## 7 Effects of Directivity on the Predictions of the Seismic Response  
For the predictions of the seismic responses of the bridges in the previous sections the ground motion records were only applied in the principal axes of the bridges, which is customary in current seismic design practice (i.e., \(\theta=0\), where the angle \(\theta\) is defined in Figure 10). For bridges that have a significantly different behavior in the transverse and longitudinal directions, the effects of applying the ground motion records along different axes need to be examined. Figure 9: Transverse displacement envelopes obtained by means of 2D elastic and inelastic analyses for: (**a**) regular bridge (**b**) irregular bridge. To investigate the influence of applying the ground motion records at different angles, each pair of ground motion records was applied to the structural models at different angles, \(\theta\), from 0 to 180\({}^{\circ}\) with increments of 5\({}^{\circ}\). The angle \(\theta\) that gives the directivity of the input motions is shown in Figure 10. It was found that the responses of the bridges studied were similar for the angles \(\theta\) and 180\({}^{\circ}\) + \(\theta\) and therefore only \(\theta\) values between 0 and 180\({}^{\circ}\) were considered. The variations of the predicted maximum radial displacements of the bridge columns are shown for 22 different pairs of records used in the analyses in Figure 11a,b for the regular and irregular bridges, respectively. As the angle \(\theta\) changes, the predicted maximum radial displacements change. The maximum predicted response for each record occurs at a critical angle, \(\theta_{\text{cr}}\), which could be significantly different from that obtained by applying the records at \(\theta=0^{\circ}\) (i.e., current practice). The variations of the predicted responses and the predicted critical angles differ for different records and they depend on the properties of both the records and the structure. In Figure 11, the average responses for 7, 14, and 22 records are also shown for different values of \(\theta\). Despite the fact that the variation of the predicted responses from different records at different angles, \(\theta\), are relatively large, the variation of the predicted average responses at different angles is much smaller and as the number of records increases, the variations become even smaller. The parameter \(\theta\) is considered as a random variable in this study and therefore different pairs of records can occur at different \(\theta\) angles. Figure 11: Maximum radial displacement of the critical central column for different records applied at different angles \(\theta\) for (**a**) regular bridge; (**b**) irregular bridge. Figure 10: Applying two horizontal components of the records, H\({}_{1}\) and H\({}_{2}\), along X\({}^{\prime}\) and Y\({}^{\prime}\) axes rotated by \(\theta^{\circ}\) relative to the principal axes of structure, X (longitudinal direction) and Y (transverse direction). One approach is to use the maximum predicted response from each record at its corresponding critical angle to predict the average maximum response of the bridge. A summary of the results obtained for this case is presented in Table 4 and Figure 12b. However, this method is overly conservative, because it is highly unlikely that all input ground motions would occur at their critical angle. For example, the mean displacement for the central column of the irregular bridge was predicted as 235 mm when the records were applied at their critical angles, \(\theta_{\text{cr}}\). The predicted mean displacement for this case is about 42% larger than the mean displacement of 166 mm (see Table 2) predicted when the records were applied along the principal axes of the bridge. Similar results were observed for the case of the regular bridge.