# Responses of Laterally Loaded Single Piles Subjected to Various Loading Rates in a Poroelastic Soil
## 1 Introduction  
Pile foundation, among many underground structures, is an embedded structure in a soil mass and is used to transfer and resist loads through its body and tip. In practice, piles may be subjected to uniaxial and combined loading systems that are tied up with a single or group system to share or move loads through its body and tip to the surrounding soil when used in conjunction with the reinforced pile cap on top of the pile. A wide range of studies have been reported on the impact of various loadings on single and group piles. Experimental, elastic theory, \(p-y\) curve, and finite element (FE) methods are some of the common methods used in tackling issues that are related to the problem of pile and pile cap subjected to various loading. Small-scale and full-scale lateral load experiments on single piles and pile groups with pile caps of different elevations from the ground surface, of varying sizes and thicknesses, and of different pile spacings in a pile group have been studied by [PERSON] and [PERSON] [1], [PERSON] et al. [2], [PERSON] and [PERSON] [3], etc. [PERSON] and [PERSON] [1] experimentally studied the lateral resistance of a pile cap subjected to different depths of cap embedment, pile group axial capacities, pile spacings, and material properties. [PERSON] et al. [2] experimentally studied the effect of pile cap elevation below the ground surface and pile spacing on lateral resistance of a single pile and pile groups driven in sand and concluded that the lateral carrying capacity of single pile and pile groups increases as the pile cap embedment depth below the ground surface increases, and as the spacing between piles in the group increases. Soil hardness and strength adjacent to the pile cap have been found to affect the pile cap's lateral resistance and overall contribution. Consistent results show thatincreasing pile cap elevation, pile cap size, and pile cap thickness have been found to have significant impact on lateral resistance and pile cap deflection. [PERSON] and [PERSON] [4] and [PERSON] [5] were perhaps the first few to use simple analytical formulae to calculate the deflection, rotation, moment, and bearing capacity of laterally loaded piles. [PERSON] et al. [6] used basic analytical formulae to evaluate the responses of laterally loaded piles. [PERSON] and [PERSON] [7] proposed an analytical model based on energy conservation of the pile-soil system to evaluate the deflection of laterally loaded piles. [PERSON] et al. [8] assessed the static behavior of vertical piles using a nonlinear analysis based on the subgrade reaction model with elastic, perfectly plastic soils for laterally loaded piles embedded in multiple-layered soil systems. Their study indicated that the behavior of a laterally loaded pile is controlled by the stiffness of the soil at shallow depth. Assuming the soil skeleton deforms elastically and under plane strain condition, [PERSON] and [PERSON] [9] obtained a closed-form analytical solution for the consolidation of soil around a laterally loaded pile. They idealized their problem as a circular rigid disk surrounded by a deformable soil zone and assumed the drainage to occur at the boundary of the assumed zone of deformable soil around the pile. The excess pore water pressure response around the pile was then obtained using an uncoupled consolidation analysis. Although the study considered the generation of excess pore-water pressure for laterally loaded piles, the result is limited to the elastic soil; however, when comparing to the result of three-dimensional (3D) analysis, the two-dimensional (2D) analysis can still reasonably predict the consolidation process along the shaft of the pile. [PERSON] and [PERSON] [10] studied the response of laterally loaded single piles embedded in cohesionless soils using the boundary element method and taking into consideration soil yielding along the pile-soil interface and linearly increasing soil stiffness with depth.