## 5 3D Nonlinear Time History Analysis of Bridges
As presented in Table 2, the predictions obtained using the absolute maximum displacements were significantly larger than those obtained by accounting for the direction of the displacement vectors for the radial displacements. To calculate the mean of absolute maximum values, the maximum response of bridges, regardless of the direction in which they occur, is determined for each record and the mean value from 22 records used in the analysis is reported (i.e., the method that is typically currently used in practice). However, for the case of the average values in different directions, the directions of the displacements are considered (i.e., displacements are treated as vectors) and the displacements of columns are determined in different directions from 0 to 360\({}^{\circ}\). For each record the average of maximum displacements in each direction was determined. For example, in this study the radial directions at 10\({}^{\circ}\) intervals were considered and therefore for each record 36 values of maximum displacements in different directions were determined around the full 360\({}^{\circ}\). Finally, the average of 22 values obtained from the 22 records used for the analysis was determined for the 36 directions under consideration. This means that 22 \(\times\) 36 = 792 values of displacements were involved in this method. The maximum displacement predicted in each direction is then reported (i.e., the maximum of the 36 values determined around the full 360\({}^{\circ}\)). To determine the displacement in each direction a computer code was developed for posts processing the analysis results, since current available software typically don't \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 & 77 (2.0) & 64 (1.7) & 95 (0.66) & 82 (0.6) \\  & Column 2 & 109 (2.8) & 93 (2.4) & 131 (3.24) & 112 (2.8) \\  & Column 3 & 77 (2.0) & 65 (1.7) & 101 (0.32) & 88 (0.3) \\ \hline \multirow{3}{*}{Radial} & Columns 1, 2 and 3 & 86 (2.2) & 74 (1.9) & 138 (1.0, 3.4, 0.4) & 124 (0.9, 3.1, 0.4) \\  & Column 1 & 101 (2.6) & 69 (1.8) & 148 (1.0) & 116 (0.8) \\  & Column 2 & 123 (3.2) & 84 (2.2) & 166 (4.1) & 114 (2.8) \\  & Column 3 & 100 (2.6) & 69 (1.8) & 151 (0.5) & 115 (0.4) \\ \hline \multirow{3}{*}{30\% Rule} & Column 1 & 109 (2.8) & 94 (2.4) & 167 (1.2) & 149 (1.0) \\  & Column 2 & 135 (3.5) & 116 (3.0) & 178 (4.4) & 158 (3.9) \\ \cline{1-1}  & Column 3 & 109 (2.8) & 94 (2.4) & 169 (0.5) & 151 (0.5) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the mean displacements (mm) of columns using 3D analysis for the regular and irregular bridges with mean ductility demands given in brackets. Figure 7: The maximum displacements obtained from 22 records using the maximum absolute displacements for: (**a**) regular bridge; (**b**) irregular bridge; and using the negative and positive displacements for: (**c**) regular bridge; (**d**) irregular bridge. report such data. It is noted this method accounts for the fact that displacements are vectors and their mean values should consider their direction. Further details regarding the logic of this method are given by [PERSON] et al. [1]. It is noted that this method has not been typically used in practice and in this paper the applicability of this method is investigated. For example, the average radial displacement demand for the central column (C\({}_{2}\)) of the regular bridge was determined as 123 mm using the absolute displacements from different records and 84 mm by considering the direction of the radial displacements (i.e., more than 45% difference).