# A Case Study on the Rehabilitation of a Fire-Damaged Structure
## 4 Numerical Analysis
Both edges of the target components were restrained to simulate axial and rotational restraint effects provided by the adjacent structural components, which reflects the actual condition of girder and column connections. A displacement-control analysis was conducted by imposing a vertical displacement in the middle of the beam and on the top surface of the column, respectively, as illustrated in the figure. The numerical modeling adopted in this study is also utilized in ([PERSON], 2012), thus confirming its validity. Figure 16a,b shows the load-displacement curves obtained from the numerical analysis for Girder #2 and Columns #3 and #4, respectively. The load-displacement curves of undamaged (_i.e._, original) components are also provided in the figures. In these figures, the ordinate axis indicates the vertical load normalized by the peak load value of the undamaged structural component. Table 3 summarizes the results of the numerical analysis. In the table, \(P_{\text{max, und\_b}}\) and \(P_{\text{max, und\_c}}\) indicate the vertical resistant capacities of the undamaged beam and column, respectively. Figure 14: Assumed depth of damaged concrete for target elements: (**a**) Girder #2, (**b**) Column #3 and (**c**) Column #4. It can be seen from the results of the table that the maximum vertical load (\(P_{\max}\)) of Girder #2 is approximately 94% of the corresponding value of the undamaged member. In structural engineering practice, the design strength of the girder is generally 90% of its demand strength. In addition, the vertical displacement values corresponding to \(P_{\max}\) of the damaged and undamaged girders are almost the same, which indicates that they have almost the same ductility capacity. Consequently, it can be concluded that Girder #2 still has enough resistant capacity. Similar to the case of Girder #2, \(P_{\max}\)'s of Columns #3 and #4 are approximately 96% and 98% of the corresponding value of the undamaged column, respectively. The vertical displacements corresponding to \(P_{\max}\) are the same in all of the damaged and undamaged cases. As a consequence, it can be also judged that Columns #3 and #4 still have enough resistant capacity. From this discussion, overall it can be observed that the behavior pattern and the maximum vertical load are hardly affected by the components that are assumed to be damaged. In other words, in spite of the loss of the sectional area and material strength of the target structural components, since the core concrete surrounded by stirrups, ties and rebar survived the fire hazard, it can be concluded that all of the target damaged girder and columns retain enough strength. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Resistant Capacity**} & \multicolumn{2}{c}{**Gider**} & \multicolumn{2}{c}{**Column**} \\ \cline{2-6}  & **Undamaged** & \begin{tabular}{c} **Damaged** \\ **(Girder \#2)** \\ \end{tabular} & **Undamaged** & \
\begin{tabular}{c} **Damaged** \\ **(Col. \#3)** \\ \end{tabular} &
\begin{tabular}{c} **Damaged** \\ **(Col. \#4)** \\ \end{tabular} \\ \hline Max. vertical load (\(P_{\max}\), kN) & 2098.3 & 1963.7 & 22,879.5 & 21,991.9 & 22,430.5 \\ \hline Vertical displacement & \multirow{2}{*}{21.36} & \multirow{2}{*}{22.00} & \multirow{2}{*}{8.63} \\ corresponding to \(P_{\max}\) (mm) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Resistant capacity of target components obtained from numerical results. Figure 15: Boundary and loading conditions of girder model (**a**) and column model (**b**).