## 3 Case Study
### FEM Creation and Updating  
3.3.1. Initial FE Model  
A FE model of Chuong Duong bridge was built, taking into account the bridge's structure (Figure 10) [54]. Top chords, bottom chords, cantilevers, gate frames, and verticals (all of which were modelled using three-dimensional beam elements), made up the principal structural components. Other components, such as wind bracing, stiffening frame, and longitudinal linkage, were modelled with truss elements. The global X-axis of the bridge was in its longitudinal direction, the Z-axis was vertical, and the Y-axis was the transverse (to the direction of the river flow). The built model consisted of 619 elements, which included 461 beam elements and 158 truss elements. Six degrees of freedom (DOF) were available for each element node, and these DOFs corresponded to translational and rotational displacements in the X, Y, and Z axes. Table 3 shows the types of elements used in the finite element model: \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Frequencies [Hz]** & **Damping Ratios [\%]** & **Modal Phase Collinearity** & **Mode Type** \\ \hline
1 & 1.79 & 1.50 & 0.999 & First vertical bending \\
2 & 2.25 & 1.06 & 0.998 & First lateral \\
3 & 3.57 & 0.77 & 0.999 & Second torsion \\
4 & 4.30 & 1.21 & 0.999 & Second vertical bending \\
5 & 4.60 & 0.40 & 0.996 & lateral movement \\
6 & 5.03 & 1.50 & 0.998 & Second lateral bending \\
7 & 8.09 & 1.06 & 0.997 & Third vertical bending \\ \hline \end{tabular}
\end{table} 
Table 2: Frequency values of the seven identified modes. Figure 10: FEM of Span 8 of the Chuong Duong bridge. The input parameters of the material (Young's modulus, specific gravity) as well as of the section (area, moment of inertia) were referenced from the as-built records. Specifically: Young's modulus of steel (beams, truss rods, cantilever) E\({}_{\text{s}}\) = 200 Mpa; the density of steel \(\rho\)s = 7850 kg/m\({}^{3}\). Nonstructural elements, such as the bridge deck, balistrades, lighting systems, and plumbing, were included in the model as additional mass. Typical cross-sections of some truss members are shown in Table 4.  
Boundary Conditions: The Dirichlet boundary conditions of the numerical model were created to accurately reflect the boundary conditions of the actual structure. The span of eight bridges in Chuong Duong included two types of bearings (Figure 11). Based on the survey results, the displacement constraints of the model were made, corresponding to the actual displacement capacity of the bearing. The first type of bearing (movable bearing) allowed for vertical and horizontal displacement (with some constrain according to the bearing design profile), while restricting longitudinal displacement. The second type of bearing (fixed bearing), which limits vertical and longitudinal displacement, allowed for small horizontal displacements. All supports allowed vertical rotation, and limited other directions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Truss Members**} & \multirow{2}{*}{**Area (mm\({}^{2}\))**} & \multicolumn{2}{c}{**Moment of**} & \multicolumn{1}{c}{**Moment of**} \\  & & & & **Inertia \(\mathbf{I_{y}}\) (mm\({}^{4}\))** & **Inertia \(\mathbf{I_{z}}\) (mm\({}^{4}\))** \\ \hline
1 & Bridge gate frame & \(4.27\times 10^{4}\) & \(2.9\times 10^{9}\) & \(1.15\times 10^{9}\) \\
2 & Top lateral bracing & \(4.75\times 10^{4}\) & \(3.31\times 10^{9}\) & \(1.75\times 10^{9}\) \\
3 & Bottom lateral bracing & \(4.75\times 10^{4}\) & \(3.31\times 10^{9}\) & \(1.75\times 10^{9}\) \\
4 & Struts & \(1.83\times 10^{4}\) & \(1.03\times 10^{9}\) & \(5.29\times 10^{7}\) \\
5 & Diagonal chords & \(4.17\times 10^{4}\) & \(2.82\times 10^{9}\) & \(1.04\times 10^{9}\) \\
6 & Vertical chords & \(1.83\times 10^{4}\) & \(1.03\times 10^{9}\) & \(5.29\times 10^{7}\) \\
7 & Top chords & \(1.83\times 10^{4}\) & \(1.03\times 10^{9}\) & \(5.29\times 10^{7}\) \\
8 & Bottom chords & \(1.83\times 10^{4}\) & \(1.03\times 10^{9}\) & \(5.29\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table}