## 5 Discussion of the Results
### The Stiffness Modified Factor \(\alpha_{D}\)  
Due to the orthotropic properties of FRP, a modified factor of stiffness \(\alpha_{D}\) has been proposed. And a reinforcement index of FRP has been defined as:  
\[\omega\ =\ \frac{E_{f}t_{f}}{E_{s}t_{s}}\ =\ \frac{n_{f}E_{f}t_{f0}}{E_{s}t_{s}} \tag{16}\]  
In order to study the variation of \(\alpha_{D}\), a large number of parametric analysis have been performed in this section, the parameters include reinforcement index \(\omega\), plate aspect ratio \(\beta\), hole size \(d/b\), and BCs. The parametric analysis' results are handled by the above approach in Equations (14) and (15), and plotted in Figures 14 and 15 for BCs of 4S and 3S1F, respectively.  
Figure 14 shows the variation of \(\alpha_{D}\) as the reinforcement index \(\omega\) increases from 0 to 0.8 for different \(d/b\) from 0 to 0.7 and different plate aspect ratios \(\beta\) from 1 to 4 with the BCs of 4S. It can be seen that the curves in Figure 14 with the same \(d/b\) and different \(\beta\) are almost the same and the values in Figure 14a are slightly smaller. Therefore, \(\alpha_{D}\) of a perforated rectangular plate strengthened by FRP with an integer aspect ratio could be conservatively taken as that of a perforated square plate, so only plates with \(\beta=1\) are analyzed in the following. From Figure 14a, it can be seen that \(\alpha_{D}\) decreases as \(\omega\) is enlarged and the curves can be coupled by Equation (17), where m, n are the undetermined second-order and first-order coefficients. For different \(d/b\), m and n are calculated in Table 3, and \(R^{2}\) is the correlation coefficient.  
\[\alpha_{D}\ =\ m\omega^{2}+n\omega+1 \tag{17}\]  
Figure 11: The model shapes of plates with \(e_{x}/b=0.5\), \(\beta=2\) and BCs of 4S. (**a**) \(d/b=0.1\); (**b**) \(d/b=0.3\); (**c**) \(d/b=0.5\); (**d**) \(d/b=0.7\).  
Figure 12: The model shapes of plates with \(e_{x}/b=0.5\), BCs of 4S and different \(\beta\) from 1 to 4. (**a**) \(\beta=1\); (**b**) \(\beta=2\); (**c**) \(\beta=3\); (**d**) \(\beta=4\).  
In Table 3, \(m\) and \(n\) can be approximate expressed as Equations (18) and (19), so \(\alpha_{D}\) with BCs of 4S can be calculated by Equation (20)  
\[m\ =\ 1.2(d/b)\ +\ 0.7 \tag{18}\]  
\[n\ =\ -1.6(d/b)-0.74 \tag{19}\]  
\[\alpha_{D}\ =\ (1.2d/b+0.7)\omega^{2}-(1.6d/b+0.74)\omega+1 \tag{20}\]  
For the plates with BCs of 3S in Figure 15, the same method can be used to study the variation in \(\alpha_{D}\) as the reinforcement index \(\omega\) increases from 0 to 0.8 for different plate aspect ratios \(\beta\) from 1 to 4. It also can be seen that \(\alpha_{D}\) decreases as \(\omega\) is enlarged. It is noted that \(\alpha_{D}\) is independent of \(d/b\) when \(\beta\) = 2, 3 and 4, so \(\alpha_{D}\) with BCs of 3S1F can be calculated by Equation (21).  
\[\alpha_{D}\ =\ \left\{\begin{array}{rrrr}(0.28d/b+0.56)\omega^{2}-(0.33d/b+0.69 )\omega+1,&\beta&=&1\\ 1.18\omega^{2}-1.45\omega+1,&\beta&=&2\\ 1.42\omega^{2}-1.73\omega+1,&\beta&=&3\\ 1.51\omega^{2}-1.84\omega+1,&\beta&=&4\end{array}\right. \tag{21}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(d\)/\(b\) & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 \\ \hline \(m\) & 0.69 & 0.76 & 0.92 & 1.09 & 1.23 & 1.34 & 1.40 & 1.47 \\ \(n\) & \(-0.74\) & \(-0.83\) & \(-1.04\) & \(-1.25\) & \(-1.44\) & \(-1.59\) & \(-1.69\) & \(-1.79\) \\ \(R^{2}\) & 0.91 & 0.92 & 0.93 & 0.94 & 0.94 & 0.95 & 0.95 & 0.95 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The relationship \(d/b\) with m and n.