Horizontal Vibration of Pile in Transversely Isotropic Saturated Soil Based on Saturated Porous Medium Theory

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Introduction
Since the study of pile foundation vibration is critical to the safety and stability of pile foundation and whole building structure, the study of pile foundation dynamic characteristics has not stopped since the 1960s.Due to the significant influence of the properties of the soil around the pile on the dynamic interaction between the pile and soil, studying the dynamic characteristics of pile foundations under various complex geological conditions has significant engineering application value.Novak et al. [1,2], Nogami and Novak [3], Koo et al. [4], Ding et al. [5], Wu et al. [6], Luan et al. [7], Zhao et al. [8], Meng et al. [9] and others have conducted systematic theoretical research on the vibration of pile foundations in single-phase homogeneous elastic and viscoelastic soil.For coastal areas and river basins, the soil is usually saturated soil, and the influence of liquid phase needs to be considered.Therefore, Maeso et al. [10], Zhou and Wang [11], Zheng et al. [12], Cui et al. [13], and Wang and Ai [14] studied the dynamic problem of pile foundations in saturated soil.It should be noted that the current research on the dynamic interaction between saturated soil and pile is mostly based on Biot's saturated porous medium theory.Although Biot saturated porous medium theory has been successfully applied to many engineering fields, the research shows that its theoretical model has some defects [15].Beor's saturated porous medium theory uses the concepts of continuum mixture axiom and volume fraction, and several microscopic properties of porous medium can be directly described by macroscopic properties, avoiding the complicated formula of hybrid mixture theory.Without additional assumptions, some effects such as dynamic, material, and geometric nonlinearity can be easily reflected in its mathematical model.Therefore, Beor's saturated porous medium theory is used to describe the mechanical properties of the soil around the pile, and the pile-saturated soil dynamic interaction model is more reasonable and accurate.In addition, in the process of soil deposition, the orientation and directionality of the arrangement of flat medium particles lead to the difference in the properties (elastic modulus, shear modulus, and Poisson's ratio) of the soil in the vertical and horizontal directions, and the soil shows various characteristics.Usually, the vertical modulus of soil is smaller than the horizontal modulus, resulting in the horizontal modulus exhibiting isotropic characteristics.At this point, treating the soil as a transversely isotropic medium is more in line with engineering practice.Due to the complexity of the mathematical solution, it is difficult to obtain analytical expression for the dynamic interaction between transversely isotropic soil and pile.Therefore, the current research on the vibration of pile foundations in transversely isotropic soil mainly focuses on torsional vibration.For example, Zheng et al. [16] studied the torsional vibration of pipe pile in transversely isotropic saturated soil based on Biot's porous medium theory, and the influence of anisotropy of internal and external soil on the torsional dynamic response of pipe pile was discussed.Chen et al. [17] studied the dynamic response of pile in transversely isotropic saturated soil under transient torsional load by means of the Laplace transform.Ma et al. [18] studied the torsional vibration of pile foundations in transversely isotropic saturated soil considering construction disturbances.In terms of vertical and horizontal vibrations of pile foundations in transversely isotropic soil, Li and Ai [19] proposed a finite elementboundary element coupling method to study the dynamic response of end-bearing pile groups in layered transversely isotropic media under transient horizontal loads.Based on Boer's porous medium theory, Zhang et al. [20] gave a simplified analytical expression for the vertical dynamic impedance of pile groups in layered transversely isotropic saturated viscoelastic soil.Yan and Liu [21] gave the analytical solution expression of soil horizontal damping factor and horizontal dynamic impedance of a single pile in transversely isotropic soil by considering the influence of soil anisotropy and three-dimensional wave effect.Liu and Yan [22] regarded the soil around the pile as a single-phase transversely isotropic medium, studied the lateral vibration of pile groups in transversely isotropic soil using Novak's plane model, and obtained the analytical solution of the problem.In this paper, based on Boer's porous medium theory, the horizontal vibration of a single pile in transversely isotropic saturated soil is studied by means of mathematical physics, and the analytical expression of the horizontal dynamic impedance at the pile top will be obtained and will explore the influence of relevant parameters on the horizontal vibration of pile foundations.

