## 6 Seismic Response Characteristics by Nonlinear Dynamic Analyses
### Analysis of Maximum Moment  
Figure 13 shows the variation in the maximum moment as the height of the bridge increases. The maximum moment was more significant when the bridge was subjected to near-fault ground motion than when subjected to far-fault ground motion. In particular, as discussed regarding the maximum displacement, near-fault ground motion with the flip-step effect showed a maximum moment larger than those with the forward-directivity effect. A maximum moment of approximately 26,600 kN-m occurred when the bridge with a height of 5 m was subjected to the EQ7_NF near-fault ground motion with the flip-step effect. For high, tall bridges, the flip-step near-fault ground motion also produced relatively high values of the maximum moment ranging from approximately 17,500 to 25,000 kN-m. On the other  
Figure 11: Relationship of the displacement spectrum (\(\mathrm{S_{d}}\)) and maximum lateral displacement with the pier height for near-fault ground motions: (**a**) Displacement spectrum at near-fault ground motion; (**b**) maximum displacement at near-fault ground motion.  
Figure 12: Relationship of the displacement spectrum (\(\mathrm{S_{d}}\)) and maximum lateral displacement with the pier height for far-fault ground motions: (**a**) Displacement spectrum at far-fault ground motion; (**b**) maximum displacement at far-fault ground motion  
hand, when the bridge was subjected to far-fault ground motion, the largest maximum moment was approximately 7600 kN-m at the bridge with a height of 20 m. The maximum moment of a bridge subjected to the forward-directional ground motion tended to decrease with increasing bridge height. That is, in the case of far-fault ground motion and near-fault ground motion with the filing-step effect, the decreasing trend of the maximum moment with increasing bridge height was indistinct. Furthermore, the ground motion with the filing-step effect remained a high maximum moment, barely affected by the bridge height.  
For a bridge, the maximum moment could have a relationship with the maximum displacement. However, considering the changes in structural configuration and ground motion, the moment is barely related to the displacement response of the structure alone. Therefore, this study proposed the non-dimensional dynamic response ratio (\(DR\)), which is defined as the ratio of the maximum dynamic response and the yield displacement of a bridge, to accurately analyze the tendency of the maximum moment occurring in a bridge according to the change in bridge height:  
\[DR(\%)=\ \frac{D_{max}}{D_{y}}\ \times 100 \tag{1}\]  
where \(D_{max}\) is the maximum displacement response of a bridge for each ground motion, and \(D_{y}\) is the yield displacement of a pier with different heights, as defined in Section 6.1.  
Figure 14 presents a scatter plot of the maximum moment and \(DR\) for all the bridges under both the near-fault and far-fault ground motions. When subjected to far-fault ground motions, the bridges showed the maximum moment within approximately 25% of the \(DR\). In the case of near-fault ground motions with forward directivity, the maximum moment reached as much as approximately 80% of the \(DR\). In particular, some near-fault ground motions with the filing-step effect were shown to be larger than 80% of the \(DR\). A \(DR\) higher than 100% means that the displacement response exceeds the yield displacement capacity of the bridge, resulting in the yielding of the bridge under an earthquake. The maximum moment tends to increase as the \