## 5 Conclusions  
In this paper, scabbing-resistant energy was defined using the energy-depth relation and scabbing theory, and the energy-based impact formula was developed for the scabbing depth, scabbing limit, and perforation limit considering various impact conditions. In particular, the scabbing depth formula was first proposed. Based on a comparison of the proposed formulae and impact test results, the following conclusions were drawn:  
1. The scabbing depth formula is derived using the same concept as that of penetration depth, based on Hertzian contact theory, but is not applied with the energy concentration effect of the impact velocity and projectile shape. The scabbing depth formula is proposed and verified using the experimental data. The mean value of the tested-to-predicted scabbing depth ratio is 1.87 and the standard deviation is 1.22. The scabbing depth formula is less accurate than the penetration depth formula. This is because it is difficult to predict the scabbing depth due to the energy amplified by reinforcing bars and/or aggregate and reflection angle. Despite these challenges, in 52 of the 81 specimens (64%), the scattering ratio is 2 and the average ratio is 1.16. Therefore, the scabbing depth formula is good overall.  
Figure 10: (**a**) Proposed formula; (**b**) modified NDRC; (**c**) [PERSON] formula; (**d**) ACE formula. Panel thickness (_h_) divided by predicted perforation limit thickness (_h_\({}_{pp}\)).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Formula** & **Perforation Failure** & **No Perforation Failure** \\ \hline \multirow{2}{*}{Proposed} & 63\% & 99\% \\  & (41 of 65 specimens) & (180 of 182 specimens) \\ \hline \multirow{2}{*}{Modified NDRC} & 77\% & 93\% \\  & (50 of 65 specimens) & (170 of 182 specimens) \\ \hline \multirow{2}{*}{Hughes} & 100\% & 67\% \\  & (65 of 65 specimens) & (125 of 182 specimens) \\ \hline \multirow{2}{*}{ACE} & 88\% & 91\% \\  & (57 of 65 specimens) & (166 of 182 specimens) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Accuracy of perforation limit thickness formula.  
The scabbing limit formula can be obtained from the scabbing depth formula. The scabbing limit thickness, which is equal to the thickness that makes the scabbing depth zero. The Hughes formula is the best for predicting perforation failure, while the proposed equation is the most conservative. However, the proposed formula is the best for predicting non-perforation, with 99% accuracy. The interface between perforation and non-perforation is properly divided.
3. The perforation limit thickness can be obtained from the penetration depth, tunneling depth, and scabbing depth, whose sum is equivalent to the minimum thickness required to prevent perforation. This is equal to the perforation limit thickness. To prevent perforation, the slab thickness should be equal to or greater than the sum of the penetration depth, tunneling depth, and scabbing depth.
4. A new formula for predicting the scabbing limit and perforation limit thickness is derived using the theoretical framework and verified using experimental data. The proposed formulae predict all scabbing failures. However, non-scabbing failure is predicted with an accuracy of 81%. This formula is found to be the most accurate.  
Conceptualization, S.K.; Formal analysis, S.K.; Funding acquisition, T.H.-K.K.; Investigation, S.K.; Methodology, S.K.; Project administration, T.H.-K.K.; Supervision, T.H.-K.K.; Validation, S.K.; Writing--original draft, S.K.; Writing--review & editing, T.H.-K.K. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Nuclear Safety Research Program through the Korea Foundation Of Nuclear Safety (KoFONS) using the financial resource granted by the Nuclear Safety and Security Commission (NSSC) of the Republic of Korea (No2003007).  
This manuscript is based on the PhD dissertation by [PERSON] [29].  
The authors declare no conflict of interest.