## 5 3D Nonlinear Time History Analysis of Bridges  
Structural analysis is typically carried out in the two perpendicular principal directions of the structure (i.e., X and Y directions) and therefore the peak displacements only in the X and Y directions are obtained. Since the peak displacements in the X and Y directions occur at different times, it is not possible to estimate the maximum radial (vectorial) displacement of the columns. In this study the radial displacements were determined at each time interval i, using the component of displacements in X and Y directions (i.e., \(\Delta_{\mathrm{i,r}}=(\Delta^{2}_{\mathrm{i,x}}+\Delta^{2}_{\mathrm{i,y}}) ^{\frac{1}{3}}\), where \(\Delta_{\mathrm{i,r}}\) is the radial displacement and \(\Delta_{\mathrm{i,x}}\) and \(\Delta_{\mathrm{i,y}}\) are the displacements in X and Y directions, respectively). Figure 6 shows the time history analysis results for the central column of the regular bridge, shown in Figure 2a, subjected to one of the selected pairs of records for the longitudinal and transverse directions. The computed radial displacements at each time step are also shown in Figure 6. It is noted that, in Figure 6, while the longitudinal and transverse displacements have a sign, the radial displacement is always positive, as calculated using the square root of the sum of the squares of displacements in the principal directions. This is because the radial displacement in Figure 6 only represents the absolute value of the displacements that can occur in a random direction around the full 360\({}^{\circ}\). Further details regarding the direction of the radial displacements are provided in Section 7. Figure 5: SRSS response spectra and mean SRSS spectrum of 22 records used for the analysis of: (a) regular bridge; (b) irregular bridge. The maximum displacements obtained for the central columns of the bridges from 22 records are presented in Figure 7 and the average values are indicated by black circles. The results are shown using two different averaging methods: using maximum absolute values (see Figure 7a,b); and considering positive and negative displacements in the longitudinal and transverse directions (see Figure 7c,d). The results presented in Figure 7 indicate that when the sign of the displacements is considered the mean predictions are around 10 to 15% smaller compared to the maximum absolute predictions. Also it is clear from Figure 7a,b that the maximum radial displacements are around 13 to 20% larger than the maximum displacements predicted in the principal directions. The predicted average displacements of the bridge columns are summarized in Table 2 for the regular and irregular bridges. The average displacements and corresponding displacement ductility demands are also given in Table 2 for the longitudinal, transverse and radial directions. To determine the ductility demands the predicted maximum displacements, \(\Delta_{\text{max}}\), were divided by the yield displacement of the columns, \(\Delta_{\text{y}}\) (i.e., \(\mu=\Delta_{\text{max}}/\Delta_{\text{y}}\)). The yield displacement of the columns was predicted using the equations provided by [PERSON] et. al. [1] that are also used by the AASHTO guide specifications [3] and Caltrans [2]. The ductility demands are also provided in Table 2, because they are better indicators of damage in columns. A larger ductility demand indicates that a larger plastic deformation, and therefore greater damage is expected in a column. It is noted that, as seen in Table 2, in the longitudinal direction the displacements obtained were similar for all bridge columns due to the large axial stiffness of the superstructure (less than 1% difference observed). Figure 6: An example of the time history results for the longitudinal, transverse and radial displacements. The average responses were predicted using different averaging methods.