## 4 Verification of the Applicability of the Unified Equation  
In this section, tests were performed on the established unified equation (Equation (3)) for the UPV and the strength evolution index to verify its applicability for estimating concrete strength. First, the aforementioned test data underwent self-verification. The specimens examined were NC1-5, NC6-20, FC1-10, and SC1-SC10. For concrete with a specific mixture proportion, dividing the UPV measured at a specific age by the terminal UPV yielded the UPV evolution index (\(V_{g}\)) of the concrete. By substituting \(V_{g}\) into Equation (3), the strength evolution index (\(S_{g}\)) of the concrete at that age was obtained. Multiplying \(S_{g}\) with the terminal strength of that mixture proportion yielded the estimation of the concrete strength. Terminal UPV and strength refer to the UPV and strength values at 28 days for normal concrete but at 70 days for pozzolanic concrete. Figure 9 presents a comparison of the estimated strengths of all specimens and the strengths measured by performing compression tests. The errors between the strength estimations and actual measurements were generally in the range of \(\pm\)15%, indicating that the unified equation (Equation (3)) could be applied for estimating concrete strength at different time points. In addition to using the established relationship curve data to perform self-applicability tests, this study also collected the test data of numerous relevant studies [24; 25; 26; 27; 28] regarding UPV and strength values at different ages, which could be adopted to examine the applicability of the proposed unified equation. The collected data was for concrete with numerous paste volume ratios, coarse aggregate contents, and pozzolanic admixture contents; these mixture proportions are listed in Table 7 for normal concrete and Table 8 for pozzolanic concrete. In total, 34 mixture proportions were documented. Tables 9 and 10 summarize all the measured UPV and strength data pairs collected from the relevant studies for normal concrete and pozzolanic concrete, respectively, at various ages. It should be noted that the UPV and strength values at 56 days instead of 70 days for pozzolanic concrete were used as the terminal UPV and strength values required for the calculation of evolution index due to the lack of test data at 70 days. Figure 9: Comparison between actual strength measurements and concrete strength estimations obtained using the unified equation. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**w/cem**} & \multirow{2}{*}{\
\begin{tabular}{c} **S/A** \\ **(\%)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **V\({}_{\text{paste}}\)** \\ **(\%)** \\ \end{tabular} } & \multicolumn{6}{c}{**Mixture Proportion, kg/m3**} \\ \cline{5-10}  & & & & & & & & & \\ \cline{5-10}  & & & & & & & & & \\ \hline PNC1 & 0.4 & & & & 201 & 502 & 0 & 0 & 0 & & \\ PNC2 & 0.5 & & & 220 & 440 & 0 & 0 & 0 & 681 & 1027 \\ PNC3 & 0.6 & & & 235 & 392 & 0 & 0 & 0 & & \\ PNC4 & 0.7 & & & 248 & 354 & 0 & 0 & 0 & & \\ \hline PNC5 & 0.4 & & & 234 & 585 & 0 & 0 & 0 & & \\ PNC6 & 0.5 & & & 257 & 514 & 0 & 0 & 0 & 608 & 926 \\ PNC7 & 0.6 & & & 275 & 458 & 0 & 0 & 0 & & \\ PNC8 & 0.7 & & & 289 & 413 & 0 & 0 & 0 & & \\ \hline PNC9 & 0.4 & & & 163 & 419 & 0 & 0 & 4.19 & & 750 & 1044 \\ PNC10 & 0.6 & & & 195 & 327 & 0 & 0 & 0.82 & & \\ \hline PNC11 & 0.4 & & & 196 & 502 & 0 & 0 & 5.02 & & \\ PNC12 & 0.6 & & 36 & 36 & 235 & 392 & 0 & 0 & 9.98 & 592 & 1044 \\ PNC13 & 0.4 & & & 163 & 419 & 0 & 0 & 4.19 & & \\ PNC14 & 0.6 & & & 195 & 327 & 0 & 0 & 0.82 & 881 & 914 \\ PNC15 & 0.4 & & & 196 & 502 & 0 & 0 & 5.02 & & \\ PNC16 & 0.6 & & & 235 & 392 & 0 & 0 & 0.98 & 723 & 914 \\ PPC1 & 0.36 & & 44 & 31 & 161 & 460 & 0 & 0 & 4.60 & 785 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Normal concrete mixture proportions collected from the relevant studies for verifying the applicability of the unified equation.