# Responses of Laterally Loaded Single Piles Subjected to Various Loading Rates in a Poroelastic Soil
## 2 Method and Materials
### Governing Equations and Boundary Conditions  
When an external load is applied to a porous medium, the volume fraction of the pores is affected. The fluid-filled pores experience a change in pressure under this mechanical stress and, in turn, it leads to fluid motion. As a reaction to the change in pore volume, the solid material repositions and deforms elastically. In short, the loading process involves at least two phenomena--fluid motion (hydraulic behavior) and a change in the material's effective stress (mechanical behavior). As these phenomena occur at the same time, they must be analyzed simultaneously; thus, a coupled hydromechanical analysis is required. In the case of, in particular, a sudden force applied to a poroelastic body, the displacement and fluid pressure with each representative elementary volume would adjust itself instantaneously to maintain a state of internal equilibrium [17]. Thus, the full combined formulation of soil-structure and fluid metaphysics may be represented by [PERSON]'s system of equations:\[G\
abla^{2}u+\frac{G}{1-2v}\bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac{ \partial^{2}v}{\partial x\partial y}+\frac{\partial^{2}w}{\partial x\partial z} \bigg{)} =F_{x}-\alpha_{B}\frac{\partial p}{\partial x}\] \[G\
abla^{2}v+\frac{G}{1-2v}\bigg{(}\frac{\partial^{2}u}{ \partial x\partial y}+\frac{\partial^{2}v}{\partial z^{2}y}+\frac{\partial^{ 2}w}{\partial x\partial z}\bigg{)} =F_{y}-\alpha_{B}\frac{\partial p}{\partial x} \tag{1}\] \[G\
abla^{2}w+\frac{G}{1-2v}\bigg{(}\frac{\partial^{2}u}{ \partial x\partial y}+\frac{\partial^{2}v}{\partial x\partial y}+\frac{ \partial^{2}w}{\partial^{2}z}\bigg{)} =F_{z}-\alpha_{B}\frac{\partial p}{\partial x}\]  
where \(G\) is the shear modulus, \(u,v,w\) is the displacement corresponding to the \(x,y,z\) direction, respectively, \(F_{x},F_{y},F_{z}\) is the body force in the respected direction, \(\alpha_{B}\) is the Biot-Willis coefficient, and \(p\) is the fluid or pore pressure. From the quasistatic linear poroelastic theory, the fluid flow through a deformable porous medium expresses the solid mechanics equilibrium and fluid mass conservation as functions of both the solid phase displacement field and pore fluid pressure as [17; 18; 19]  
\[\frac{1}{M}\frac{\partial p}{\partial t}+\
abla\bigg{[}-\frac{k}{\mu}\
abla \Big{(}p+\rho_{f}gz\Big{)}\bigg{]}=-\alpha_{B}\frac{\partial}{\partial t} \varepsilon_{\text{vol}} \tag{2}\]  
where \(\mu\) is the viscosity of the fluid, \(\
abla\) is the gradient operator, \(k\) is the intrinsic permeability, \(\rho_{f}\) is the fluid density, \(g\) is the acceleration of gravity, \(z\) is the elevation potential, \(\varepsilon_{\text{vol}}\) is the volumetric strain of the porous medium, and \(M\) is the Biot's modulus, which is given by  
\[\frac{1}{M}=\frac{\alpha_{B}-n}{K_{s}}+\frac{n}{K_{f}} \tag{3}\]  
where \(K_{s}\) is the solid bulk modulus, \(K_{f}\) is the fluid bulk modulus, and \(n\) is the porosity of the porous material. For an uncoupled analysis, the Biot-Willis coefficient is set to zero. To evaluate the deformation of the porous soil as a result of fluid motion, the solution of the above Biot's system of Equation (1) and the fluid mass flow Equation (2) were simultaneously computed using a commercial partial differential equations (PDE) solver COMSOL [20] through its built-in poroelasticity interface. The pile-soil interaction of a capped pile in a completely saturated soil is studied here using an integrated multiphysics model. The study pile was 0.4 m wide by 18.0 m long and the pile cap was 1.0 m \(\times\) 1.0 m \(\times\) 0.3 m thick. A schematic layout of the study problem is shown in Figure 1a. To solve the solid mechanics governing equations, sufficient boundary conditions must be provided: the left and right boundaries of the model were constrained by roller support, the bottom boundary was constrained by fixed pinned, while the top boundary was left free.