# Study on the Time Domain Semi Analytical Method for Horizontal Vibration of Pile in Saturated Clay
## 1 Introduction  
Saturated clay is a typical weak soil that cannot be directly used as a bearing foundation for structures. Therefore, in practical engineering, pile foundations are usually used to pass through saturated clay layers to directly transmit loads to the actuating layer, while pile foundations are buried in saturated clay. During the service process, pile foundations not only need to bear the vertical loads transmitted by the upper structure but also bear various horizontal dynamic loads, such as environmental vibration, earthquakes, and impacts [1; 2; 3; 4; 5]. Therefore, the horizontal vibration response of pile foundations has always been a key issue of concern in engineering. At present, the design and analysis methods for the bearing capacity of pile foundations are very mature, while dynamic analysis is still in the development stage. [PERSON] [6] established a coupled three-dimensional dynamic analysis model for piles in a liquefied foundation, studied the dynamic characteristics of piles, and analyzed the reliability of the model under dynamic loads using centrifugal tests. [PERSON]. [7] studied the influence of seismic body wave types and their incidence angles on the dynamic response of pile foundations. [PERSON] [8] used numerical simulation methods to simulate the dynamic behavior of soil, pile foundations, and structures under seismic excitation while considering the seismic performance of buildings located on soft soil. [PERSON] [9] used finite difference software to analyze the deflection and bending moment behavior along the pile length in non liquefied and liquefied soils under seismicloading conditions, and studied the horizontal and vertical inertial interactions as well as the kinematic interactions caused by free field motion. [PERSON] [10] used numerical simulation methods to study the response of adjacent pile foundations, strong grouting protection, and soil interactions induced by tunnel construction. It can be seen that further research on the horizontal vibration characteristics of pile foundations in saturated clay is of great significance for engineering construction, seismic resistance, and the safe service of saturated clay sites. Based on the Biot porous medium theory, scholars, such as [PERSON] [11], have conducted many studies on the horizontal vibration characteristics of pile foundations in saturated soil. According to the mixed boundary value conditions, the dual integral equations of the horizontal vibration of a rigid disk on a saturated poroelastic half-space are established, analytically examining the horizontal vibration of a rigid disk on a saturated poroelastic half-space. The dynamic response of pile groups partially buried in layered saturated soil under horizontal harmonic loads was studied [12]. The Winkler model was used to derive the soil-pile dynamic interaction factor. Based on the dynamic interaction factor, the horizontal impedance of the pile group was obtained using the superposition principle, providing a valuable reference for geotechnical and seismic engineers in pile foundation design. [PERSON] [13] developed a simplified analytical solution for the impedance function of viscoelastic pile groups partially embedded in a layered, transversely isotropic, liquid-saturated, viscoelastic soil medium under a vertical time-harmonic load. [PERSON] [14] provided a closed-form solution for the horizontal vibration of pipe piles in saturated soil, considering radial heterogeneity effects. [PERSON] [15] made beneficial extensions to the horizontal vibration of piles in saturated soil by considering the variable cross-section and defects of the pile body. In addition, [PERSON] [16] studied the horizontal vibration of partially buried piles for offshore structures. In the latest reported study, [PERSON] [17] derived the dynamic impedance frequency domain analytical formula for horizontal vibrations of pile foundations in unsaturated soil using integral equation theory.