## 4 Results and Discussion  
### Displacement of Abutments  
From the time-history analyses on all SIAB models, it was found that the maximum longitudinal and transverse displacements of the abutment were always located at the abutment top. Therefore, only the abutment top displacement is discussed in this section. Table 4 shows the maximum abutment displacements from the time-history analyses on the prototype bridge model (Model 10) with a skew angle of 15\({}^{\circ}\) and densely compacted abutment backfill. Figure 5: Acceleration, velocity, and displacement time-histories of pulse and non-pulse type ground motions. Figure 6: Acceleration response spectra of pulse and non-pulse type ground motions. From Table 4, it is clear that there are longitudinal as well as transverse displacements of the abutment under earthquake ground motions inputted in the longitudinal direction of the bridge, which reflected an in-plane rotation of the bridge superstructure, as shown in Figure 7 below. This is mainly because when there is a skew angle between the longitudinal direction of the bridge and the normal direction of the abutment, the rigid center of the bridge deck does not coincide with the mass center, so there will be coupling between the longitudinal and the transverse displacements of the bridge deck, which is directly reflected in the abutment displacements. Furthermore, abutment displacement at the acute corners is always higher than the obtuse corners, especially for Abutment 1, for which the longitudinal displacement at the acute corner is about twice as much as that of the obtuse corner under GM1. Therefore, only the abutment displacement at the lower left acute corner of Abutment 1 is discussed in the following sections. In order to quantify the rotation angle of the bridge deck, Equation (4) below was used with orientation and sign conventions defined in Figure 7.  
\[\omega=\arctan(|U_{2}-U_{1}|/L) \tag{4}\]  
where \(U_{1}\) represents transverse displacement of abutment 1 at obtuse corner, \(U_{2}\) represents transverse displacement of abutment 2 at acute corner, and \(L\) is the bearing to bearing distance of the skewed integral abutment bridge. Time-history curves of the longitudinal and transverse displacement of the abutment are shown in Figures 8 and 9 below for Model 12 (skew angle is 45\({}^{\circ}\), densely compacted backfill) and Model 16 (skew angle is 45\({}^{\circ}\), no backfill), respectively. As shown in the Figures, the abutment displacement time-history under the pulse and non-pulse type ground motions are quite different, although the PGA was the same, and the pulse type ground motions obviously induced much higher abutment displacements, which is more destructive for the bridge. In addition, when there was backfill behind the abutment, the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Abutment**} & \multirow{2}{*}{**
\begin{tabular}{c} **Lateral** \\ **Displacement** \\ \end{tabular}**} & \multirow{2}{*}{**Location**} & \multicolumn{3}{c}{**Pulse Ground Motion (mm)**} & \multicolumn{3}{c}{**Non-Pulse Motion (mm)**} \\ \cline{3-10}  & & & **GM1** & **GM3** & **GM5** & **GM2** & **GM4** & **GM6** \\ \hline \multirow{4}{*}{Abutment 1} & Longitudinal & Acute & 9.1 & 9.3 & 8.7 & 8.5 & 8.9 & 11.2 \\  & & Obtuse & 4.6 & 6.8 & 6.1 & 6.0 & 6.3 & 8.8 \\ \cline{2-10}  & Transverse & Acute & 15.5 & 21.8 & 15.5 & 12.9 & 14.1 & 15.4 \\  & & Obtuse & 13.9 & 20.0 & 14.0 & 11.4 & 12.5 & 13.7 \\ \hline \multirow{4}{*}{Abutment 2} & Longitudinal & Acute & 7.4 & 9.6 & 8.9 & 8.6 & 8.9 & 11.3 \\  & & Obtuse & 6.3 & 6.7 & 5.9 & 5.8 & 6.1 & 8.4 \\ \cline{1-1} \cline{2-10}  & Transverse & Acute & 14.3 & 15.0 & 14.5 & 12.8 & 15.0 & 14.1 \\ \cline{1-1}  & & Obtuse & 12.6 & 13.3 & 12.9 & 11.2 & 13.4 & 12.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum abutment displacement for Model 10. Figure 7: Abutment displacement and bridge deck in-plane rotation.