## 3 Test Results
### Dynamic Test Results at Different Damage Levels  
Using the collected dynamic data from various beams (i.e., Beam_GFRP_PreStr, Beam_CFRP_PreStr, Beam_GFRP_PostStr, and Beam_CFRP_PostStr) at different damage levels (i.e., the undamaged state, D1, D2, D3, and D4), modal parameters were estimated. In this context, the first three vibration modes of the beams in both vertical and horizontal directions were considered. Table 5 provides the modal frequency estimates for all beams in their undamaged states. In the table above, f1, f2, and f3 indicate modal frequency estimates for the first, second, and third vibration modes, respectively, in both vertical and horizontal directions. As previously mentioned, Beam_GFRP_PostStr and Beam_CFRP_PostStr are bare in the undamaged state. Since the modal frequencies of these two identical beams are found to be very close to each other, they are presented in a single row in Table 5. The results denote that strengthening with both GFRP and CFRP leads to noticeable increases in the frequency estimates of the beams, reflecting an improvement in their stiffness values. Among the two strengthening materials, CFRP provides a more significant increase in the frequency estimates in both directions, suggesting that it is more effective in enhancing structural rigidity compared to GFRP. The bare beams exhibit lower frequencies, confirming the baseline stiffness characteristics in the absence of any strengthening. The near-identical values between the two bare beams validate the consistency and reliability of the testing setup. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Vertical Direction**} & \multicolumn{3}{c}{**Horizontal Direction**} \\ \cline{2-7}
**Beam Name** & **f1 (Hz)** & **f2 (Hz)** & **f3 (Hz)** & **f1 (Hz)** & **f2 (Hz)** & **f3 (Hz)** \\ \hline Beam\_GFRP\_PreStr & 26.73 & 93.95 & 325.15 & 11.68 & 27.93 & 175.11 \\ Beam\_CFRP\_PreStr & 33.52 & 99.56 & 432.84 & 13.34 & 28.42 & 229.75 \\ Beam\_GFRP\_PostStr & 25.19 & 89.59 & 316.48 & 9.50 & 27.63 & 149.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Undamaged state modal frequency estimates for all beams. Figure 11 shows the mode shapes of Beam_GFRP_PostStr and Beam_CFRP_PostStr in the undamaged state, along with the corresponding modal frequencies. Additionally, Table 6 presents the Modal Assurance Criterion (MAC) values of the mode shapes of strengthened beams compared to those of the bare beams (i.e., the reference mode shapes). The MAC value quantifies the similarity between two mode shapes, with values closer to 100% indicating greater similarity [51]. The high MAC values, particularly in the first modes exceeding 90%, indicate that the mode shapes of the strengthened beams closely resemble those of the bare beams. This suggests that the strengthening process does not significantly alter the first mode shapes in both the vertical and horizontal directions, but rather enhances the stiffness values. The reduction in MAC values in the second and third modes reflects certain deviations in the mode shapes, likely due to the effects of the strengthening process. Modal frequency ratios were calculated by dividing the modal frequency estimates at different damage levels by those in the undamaged state, enabling the proportional change in modal frequencies to be monitored. The focus was on the relative changes in frequency ratios rather than the absolute frequency values, highlighting the influence of damage and strengthening on the dynamic behavior of the beams. Table 7 provides the modal frequency ratios for Beam_GFRP_PreStr at various damage levels, encompassing both the vertical and horizontal directions. These ratios offer insight into the effect of progressive damage on the modal frequencies of the beam. Figure 12 illustrates the variation in these ratios for the vertical and horizontal directions.