## 3 Analysis of Acceleration Response Results
### Acceleration Response of Beams  
Figure 8 shows that the response waveforms for unidirectional and bidirectional excitations could be directly superimposed with the response waveform for multidirectional excitation. The waveforms were close at L/4 of the mid-span for beam 3 and slightly differed at the mid-span for beam 1, and the differences increased over time. However, the waveforms generally maintained their original shapes with a large acceleration value. The peak variations in the acceleration responses of beam 1 and beam 3 were 8.1% and 6.6%, respectively. For practical application, this error meant that only bidirectional excitation could be considered and vertical excitation could be ignored for the seismic design of the main beam of such Y-shaped bridges.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Waves**} & **Input** & \multirow{2}{*}{**X-Direction**} & **Y-Direction** & **X-Direction** & **Y-Direction** & **X-Direction** & **Y-Direction** \\ \cline{3-8}  & **Direction** & & **Pier 1** & & **Pier 2** & & **Pier 3** \\ \hline \multirow{2}{*}{El Centro} & X + Y & 0.470 & 0.636 & 0.390 & 0.587 & 1.156 & 1.531 \\  & X + Y + Z & 0.548 & 0.874 & 0.406 & 0.56 & 1.220 & 2.000 \\ \hline \multirow{2}{*}{Imperial} & X + Y & 0.391 & 0.823 & 0.215 & 0.84 & 0.530 & 0.859 \\  & X + Y + Z & 0.267 & 0.383 & 0.099 & 0.323 & 0.284 & 0.371 \\ \hline \multirow{2}{*}{El Centro} & \multicolumn{4}{c}{**Pier 4**} & \multicolumn{4}{c}{**Pier 5**} & \multicolumn{4}{c}{**Pier 6**} \\ \cline{2-8}  & X + Y & 1.186 & 1.23 & 0.652 & 0.677 & 3.870 & 2.695 \\  & X + Y + Z & 1.007 & 1.1 & 0.738 & 0.693 & 3.466 & 3.200 \\ \hline \multirow{2}{*}{Imperial} & X + Y & 0.458 & 0.67 & 0.267 & 0.402 & 0.425 & 1.402 \\  & X + Y + Z & 0.177 & 0.23 & 0.79 & 0.817 & 0.762 & 2.465 \\ \hline \end{tabular}
\end{table}
Table 5: Peak acceleration response of the pier top under multidirectional excitation (_g_).  
Figure 7: Acceleration response at the top of the piers. (**a**) X-direction (**b**) Y-direction.  
Figure 9 shows the vertical accelerations for the mid-spans of beam 1, beam 2, and L/4 of beam 3 subjected to tridirectional (X + Y + Z) El Centro and Lanzhou waves. The mid-span response was greater with the El Centro wave than with the Lanzhou wave, which was related to the frequency spectral characteristics of the excitation waves and structural properties of the bridge. Each excitation had its own frequency spectrum distribution characteristics. Meanwhile, it was verified that the dynamic responses of structures were closely related to the frequency characteristics of the input ground motions. Beam 1 had the largest mid-span acceleration response, and L/4 of beam 3 had the smallest. This was because beam 1 was straight and was simply supported without redundant constraints, so the span and weight were relatively large. Beam 2 was straight and was simply supported as well, but it was lighter than beam 1. Beam 3 was curved with redundant constraints and lightweight. Thus, beam 1 had a greater mid-span stress, and transverse cracks appeared earlier and in more abundance in the test process, than for the curved beam 3, as shown in Figure 10. This indicated that, for irregular bridges, simply supported straight beams did not provide better seismic performance than curved beams in some cases. In addition, redundant constraints could also reduce the mid-span acceleration response.  
Figure 8: Vertical acceleration response at the mid-span. (**a**) Beam 1; (**b**) Beam 3.  
Figure 10: Cracks in the mid-span. (**a**) Beam 1; (**b**) 1/4-span of Beam 3.  
Figure 9: Acceleration response at the mid-span with multidirectional excitation. (**a**) Beam 1; (**b**) Beam 2; (**c**) 1/4-span of Beam 3.