## 1 Introduction  
In 1957, pulse-like ground motions were first recorded during the Port Hueneme earthquake in the United States. Since a large amount of seismic energy is concentrated in a single pulse and is then released within a short period of time, the losses caused by the Port Hueneme earthquake, which had a magnitude of only 4.7, were much more severe than those from other earthquakes of similar magnitude [1]. Subsequently, earthquakes such as the Imperial Valley earthquake in 1979, the Kobe earthquake in 1995, the Chi-Chi earthquake in 1999, and the Wenchuan earthquake in 2008 all caused extensive damage to engineering structures and substantial human casualties in the earthquake areas, due to the pulse effect of ground motions. The rupture's directional propagation and the fling-step effect are the main causes of pulse-like ground motions, making them dependent on the rupture of faults and the development process of dislocations on fault planes, the sliding direction and speed of the fault planes, the relative position between the observation points and faults, etc. This leads to the inclusion of low-frequency pulses with high energy in pulse-like ground motions [2; 3]. Therefore, the dynamic characteristics of pulse-like ground motions are significantly different from non-pulse ground motions [4]. Scholars have also conducted research into the seismic performance of engineering structures experiencing pulse-like ground motions, indicating that due to the short duration and high-energy ground motion energy input, their impact on the dynamic response of structures, such as force and deformation, is different from that of non-pulse ground motions, and amplifies the dynamic response of medium- to long-term engineering structures [5; 6; 7; 8]. The southwestern region of China is located at the junction of the Indian Ocean tectonic plate and the Eurasian tectonic plate. The Indian Ocean plate is still continuously moving northward and squeezing the Eurasian plate, resulting in the formation of extensive fault zones in the region. This has led to the frequent occurrence of strong earthquakes in southwestern China, such as the Wenchuan earthquake (8.0 Mw) in 2008 and the Lushan earthquake (7.0 Mw) in 2013, both of which occurred around the area of the Longmenshan fault belt in the western Sichuan Province of China, has increased the probability of pulse-like ground motions in the region. Moreover, as a result of the collision of the two plates, the topography of southwestern China is complex, with plateaus, mountains, and canyons that are widely distributed. The region has greatly undulating terrain, making it one of the most precipitous areas in the world [9]. With the continuous economic development of southwestern China, more and more bridges are being constructed to further improve the transportation network in the region. However, due to the complexity of the local topography and landforms, an increasing number of high-pier bridges are being constructed [10]. According to the official statistics, more than 40% of the total number of bridges in the region are high-pier bridges, with pier heights exceeding 40 m. Due to the greater slenderness ratio of these high piers, high-pier bridges typically have lower stiffness and greater natural vibration periods than medium-low-pier bridges. Therefore, the dynamic performance of high-pier bridges is different from that of medium-low-pier bridges. However, the current specifications regarding the seismic-related design of bridges are not suitable for bridges with a pier height that exceeds 40 m [11]. [PERSON] et al. [12] first proposed the concept of seismic fragility analysis in 1975, which can be used to reflect the damage probability of an engineering structure exceeding a certain damage stage under seismic loading. Thereafter, the seismic fragility analysis approach was demonstrated to be an efficient and useful tool for evaluating the safety of structures and infrastructures [13].