## 9 Conclusions  
This study, while limited to two bridges, demonstrates that the influence of the directivity of ground motion is important and guidance is given on predicting the mean response considering the effects of directivity of horizontal ground motions. Conclusions from this study are summarized as follows:  
(1) The use of the 30% rule using 2D analyses resulted in good predictions of the maximum radial displacement of columns for the case of the straight bridges studied in this research, when the average displacements were computed using the maximum absolute displacements for each ground motion record. No general conclusion can be made in this regard without further research. (2) When the averaging procedure involving the prediction of the mean displacements in different radial directions was adopted for the bridges in this study, the maximum radial displacements obtained using 3D analyses were smaller than the maximum displacements obtained in the longitudinal and transverse directions using 2D analysis. However, whenthe averaging procedure based on the maximum absolute values was adopted to determine the mean displacements, the displacements in the transverse and longitudinal directions needed to be combined using the 30% rule resulting in predictions similar to those obtained using 3D analyses. (3) The use of the averaging method in different radial directions, which is more logical than the averaging methods using the maximum absolute displacements, resulted in 40-50% smaller predicted displacements for the 3D analyses. For this approach the maximum possible mean radial displacements predicted, that is, when all records were applied at their critical angles at the same time, was in the order of the maximum response predicted in the principal axes for a 3D analysis using absolute maximum values and without considering the directivity effects (i.e., simple method that typically used in practice). This means that when the concepts of averaging in different directions is adopted, the use of a simple 3D analysis without considering directivity effects and without determining the radial displacements may predict the maximum response, when averaging based on maximum absolute values are used only for the displacements along the principal axes of the bridge. (4) By comparing the seismic response of the regular and irregular bridges predicted using different approaches, it was observed that for the irregular bridge significantly larger ductility demands and plastic rotations occur in the shortest column compared to the case of the regular bridge. This can significantly affect the vulnerability of the irregular bridges. (5) When all of the records were applied at their critical angles (overly conservative case), the predicted displacements increased by 40-50% compared to the predictions determined by applying the records in the principal axes of the structure. (6) A method was proposed to predict the variability in the mean response due to the application of the pairs of records at different angles \(\theta\). Although the response of the individual records was found to be sensitive to the variations of angle \(\theta\), when the mean response was considered the variations were relatively small. (7) Although the use of the 30% rule provides conservative predictions of the radial displacements when compared with the predictions using 3D analyses, when directivity effects are not considered, this conservatism allows for some increases in the radial displacements due to different excitation angles, \(\theta\). The probability of exceeding the mean radial displacements of the columns, obtained from the 30% rule, was about 15 to 20%, for the cases studied when directionality effects were considered. This probability may be higher for different structures especially some highly irregular structures. More research is needed on this subject. (8) The combination rules such as 100-30% rule in the codes are typically recommended and used for the elastic analysis methods, when a bridge is designed.