# Stochastic Analysis of a Large-Span Continuous
## 3 Stochastic Seismic Response of the Large-Span CGHSR Bridge  
### Finite Element Model (FEM) of the CGHSR Bridge  
A large-span CGHSR bridge is employed here, with a total span of 268 m, consisting of a cast-in-place, pre-stressed concrete box girder made of C55, with three spans of (68 + 132 + 68) m [48]. The bridge configuration is presented, in detail, in Figure 2. The single-box, single-cell continuous girder with variable height and cross-section is 8.5 m wide, while the maximum and minimum girder height is 9.6 m and 5.6 m, respectively. The girder softit varies in terms of a quadratic parabola. Four solid reinforced concrete shaft piers made of C35 concrete consist of the substructure of the CGHSR bridge. Spherical steel bearings are utilized to connect the girder and piers, the prototype of which is shown in Figure 2. The fixed bearing is employed at the top of 2# pier, while others remain unfixed. The total weight of the CGHSR bridge is 214,800 kN. The 3D model of the bridge, shown in Figure 2, is developed using MATLAB, based on the FEM method. In contrast to the general highway and railway bridge, the HSR bridge is well-designed, with a more rigid and relatively strong structure to ensure safety and comfort in the operating of high-speed trains on the railway line. The following assumptions are considered to reasonably simplify the FEM and highlight the main objective of this work, as much as possible: (i) the bridge structure is within the elastic range under seismic excitation; (ii) the stiffness contribution of the auxiliary structures such as curbs, parquet walls, deck pavement, and electric system, are ignored, whereas the mass of these facilities is added in the model. Linear elastic material is employed. The Young's modulus and Poisson ratio of concrete are taken as 3.45 \(\times\) 104 MPa and 0.2, respectively [49; 50]. The girder and piers are modelled by elastic beam elements. The fixed boundary conditions are adopted to the bottom of pier 1# to pier 4#. The whole model consists of 67 nodes and 374 DoFs. The modal analysis was conducted based on the initial FEM. For brevity, Table 2 lists only the first ten natural frequencies and vibration modes of the bridge. More details of the FEM can be found in [48]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Frequency (Hz)**} & \multicolumn{2}{c}{**Modal Mass**} \\  & **FEM** & **Design** & **Characteristics of Modes** & **Participation Ratios (\%)** \\ \hline
1 & 1.1291 & 1.0096 & Symmetric lateral bending vibration of the girder & 13.06 (y direction) \\
2 & 1.1774 & 1.1334 & Longitudinal bending vibration of the pier & 91.89 (x direction) \\
3 & 1.1824 & 1.2705 & Symmetric vertical bending vibration of the girder & – \\
4 & 1.9482 & 1.5872 & Lateral bending vibration of one side span & 14.13 (y direction) \\
5 & 2.1495 & 1.7277 & Lateral bending vibration of the other side span & 55.22 (y direction) \\
6 & 2.6168 & 2.3257 & Anti-symmetric lateral bending vibration of the girder & – \\
7 & 2.7394 & 2.6887 & Anti-symmetric vertical bending vibration of the girder & 0.11 (y direction) \\
8 & 3.5933 & 3.2899 & Second symmetric lateral bending vibration & – \\
9 & 3.8228 & 3.6131 & Second symmetric vertical bending vibration & 7.00 (y direction) \\
10 & 4.0072 & 4.0078 & Second Anti-symmetric lateral bending vibration of the girder & 2.16 (x direction) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the first 10 modes of the CGHSR bridge. Figure 2: The structure layout of the CGHSR bridge (unit: m). As shown in Table 2, most of the numerical results are close to the design value, which to some extent verify the validity of the developed FEM. The first natural frequency of the bridge, corresponding to the symmetric lateral bending vibration of the girder, is 1.1291 Hz. This indicates that the CGHSR bridge is much more rigid than other large-span highway or railway continuous girder bridges [15; 51]. Figure 3 schematically shows several typical modes.