## 4 Case Study
### Analysis Model and Model Updating  
LS-DYNA software [36] was used to study the failure process of continuous girder bridge specimen under seismic action. The concrete was simulated by solid elements. The cube compressive strength of the concrete was 32 MPa and uniaxial compressive strength was 23 MPa. The uniaxial tensile strength of the concrete was 2.24 MPa while elastic modulus was \(2.75\times 10^{4}\) MPa. The material number of concrete was Mat_024, which was an elastoplastic material with an arbitrary stress-strain curve (see Figure 5a). If the compressive strain reached 0.0033 or the tensile stress reached 2.24 MPa, the concrete would fail. The material number of the steel bar was Mat_003, which had a bilinear kinematic hardening (see Figure 5b). \(E\) and \(E_{t}\) were the slopes of the bilinear stress-strain curve. Specific parameters are shown in Table 2. The connection between steel bars and concrete was simulated by common nodes, without considering the slip and fracture of steel bars. The counterweight block was specified as mass elements which were assumed to be lumped at the nodes, and each mass node was 9.3 kg weight. Discrete spring elements were chosen to model the laminated rubber bearing. The material number was MAT_S01. The longitudinal, horizontal, and vertical stiffness of the middle bearing was 955 kN/m, \(1\times 10^{3}\) kN/m, and \(5\times 10^{5}\) kN/m. The longitudinal, transverse, and vertical stiffness of the side bearing was 550 kN/m, 800 kN/m, and \(2\times 10^{5}\) kN/m, respectively. If the spring force exceeded the maximum shear force of the bearing, it would withdraw from working, and the cap beam slid on the top of the girder, which was simulated by the contact element with static and dynamic friction coefficients of 0.3 and 0.15. To model the HDR bearing, solid elements and discrete spring elements were used. The material numbers of the vertical discrete spring element and horizontal discrete spring element were MAT_S01 and MAT_S03, respectively. The vertical stiffness of the side bearing and middle bearing were \(1\times 10^{5}\) kN/m and \(4\times 10^{5}\) kN/m, respectively. The horizontal constitutive relation of the HDR bearing has been tested [42] and illustrated in Figure 6. The longitudinal stiffness and transverse stiffness of the steel plate fixing the middle bent were \(3\times 10^{3}\) kN/m and 600 kN/m, respectively. The steel plate would fail if the shear force exceeded the ultimate shear force (40 kN). The bearing would slide when the sheer force of the support was more than the static friction. Contact simulation was used as the interaction between the pier and the girder after the bearing slid. The static and dynamic friction coefficients were 0.3 and 0.15, respectively. Figure 7 shows the finite element model. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Concrete**} \\ \hline Concrete & Compressive & Elastic & Tensile \\ Grade & Strength (MPa) & Modulus (MPa) & Strength (MPa) \\ \hline
30 & 32 & \(2.75\times 10^{4}\) & 2.24 \\ \hline \multicolumn{4}{c}{**Steel**} \\ \hline Reinforcement & Yield & Elastic & Tensile \\ Diameter & Strength (MPa) & Modulus (MPa) & Strength (MPa) \\ \hline \(\varphi\)10 & 351 & \(2.01\times 10^{5}\) & 558 \\ \(\varphi\)6.5 & 350 & \(2.04\times 10^{5}\) & 531 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the specimen materials. Figure 5: Constitutive relationship: (**a**) the concrete (**b**) the steel bar. BC system and CHDR system were tested in the shaking table array test and the ambient vibration test was carried out before the shaking table test. The stochastic subspace identification (SSI) method was chosen for modal parameter identification. Self-developed data analysis software called MACES was used to identify the frequency of the bridge model using the SSI method. Finally, the natural frequencies of the bridge specimens in longitudinal, transverse, and vertical directions were obtained and used for model updating.