# Collapse Analysis of a Two-Span Reinforced Concrete Bridge Model
## 4 Analysis of Collapse Mode
### Influence of Multidirectional Seismic Coupling  
In the previous shaking table test, due to equipment constraints, only horizontal ground motion input was carried out. The numerical analysis also followed the input mode of the shaking table test in the horizontal direction, so the collapse mode obtained did not consider the influence of vertical ground motion input. However, in an actual earthquake, the input of a seismic wave imposes both vertical and horizontal loading at the same time. Therefore, the coupling effect of vertical ground motion and horizontal ground motion was considered in this paper, and the longitudinal collapse mode of a two-span continuous bridge model was analyzed. The test waves were the El Centro wave and Chi\(-\)chi wave. According to the information of original ground motion, the peak ratio of vertical ground motion and horizontal ground motion was adjusted to the peak ratio of the original wave; thus, the peak ratios of the El Centro wave and Chi-chi wave were 0.6 and 0.8, respectively. The case of the El Centro wave with a 0.4 g longitudinal PGA was selected for analysis. The time history curve of vertical ground motion in this case is shown in Figure 10, and the PGA was 0.24 g. After adding the vertical ground input, the longitudinal relative displacement between the superstructure and substructure was basically not changed. From the comparison curve of Figure 11, it can be seen that the two curves almost coincided, and the maximum longitudinal relative displacement increased from 12.14 mm to 12.45 mm. The peak acceleration response of the middle pier cap increased slightly from 6.75 m/s\({}^{2}\) to 7.68 m/s\({}^{2}\) under unidirectional input. The main compressive strain at the bottom of the middle pier also increased from 519 \(\mu\epsilon\) to 531 \(\mu\epsilon\), as shown in Figures 12 and 13. Figure 9: Longitudinal collapse mode of bridge model under Chi-chi wave. In the analysis of the 0.4 g El Centro bidirectional wave case, it could be seen that the coupling effect of vertical ground motion and horizontal ground motion was not significant. The bridge structure finally failed in the form of the collapse of the main girder due to the failure of the middle pier. The PGA was still 1.2 g, and the specific collapse process was similar to that in Figure 8.  
Figure 11: Time history curve of the relative displacement between the side pier and the girder under El Centro wave. Figure 10: Time history curve of vertical seismic input of El Centro wave. Then, the longitudinal Chi\(-\)chi wave of 0.4 g PGA was selected for analysis. The time history curve of vertical ground motion in this condition is shown in Figure 14, and the PGA was 0.32 g. After adding vertical ground motion in the longitudinal seismic condition, the response of the structure showed no obvious change. From Figure 15, the maximum longitudinal relative displacement was 15.47 mm, which was basically consistent with the value of 15.53 mm under unidirectional input. The acceleration response of the middle pier increased from 6.59 m/s\({}^{2}\) to 7.52 m/s\({}^{2}\) under unidirectional input, as shown in Figure 16. The response diagram of the maximum principal compressive strain at the bottom of the middle pier showed that the force of the pier column was slightly weakened, from the maximum compressive strain of 611 \(\mu\epsilon\) to 602 \(\mu\epsilon\) (see Figure 17). Figure 14: Time history curve of vertical seismic input of Chi\(-\)chi wave. Figure 13: The time history curve of principle compressive strain of middle pier at the bottom under El Centro wave. Figure 15: Time-history curve of the relative displacement between the side pier and the girder under Chi\(-\)chi wave. From the analysis of the bidirectional input condition of the Chi-chi wave with 0.4 g PGA, it was found that the vertical and horizontal coupling of ground motion was not obvious. The two-span continuous beam bridge model finally collapsed in the 1.2 g Chi-chi wave bidirectional input condition, and the collapse mode was a falling beam.