## 5 Analytical Study and Model Calibration
### Calibration of the Analytical Model  
This section presents a comparative analysis of the results obtained from the analytical model against the outcomes of the shake table tests. Initially, the eigenvalue results of the specimens subjected to testing are examined. This is subsequently followed by an assessment of the LVDT results in relation to the nonlinear time history analysis derived from the analytical models. After calibrating the analytical models with the empirical results from the shake table tests, the damped hysteretic energy values of these models are computed and discussed in relation to the established code threshold limits.  
The natural vibration periods were computed by taking into account the rigidities of the cracked sections in both longitudinal and transverse directions, and these calculations for both specimens are juxtaposed with the experimental data, as shown in Table 2. The calculated eigen periods demonstrate concordance with the results obtained from the shake table tests for each specimen. An evaluation of the LVDT data obtained from the shake table test juxtaposed with the computed displacement trace derived from the analytical model is presented in Table 3. The data presented in the tables indicates a strong correlation between the outcomes of the analytical model analysis and the shake table experiments. This observation suggests that the analytical model has been effectively calibrated.  
To further substantiate the coherence between the experimental shake table results and the numerical simulations, Figure 11 presents a direct time-history comparison of displacement responses obtained from both the physical tests and the calibrated FE model of the two specimens, considering 0.35 g and 1 g ground excitations. Each subplot illustrates  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen No** & **Shake Table Test (s)** & **Analytic Model (s)** & **Error (\%)** \\ \hline
1 & 0.495 & 0.463 & 6.45 \\
2 & 0.151 & 0.147 & 2.65 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the natural vibration periods.  
Figure 10: Three-dimensional finite element models of tested specimens and fiber-based discretization of beam and column elements: (**a**) Specim 1, (**b**) Specim 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specimen No** & **Ground Motion** & **Shake Table Test (mm)** & **Analytic Model** (mm)** & **Error (\%)** \\ \hline
1 & 0.35 g & 53.14 & 53.57 & 0.43 \\
2 & 0.35 g & 1.87 & 2.06 & 9.22 \\
2 & 1 g & 19.86 & 18.97 & 4.48 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the top displacements.  
overlaid displacement traces, with the red lines representing the measured response from the shake table tests and the blue lines corresponding to the simulated output from the FE model.  
Across all intensity levels, the experimental and numerical results exhibit strong agreement not only in peak amplitudes but also in waveform shape, frequency content, and phase alignment. The fidelity of this match reinforces the robustness of the fiber-based FE modeling approach adopted in this study, particularly in capturing nonlinear dynamic response characteristics of the retrofitted RC frame. These results confirm that the numerical model is sufficiently accurate to reflect the global seismic behavior observed experimentally, thereby providing a validated basis for subsequent parametric studies and design-oriented evaluations.