# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 1 Introduction  
The analytical solution of the dynamic response of surface foundation in a poroelastic medium under horizontal, vertical, and torsional vibration is widely studied by many researchers [1; 2; 3]. Generally, the foundation is embedded in a certain depth and the horizontal force from wind, earthquake ground motion, and mechanical vibrations that governs the dynamic response of embedded foundations. A great deal of research has been done in order to study the dynamic response of surface foundation under horizontal vibration. [PERSON] et al. [4] presented the impedance of foundation resting on an elastic soil medium. Numerical methods, that is, the finite element method and boundary element method were employed in order to investigate the dynamic response of embedded footings. [PERSON] et al. [5] presented an extensive investigation on the horizontal impedance of square footing in layered soil using a boundary element method. [PERSON] [1] investigated the dynamic response of suction caissons by 3D element models in the frequency domain. [PERSON] and [PERSON] [6] studied the dynamic response of rigid strip footing foundations using the indirect boundary integral equation method. The soil is treated as a single-phase linear elastic material in most of the existing studies. Generally, the majority of the foundations are embedded in two-phase soils consisting of a solidskeleton and pores full of groundwater, which is so-called poroelastic medium. [PERSON] [7; 8] first derived the wave propagation equations in poroelastic material. Then, many of the researchers studied the dynamic responses of a disk resting on or embedded in a poroelastic medium [9; 10; 11; 2; 12; 13; 14]. [PERSON] et al. [15] studied the dynamic response of embedded foundation resting on a uniform half-space under torsional vibration. [PERSON] and [PERSON] [4] presented an integration technique to calculate the dynamic response of embedded foundations subjected to external forces and moments. Different embedment ratios were also studied. [PERSON] and [PERSON] [16] presented the dynamic response of forced vertical vibrations of an elastic circular plate embedded in poroelastic soils. The dynamic response of an arbitrary buried load was also studied by virtue of Green's function by [PERSON] et al. [17]. Indirect boundary integral equation method was adopted to investigate the vertical vibration of the massless foundation embedded in a homogeneous poroelastic medium by [PERSON] et al. [18]. [PERSON] et al. [19] studied the vertical vibration of embedded massive foundation resting in a poroelastic medium using a simplified analytical method. [PERSON] and [PERSON] [20] also extended their research to the finite thickness of poroelastic soil. [PERSON] et al. [21] had done an excellent exploration on addressing the importance of seismic site response and soil-structure interaction of a tall building in Italy. [PERSON] and [PERSON] [22] further conducted the feasibility study of a loss-driven earthquake early warning and rapid response systems for tunnels in Italy high-speed railway network. [PERSON] et al. [23] investigated the effect of the rotational component of motion induced by the interaction between a pile group. An application highlighting the rotational component of input motion is important for tall structures on small pile groups. In conclusion, with the rapid development of underground structures in recent years, the dynamic response of embedded foundations is still a matter of great concern. Seismic vibration, ground shaking due to machine or high-speed train can easily induce damages to the infrastructure which may cause great economic losses. Soil-structure interaction has been an important issue in the assessment of the seismic vulnerability of a building. Considering the great risk of structures under seismic risk, a number of techniques have been developed to deal with this problem.