## 5 Response of Pulse Seismic Waves to Bridge Piers
### Response of a Bridge Pier Under Pulse Ground Motion  
The acceleration at the bridge pier height of 10 m and the stress at 3 m were taken as the investigation items. Figure 5 illustrates the acceleration and displacement time histories of the bridge pier under various characteristic periods of pulse earthquake and non-pulse earthquake actions. As can be observed in Figure 5, the acceleration time histories of bridge piers subjected to varying characteristic period seismic actions are extremely different, and the displacement response time histories are also extremely different. When subjected to identical characteristic periods, the bridge pier's acceleration and displacement responses differ markedly between pulse and non-pulse motions. Pulse-type ground motions produce greater acceleration and displacement responses compared to non-pulse motions. This disparity highlights the substantial symptom of the motion type on the dynamic behavior of the structure, with pulse motions inducing higher peak forces and displacements. Such findings stress the paramount need to adopt motion characteristics in seismic events to ensure the resilience of infrastructure, particularly in regions prone to pulse-like ground motions. When Tg = 0.45 s, the peak acceleration values of the bridge pier under non-pulse seismic motion and pulse ground motion are 1.0694 m/s\({}^{2}\) and 1.3252 m/s\({}^{2}\), respectively, and the discrepancy between the two earthquake conditions is 23.92%. The peak stress values are 10.7879 MPa and 20.2414 MPa, respectively, with an error of 87.63%. When Tg = 0.40 s, the peak acceleration values of the bridge pier under non-pulse seismic motion  
Figure 4: The acceleration and stress time histories of bridge piers with different characteristic periods in an anhydrous environment. and pulse ground motion are 1.1844 m/s\({}^{2}\) and 1.4778 m/s\({}^{2}\), respectively, with an error of 24.77%. The peak stress values are 10.8770 MPa and 19.6423 MPa, respectively, and the error is 80.59%. When Tg = 0.30 s, the peak acceleration values of the bridge pier in response to non-pulse seismic motion and pulse ground motion are 1.1566 m/s\({}^{2}\) and 1.4228 m/s\({}^{2}\), respectively, and the discrepancy between the two conditions is 23.02%. The peak stress values are 12.7339 MPa and 20.0871 MPa, respectively, with an error of 57.75%. Among the three characteristic periods, the errors in the peak acceleration and peak stress of a bridge pier are different under pulse and non-pulse seismic motions. The largest peak stress error occurs at Tg = 0.45 s, while the highest peak acceleration error is observed at Tg = 0.40 s. The smallest errors for both peak acceleration and peak stress are found at Tg = 0.30 s. This indicates that under varying characteristic periods of ground motion,  
Figure 5: Acceleration and stress time histories of bridge pier under different characteristic periods of pulse earthquake and non-pulse earthquake motions. the non-pulse seismic motion and pulse ground motion have a considerable effect on the acceleration and stress responses of the bridge piers. This is mainly because different characteristic periods are associated with distinct ground motion spectrum characteristics. To substantiate the conclusion with greater clarity, a fast Fourier transform was applied to analyze the acceleration and stress time histories presented in Figure 5 and to obtain the frequency domain representation of the acceleration and the stress response time histories, as illustrated in Figure 6. From Figure 6, when the period of the seismic motion Tg = 0.45 s, the peak acceleration amplitudes of the bridge pier under non-pulse seismic motion actions and pulse ground motion actions are 0.1915 m/s\({}^{2}\) and 0.2572 m/s\({}^{2}\), respectively, showing a discrepancy of 34.31%. The peak values of stress are 3.8756 MPa and 4.9359 MPa, respectively, showing a discrepancy of 27.36%.