# Genetic Algorithm-Based Model Updating in a Real-Time Digital Twin for Steel Bridge Monitoring
## 5 Model Updating and Results  
The optimization of the bridge model is performed by systematically adjusting the elastic modulus of steel components within the finite element (FE) framework to minimize discrepancies between the numerical model and the real-world structural behavior. This calibration process focuses on steel properties. The elastic modulus of steel \((E_{s})\), a key parameter governing material stiffness, is iteratively modified across the various members of the bridge--such as the diagonals, verticals, upper and lower currents, etc.--to reflect potential material degradation, manufacturing variability, or operational aging. The baseline mechanical properties of the bridge, including the elastic modulus of steel (\(E_{s}=200\) GPa), were derived from material certificates and design specifications, representing the undamaged state of the structure. To account for aging and localized defects, the elastic modulus was allowed to vary within a \(\pm 30\%\) range during the GA-based optimization. This range reflects potential variability in material properties due to factors such as corrosion, fatigue, and joint loosening. The GA-based approach uses the global dynamic response of the structure (e.g., natural frequencies and mode shapes) to infer changes in material properties, providing a practical and efficient method for model updating. This approach is consistent with established practices in SHM literature [22, 23] and ensures that the numerical model accurately reflects the current condition of the bridge. The optimization focuses on adjusting the steel elastic modulus \(E_{s}\) of members because it serves as a practical and effective parameter to capture the global effects of localized phenomena such as joint degradation, fatigue, and stress redistribution. In typical finite element (FE) models of steel truss bridges, connections and joints are often idealized, making it challenging to explicitly model their complex behavior. By calibrating the elastic modulus, these localized effects are synthetized into a single parameter that governs the overall stiffness of the structural members. This approach allows the digital twin to identify members exhibiting anomalous behavior, which may indicate potential damage. Once identified, detailed inspections can determine the specific nature of the damage. The objective function is formulated as a multi-criteria error metric that quantifies the alignment between the experimentally measured dynamic properties of the real bridge and the numerically simulated results. Specifically, the function incorporates two principal components:  
1. Natural Frequency Error: The relative difference between the first \(n\) natural frequencies (\(f_{real,i}\)) obtained from OMA using accelerometer data and the corresponding FEM-derived frequencies (\(f_{FE,i}\)): \[E_{freq}=\sum_{i=1}^{n}\left(\frac{f_{real,i}-f_{FE,i}}{f_{real,i}}\right)^{2}\] (4) This term ensures the model replicates the global vibrational characteristics of the structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mode** & \(f_{real}\) **(Hz)** & \(f_{FE}\) **(Hz)** & \(\Delta_{f}\) **(\%)** \\ \hline \multirow{2}{*}{Span 1} & 1 & 8.85 & 10.86 & 18.51 \\ \cline{2-5}  & 2 & 16.03 & 19.50 & 17.79 \\ \hline \multirow{2}{*}{Span 2} & 1 & 6.48 & 7.48 & 13.37 \\ \cline{2-5}  & 2 & 14.38 & 17.98 & 20.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the frequencies of the real bridge (\(f_{real}\)) and the uncalibrated finite element model (\(f_{FE}\)). 2.