# A Simplified Analysis Method for Seismic Response of
## 1 Introduction  
Because the pile foundation can be adapted to different load types and geological conditions, it is widely used in high-rise buildings, bridges, whaves, nuclear power plants, and other large-scale projects. With the rapid development of pile foundation engineering and the need for earthquake prevention and disaster reduction, the dynamic response of pile foundations has become an important research highlight in the engineering field. The earthquake damage data [1, 2] show the pile foundation has good seismic performance. But even if the stiffness of the pile is large and sufficient reinforcement for piles, it is hard to avoid damage under strong earthquakes. For bridge structures, pile foundation damage due to earthquakes often leads to serious consequences, such as falling beams [3, 4, 5, 6]. For buildings, although it generally does not cause disastrous consequences such as the collapse of the superstructure, the concealment of pile foundation also makes it difficult to grasp the earthquake damage [7, 8]. The setting of the pile foundation significantly improves the bearing capacity of the original foundation, so the failure of the pile may cause foundation instability. Therefore, it is necessary to study the dynamic characteristics of the pile foundation under the earthquake load. At present, many methods have been accumulated in the research field of pile foundation dynamic response. According to the different simulation methods and calculation models of the soil around the pile, it can be divided into analytical method [9, 10], experimental method [11, 12], discrete model method [13, 14], and numerical method [15, 16]. For pile group foundation, the latter two methods are more widely used. The dynamicWinkler model of the foundation beam [17; 18] belongs to the discrete model method. It uses continuously distributed and independent springs and dampers to replace the dynamic effect of the soil around the pile on the pile. It has the characteristics of a clear mechanical concept, simple and practical. The disadvantage is that the wave in the foundation cannot be considered, and the application is easily limited when the number of piles is large. The numerical analysis methods such as finite element [19; 20] and boundary element [21; 22] have higher accuracy than the discrete model method and can simulate more complex situations according to the actual situation, such as the heterogeneity of soil, the pile-soil interface, the coupled vibration of pile groups, and other complex vibration problems. Nevertheless, the numerical method has a huge amount of calculation and complicated boundary conditions, which makes it difficult to implement in practical engineering. Existing tests and numerical calculations show that the deformation and internal force of the pile foundation are jointly affected by the motion interaction (free field deformation of the soil layer) and the inertial interaction (inertial force of the superstructure) [23; 24]. In general, the two effects play equally important roles and should be considered. At present, the seismic design of the superstructure and underground pile foundation is calculated separately. In the seismic design of a pile foundation, the consideration of the horizontal shear force transmitted by the superstructure and the soil around the pile foundation at the same time is not well solved. The dynamic time history analysis method can accurately analyze the seismic response of pile foundations and superstructures. However, due to the complexity of practical engineering problems, the use of time history analysis methods often requires a long computational time [25; 26]. From the seismic calculation results of underground structures such as superstructures and rail transits, the quasi-static calculation method can give more practical calculation results, which is simple and has been widely used in engineering practice.