# Retrofitting of a Multi-Span Simply Supported Bridge into a Semi-Integral Bridge
## 5 Finite Element Analysis
### Response Spectrum Analysis
#### 5.2.2 Limit State Checks  
According to the guidelines for the seismic design of highway bridges [41], an elastic seismic design is employed under the E1 seismic action, primarily focusing on the strength assessment of bridge components, including piers, beams, and foundations. The requirement stipulates that bridge structures must remain in an elastic state to ensure that the structure and its components meet the specified criteria. In contrast, for bridges designed with a ductile seismic design under the E2 seismic action, the primary evaluations include structural displacement and the strength of capacity-protecting components. The guideline specifies that bridges should effectively absorb and dissipate seismic energy through ductility or isolation measures during significant seismic events, thereby ensuring the safety of the bridge structure. Consequently, in accordance with these guidelines, it is necessary to check the shear strength of the pier and the displacement of the pier top under the dead load and earthquake action.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loading Combination** & \begin{tabular}{c} **Earthquake** \\ **Direction** \\ \end{tabular} & \begin{tabular}{c} \(\Delta_{1}\) \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} \(\Delta_{2}\) \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} \(\Delta_{2}-\Delta_{1}\) \% \\ \(\Delta_{2}\) \\ \end{tabular} \\ \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} Dead load and E1 \\ \end{tabular} } & X & 3.9 & 28.1 & 86.1\% \\ \cline{2-5}  & Y & 3.2 & 27.9 & 88.5\% \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} Dead load and E2 \\ \end{tabular} } & X & 11.6 & 82.6 & 80.1\% \\ \cline{2-5}  & Y & 9.3 & 82.1 & 88.6\% \\ \hline \hline \end{tabular}  
* Notes: \(\Delta_{1}\) = maximum displacements at beam ends of the semi-integral bridge; \(\Delta_{2}\) = maximum displacements at beam ends of the simply supported bridge.  
\end{table}
Table 2: Maximum displacements of beam ends.  
Figure 13: Design response spectrum functions. (**a**) Type E1. (**b**) Type E2.  
Checks on Bending Moments and Shear Forces  
Table 3 shows the calculated internal forces and ultimate strengths of the pier of the semi-integral bridge under the dead load and earthquake action. In Table 3, \(M_{u}\) and \(M_{r}\) are the calculated factored applied moments and resisting ultimate moments, respectively; \(V_{u}\) and \(V_{r}\) are the calculated factored applied shear force and resisting ultimate shear forces, respectively. It can be seen from Table 3 that the maximum calculated bending moment and shear force are 984.7 kN\(\cdot\)m and 312.6 kN, respectively, which are less than the corresponding ultimate moment and shear force. This shows that the strength of the pier of the semi-integral bridge meets the current design and safety requirements.  
Checks on Displacements  
Table 4 shows the longitudinal (X-direction) and transverse (Y-direction) displacements and the allowance displacement under the dead load and earthquake action. In Table 4, \(\Delta_{u}\) and \(\Delta_{limit}\) are the calculated longitudinal displacements and the limiting displacement, respectively. It can be seen from Table 4 that the maximum calculated displacement is 3.9 mm, which is much smaller than the allowance displacement of 52.4 mm. This shows that the displacement of the pier of the semi-integral bridge meets the current design and safety requirements.