Dynamic Control Equations of Transversely Isotropic Saturated Soil
The horizontal dynamic problem of a single pile in transversely isotropic saturated soil as shown in Figure 1 will be investigated, the radius of the pile is r 0 , the pile length of the pile is H, E p is the elastic modulus of the pile, and ρ p is the density of the pile.A horizontal harmonic load P t = P 0 e iωt acts on the pile top, where P 0 is the amplitude of the load, ω is the harmonic load frequency, and i is the virtual unit.In order to consider the influence of the liquid phase and anisotropy of soil around the pile, the soil around the pile is regarded as transversely isotropic saturated porous medium.Here, Boer's saturated porous medium theory and the constitutive equation of transversely isotropic elastic media proposed by Ding et al. [23] are used to build the motion equations of transversely isotropic saturated porous medium and the pile-soil dynamic interaction model.When ignoring the mass exchange and energy exchange between 2 Journal of Applied Mathematics liquid and solid phases, according to Boer's saturated porous medium theory, it can be known that the momentum equation of fluid solid mixture, pore fluid momentum equation, and volume fraction equation of saturated soil around the pile is [24,25] In the formula, σ S and σ F , respectively, represent the macroscopic stress tensors of the solid soil skeleton and the liquid phase, u S and u S represent the velocity and acceleration of the solid soil skeleton, u F and u F represent the velocity and acceleration of the liquid phase, ρ S and ρ F represent the volume density of the solid soil skeleton and the liquid phase, b S and b F represent the volume force per unit mass of the solid phase and liquid phase, and pS and pF represent the effective interaction force between the solid soil skeleton and the liquid phase and meet the requirement pS + pF = 0. Considering the incompressible condition, using the volume fraction theory, in the relationship between the stress tensors σ S and σ F and the interaction force p S , p F can be obtained as [26] σ S = −n S pI + σ SE , where σ SE is the effective stress tensor of the solid soil skeleton, pLE is the additional quantity, p is the effective pore water pressure of the incompressible fluid in saturated soil, S v is the coupling coefficient describing the coupling effect between the solid phase and liquid phase, and n S and n F are the volume fraction and satisfy n S + n F = 1.
In order to consider the anisotropy of saturated soil around the pile, the constitutive equation of transversely isotropic elastic medium proposed by Ding et al. [23] is used here to describe the stress and strain relationship of the solid skeleton of saturated soil around the pile, that is where σ SE rr , σ SE θθ , σ SE zz , σ SE θz , σ SE rz , and σ SE rθ are, respectively, the radial, circumferential, vertical, and shear effective stresses of the solid skeleton of transversely isotropic saturated soil; ε S rr , ε S θθ , ε S zz , ε S θz , ε S rz , and ε S rθ are, respectively, the radial strain, circumferential strain, vertical strain, and shear strain of the solid skeleton of transversely isotropic saturated soil; C 11 , C 12 , C 13 , C 33 , C 44 , and C 66 are the elastic constants of transversely isotropic saturated soil and meet C 66 = 1/2 C 11 − C 12 = G h , and G h and G v are, respectively, the shear modulus on the horizontal plane and vertical plane.
The strain-displacement relationship of transversely isotropic saturated soil is In which u S r , u S z , and u S θ are the radial, vertical, and circumferential displacements of the solid skeleton of transversely isotropic saturated soil.Ignoring the volumetric force (b S = b F = 0), the motion control equation of transversely isotropic saturated soil represented by displacement can be obtained from Eq. (1) to Eq. ( 4) as follows: where u F r , u F z , and u F θ are the radial, vertical, and circumferential displacements of the liquid phase of transversely isotropic saturated soil.Eq. ( 5)-Eq.( 10) are the motion control equations of transversely isotropic saturated soil represented by displacement.

