## 1 Introduction  
The goal to build a country with a strong transportation network has encouraged the construction of China's modern, comprehensive transportation system to flourish. Karst landforms constitute a kind of harmful geological condition that has an adverse impact on traffic engineering construction, especially the construction of the deep foundation of bridges. They not only induce many technical difficulties to the construction of the bridge infrastructure, but also bring risks to the safe operation of the infrastructure and even serious economic losses [1, 2, 3].  
During earthquakes, the seismic performance of piles influences the effects on the upper bridge, and, therefore, the investigation on the seismic responses of piles is one of the most important research objects in geotechnical and earthquake engineering [2, 4]. Focusing on the slope angle and the elastic modulus of the slope-pile, [PERSON] investigated the seismic response characteristics of a slope rock-socketed pile, employing pseudo-static tests [5]. [PERSON] and [PERSON] investigated the seismic behavior of pile groups embedded in liquefiable soil, and the performance of pile groups was investigated for the fixed and pinned pile to cap connections for both floating and end bearing types of pile groups [6]. [PERSON] et al. investigated the free vibration analysis on the seismic responses of piles,and promoted a new method for road and bridge engineering [7; 8; 9]. According to these excellent papers, the seismic responses of piles in the general soil layers have been widely investigated [4]. However, the investigation of seismic responses of rock-socketed piles in karst areas are very rare [10; 11].  
Since the geological structure of karsts is complicated, piles are widely used for the substructure of important projects in karst areas [4]. In a karst landform, there is a number of karst caves in the field, which can influence the bearing capacity of the foundation; additionally, the scales of karst caves (such as the height and span) are most important influencing factors [12]. Due to the different lithology of the rock mass, the different development of a karst cave, and the complexity and diversity of the filling, the pile in the karst section is prone to large displacement to affect the stability of the superstructure during earthquakes [4]. Furthermore, the seismic responses of rock-socketed piles would also be complicated due to the existence of karst caves. Were rock-socketed piles built passing through karst caves, the scales and positions of karst caves would highly influence the seismic response characteristics and failure mechanism of the rock-socketed piles. Therefore, the investigations on the seismic responses of rock-socketed piles are revealing and essential.  
In this paper, the seismic responses of the rock-socketed pile under complicated karst conditions and vertical loads were investigated, referring to a major bridge structure in a karst area in China. The finite element models of rock-socked piles with different karst conditions were established, and the seismic characteristics and failure mechanisms of rock-socketed piles were investigated considering the effect of the vertical load. The comprehensive strain analyses of the pile under the conditions of different karst cave heights, different karst cave roof heights, the through-pile karst cave, and the underlying karst cave were carried out. The list of symbols in paper is shown in Table 1. The research results are helpful to develop the key construction technology of deep-pile foundations in strong development karst areas, and have theoretical significance and engineering value for improving the design and construction of bridge pile foundations in karst areas.