues with cracking paths. Existing studies have shown little difference between the above two models in the prediction of mechanical results [23]. For the case of predicting the long-term performance and post-crack behavior of the CPCR bridge, the crack position is already determined, and the discrete crack model is usually appropriate to use. The sketches of the above two methods are presented in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nominal Element Length (mm)** & **Predicted Mid-Span Deflection (mm)** & **\% Change in Deflection** & **Time Cost (Hours)** \\ \hline
2000 & 87.73 & \(-\)3.60 & 0.33 \\
1000 & 89.90 & \(-\)1.22 & 0.69 \\ Non-uniform (Mix 500 and 1000) & 90.78 & \(-\)0.25 & 0.89 \\
500 & 90.92 & \(-\)0.10 & 1.67 \\
250 & 91.01 & - & 4.41 \\ \hline \hline \end{tabular} System environment: Operation system: Windows 10. RAM: 16.0 GB. Processor: Intel® CoreTM i7-6700K CPU @ 4.00 GHz 4.01 GHz.  
\end{table} 
Table 2: Mesh convergence study results for element length along the bridge direction. Figure 3: Discretization of Bridge A in ABAQUS. The 270 m main span consists of 65 segments in total. Except for the mid-span closure segment, two identical segments are arranged symmetrically along the left and right half, respectively. For ease of distinction, the segment located in the right half of the bridge is labeled X, and that in the left half of the bridge is labeled X'. The numbering rules for each segment in the main span of Bridge A are shown in Figure 5.  
In order to accurately simulate the structural behavior, on-site crack monitoring was carried out. According to the field investigation, more than 1000 cracks were found based on the ultrasonic testing method, and most of them were shallow cracks. Considering the computational efficiency of the model, it was not feasible to incorporate all the cracks into the model. Given that the box girder structure is a bending-shear coupling force system, two obvious cracks near the bend-shear coupling region were selected for analysis. These cracks are indicated by red lines in Figure 5 and were chosen to maximize their impact on the bridge beam structure. To investigate the characteristics of cracks, four concrete core samples were drilled outside both webs near the location of the crack. The core drilling process and the core sample are presented in Figure 6, and it can be seen that the crack had penetrated the concrete core samples. The crack dimensions obtained using the drilling method are listed in Table 3, and the preset crack zone of the box beam can be built initially in the Part module in ABAQUS, as can be seen in Figure 7.  
Figure 4: Concrete cracking models: (**a**) discrete crack model; (**b**) smeared crack model (reproduced from [24]).