Solution to Horizontal Vibration of Transversely Isotropic Saturated Soil
Here, we only study the horizontal vibration problem of pile foundation in transversely isotropic saturated soil under the action of the horizontal harmonic load P t = P 0 e iωt at the pile top (as shown in Figure 1).We ignored the influence of vertical displacement and only considered radial and circumferential displacements, and radial and circumferential displacements are independent of coordinate z.At this point, Eqs. ( 5), (6), and (10) are simplified as follows: Eqs. ( 8), (9), and ( 11) to ( 13) are the horizontal vibration control equations for transversely isotropic saturated soil.
The whole system makes a steady simple harmonic vibration under the action of horizontal harmonic load P t = P 0 e iωt at the pile top.Considering the harmonic nature of the problem, the form of each parameter is f = f e iωt and is substituted into the horizontal movement control equations of transversely isotropic saturated soil (Eq.( 8), Eq. ( 9), and Eqs. ( 11)-( 13)), and dimensionless quantities r and u S θ are the amplitudes of radial and circumferential displacement of the solid phase skeleton of the transversely isotropic saturated soil, and u F r and u F θ are the amplitudes of radial and circumferential displacement of the liquid phase fluid of the transversely isotropic saturated soil; v = C 66 /ρ S is the vertical shear wave velocity of soil.By performing dimensionless operations on the horizontal movement control equations of transversely isotropic saturated soil (Eq.( 8), Eq. ( 9), and Eq. ( 11)-Eq.( 13)), it can be obtained that In which, δ 1 = C 11 /C 66 is the anisotropic parameter that reflects the degree of anisotropy of the saturated soil.
To solve the horizontal vibration of transversely isotropic saturated soil layers, Eq. ( 14)-Eq.( 18) need to be decoupled, 4 Journal of Applied Mathematics and the following potential function is introduced for this purpose: 19 Here, φ S , ψ S and φ F , ψ F are the amplitude of the displacement potentials of the solid and liquid phases, respectively.
By decoupling, it can be obtained that In which, ∇ 2 = ∂ 2 /∂r 2 + 1/r ∂/∂r + 1/r 2 ∂ 2 /∂θ 2 and a 1 = iωs v − ω 2 /ρ .From Eq. ( 22) and Eq. ( 24), it can be concluded that In which, q 2 = ω 2 + iω 3 s v /ρa 1 .Using the method of separating variables to solve Bessel's Eq. ( 26) while considering that the displacement of saturated soil at infinity is zero, it can be obtained that where A 1 is the undetermined coefficient.From Eqs. ( 21), (23), and (25), it can be concluded that Similarly, using the separation of variable method to solve Bessel's equation (29) and considering that the displacement of saturated soil at infinity is zero, it can be obtained that where A 2 is the undetermined coefficient.By solving Eq. ( 30) and considering that the displacement of saturated soil at infinity is zero, it can be obtained that where A 3 is the undetermined coefficient; A 1 , A 2 , and A 3 can be determined by boundary conditions.Further consideration of Eqs. ( 21), (22), and (31) yields From Eqs. ( 19), ( 20), ( 27), ( 28), (31), and (32), it can be obtained that the radial and circumferential horizontal dynamic displacements of the solid skeleton of transversely isotropic saturated soil are

Solution to Horizontal Vibration of a Single Pile in Transversely Isotropic Saturated Soil
In order to solve the horizontal vibration of pile foundations in transversely isotropic saturated soil, the horizontal dynamic interaction between pile and soil is equivalent to spring and damper distributed around the pile; that is, the horizontal dynamic interaction between pile and soil is described using the Winkler spring-damper model.In order to determine the stiffness and damping coefficients of the Winkler spring-damper model, assuming that the dimensionless horizontal displacement of the pile body is 1, the stiffness and damping coefficients are determined by solving the horizontal force of the soil around the pile on the pile body.Assuming that the dimensionless horizontal displacement of the pile body is 1 and the pile-soil contact surface is 5 Journal of Applied Mathematics impermeable, the following boundary conditions can be obtained: In which, r 0 = r 0 /H, r 0 and H are the pile radius and pile length, respectively.
In the equations, A 1 , A 2 , and A 3 are the undetermined coefficients, which can be determined from Eq. ( 14)-Eq.(18).
In the equation, δ 2 = C 12 /C 66 .According to the radial stress (Eq.( 39)) and shear stress (Eq.( 39)) of the solid skeleton of the saturated soil, the dimensionless force per unit thickness of the soil layer on the pile body can be obtained, that is In the formula, the real and imaginary parts of F x are the stiffness and damping coefficients of the Winkler springdamper model, and Taking the micro element of the pile body as the research object, while considering the force (equation ( 41)) of the soil around the pile on the pile body, the dimensionless horizontal vibration equation of the pile foundation in transversely isotropic saturated soil can be obtained as follows: where u p z is the dimensionless horizontal displacement amplitude of the pile foundation and λ 4 = ρf x − ρ p r 2 0 ω 2 /ρE p r 4 0 , ρ p = ρ p /ρ F ,, E p = E p /C 66 , E p , and ρ p are the elastic modulus and density of the pile, respectively.
The displacement, rotation angle, shear force, and bending moment at the top of the pile are U 0 , θ 0 , Q 0 , and M 0 , respectively, using the initial parameter method [27], and the horizontal displacement of the pile body can be obtained from equation (43) as 6 Journal of Applied Mathematics
Taking an end-bearing pile as an example, due to the fixed bottom of the pile, the displacement and rotation angle at the pile bottom are both zero, and there are the following boundary conditions: Considering the definition of the horizontal dynamic impedance at the pile top, the required horizontal shear force is the horizontal dynamic impedance at the pile top for generating unit horizontal displacement when constraining the rotation angle at the pile top.Therefore, the horizontal dynamic impedance of the pile foundation in transversely isotropic saturated soil can be obtained as

