## 7 Effects of Directivity on the Predictions of the Seismic Response
One interesting point that can be noticed in Table 4. is that for the averaging method based on different directions the radial displacements predicted for the central critical columns are 100 mm and 133 mm for the regular and irregular bridges, respectively. These displacements are comparable to the maximum displacements of 109 mm and 138 mm, respectively, predicted in Table 2 for the displacements in the principal directions using the averaging method based on maximum absolute values for the same bridges. This may \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Regular Bridge**} & \multicolumn{2}{c}{**Irregular Bridge**} \\ \cline{3-6}  & & **Average of Absolute** & **Average in Different** & **Average of Absolute** & **Average in Different** \\  & & **Maximum Values** & **Directions** & **Maximum Values** & **Directions** \\ \hline \multirow{3}{*}{Transverse} & Column 1 & 81 (2.1) & 71 (1.8) & 88 (0.6) & 76 (0.5) \\  & Column 2 & 156 (4.0) & 129 (3.4) & 147 (3.6) & 126 (3.1) \\  & Column 3 & 81 (2.1) & 71 (1.8) & 98 (0.3) & 84 (0.3) \\ \hline \multirow{3}{*}{Radial} & Columns 1, 2 and 3 & 133 (3.5) & 105 (2.7) & 196 (1.4, 4.9, 0.7) & 159 (1.1, 3.9, 0.6) \\ \hline \multirow{3}{*}{Radial} & Column 1 & 148 (3.9) & 93 (2.4) & 220 (1.5) & 143 (1.0) \\  & Column 2 & 175 (4.5) & 100 (2.6) & 235 (5.8) & 133 (3.3) \\  & Column 3 & 148 (3.8) & 90 (2.3) & 221 (0.7) & 142 (0.5) \\ \hline \multirow{3}{*}{30\% Rule} & Column 1 & 157 (4.1) & 127 (3.3) & 222 (1.6) & 190 (1.3) \\  & Column 2 & 180 (4.7) & 154 (4.0) & 240 (5.9) & 197 (4.9) \\ \cline{1-1}  & Column 3 & 157 (4.1) & 125 (3.2) & 225 (0.7) & 191 (0.6) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of the mean displacements (mm) of columns using 3D analysis for the regular and irregular bridges by applying the records at their critical angle (i.e., maximum possible seismic demands). Mean ductility demands given in brackets. Figure 12: Mean displacement of the central column of the bridges along different directions for: (**a**) records applied along the principal axes of bridges; (**b**) records applied at their critical angle, \(\theta_{\text{cr}}\). indicate that the use of a simple 3D analysis using the current simple averaging methods can predict the actual radial displacement including directivity effects, when the concepts of averaging in different directions are adopted. In Figure 12 the average displacement of the critical central columns of the bridges in different directions are shown for two cases including when the records are applied only at the principal axes of the bridges (e.g., see Table 2 and Figure 12a) and when the records are applied at their critical angles, \(\theta_{\text{cr}}\) (see Table 4 and Figure 12b). It is interesting to note that for the regular bridge the maximum displacements occur at about 270\({}^{\circ}\) while for the irregular bridge the maximum displacement occurs in a range close to 180\({}^{\circ}\). This indicates that the transverse response is critical for the regular bridge and the longitudinal response is critical for the irregular bridge. It is overly conservative to predict the seismic response deterministically by applying records at their critical angle and hence the uncertainty due to the angle \(\theta\) should be considered probabilistically. By applying each record at different \(\theta\) angles a set of maximum radial displacements can be determined. For example, in this study, each record was applied at 37 different \(\theta\) angles from 0\({}^{\circ}\) to 180\({}^{\circ}\) with increments of 5\({}^{\circ}\). To investigate if the data is Normally distributed, the Quantile-Quantile plots (Q-Q plots) [62] were developed for the data obtained from different records. The results demonstrated that the normal distribution assumption was satisfactory.