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Journal of Applied Mathematics Here, K hh is the horizontal dynamic impedance at the pile top, f 1 is the horizontal dynamic impedance stiffness factor of the pile in transversely isotropic saturated soil, and f 2 is the horizontal dynamic impedance damping factor.Overall, as the frequency increases, the horizontal dynamic impedance stiffness factor gradually increases, while the horizontal dynamic impedance factor rapidly decreases at low frequencies and gradually stabilizes.The influence of anisotropic parameter δ 1 of transversely isotropic saturated soil on the horizontal vibration of a single pile is shown in Figure 2. The influence of anisotropic parameters δ 1 on the horizontal dynamic impedance of pile in transversely isotropic saturated soil is significant, whether it is the horizontal dynamic stiffness factor or the horizontal dynamic impedance factor, and the transversely anisotropic parameter δ 1 has a significant impact on them, and with the transverse anisotropy parameter δ 1 increases, its impact gradually decreases.It can be seen that in practical engineering, the influence of anisotropy on the horizontal vibration of pile foundations should not be ignored.The influence of the liquid-solid coupling coefficient s v of transversely isotropic saturated soil is shown in Figure 3.The influence of the   Journal of Applied Mathematics liquid-solid coupling coefficient s v on the horizontal dynamic impedance damping factor f 2 is greater than that of the horizontal dynamic impedance stiffness factor f 1 .The larger the liquid-solid coupling coefficient, the greater the horizontal dynamic impedance stiffness factor.The effect of the liquidsolid coupling coefficient on the horizontal dynamic impedance factor is related to frequency.At high frequencies, the larger the liquid-solid coupling coefficient, the smaller the horizontal dynamic impedance damping factor, while at low frequencies, the opposite is true.At high frequencies, the relative motion between the liquid and solid phases weakens, resulting in a decrease in damping.

Numerical Examples and Discussions
The effect of the diameter-length ratio r 0 /H of the pile on the horizontal vibration of the pile foundation in transversely isotropic saturated soil is shown in Figure 4. Like homogeneous soil, the diameter-length ratio of the pile has a significant impact on the vibration of the pile foundation.The smaller the diameter-length ratio of the pile, that is, the longer the pile, the smaller the stiffness factor and damping factor of the horizontal dynamic impedance.This is because the longer the pile, the easier it is to deform, the greater the displacement at the pile top, and the smaller the horizontal dynamic impedance at the pile top.At the same time, it can also be seen that when the pile length is longer, the impact of the pile length gradually decreases.
Since C 66 = G h , G h is the shear modulus on the water plane of transversely isotropic saturated soil, and the modulus ratio E p /C 66 is the ratio of the elastic modulus of the pile body to the shear modulus on the water plane of transversely isotropic saturated soil.From Figure 5, it can be seen that the modulus ratio E p /C 66 has a greater impact on the horizontal dynamic impedance of a single pile in transversely isotropic saturated soil, and its impact on the horizontal stiffness factor is much greater than that of the damping factor.The larger the modulus ratio E p /C 66 , the greater the horizontal dynamic impedance stiffness factor and damping factor.Because when the modulus ratio E p /C 66 is larger, the pile body stiffness is relatively larger, the deformation is smaller, and the displacement at the pile top is smaller, resulting in larger dynamic impedance at the pile top.

Conclusions
Considering the influence of the anisotropy and the liquid phase of the soil around the pile, the horizontal vibration of the transversely isotropic saturated soil layer is solved by means of mathematical and physics, and the analytical expression of the horizontal dynamic impedance of a single pile in transversely isotropic saturated soil is obtained.The numerical example analyzed the influence of the transversely isotropic saturated soil anisotropy parameter, liquid-solid coupling coefficient, etc. on the horizontal vibration of a single pile, and the main conclusions are as follows: (1) Anisotropic parameter δ 1 of soil has a significant impact on the horizontal dynamic impedance of a single pile in transversely isotropic saturated soil.In practical engineering, the influence of soil anisotropy on the horizontal vibration of pile foundations should not be ignored.(2) The influence of the liquid-solid coupling coefficient on the horizontal dynamic impedance damping factor f 2 is greater than that of stiffness factor f 1 .The influence of the liquid-solid coupling coefficient on the horizontal dynamic impedance factor is related to frequency to a certain extent.(3) The diameter-length ratio has a significant impact on the horizontal vibration of pile foundations in transversely isotropic saturated soil.The longer the pile, the smaller the horizontal dynamic impedance at the pile top.When the pile length is longer, the influence of the pile length gradually decreases.(4) When the stiffness of the pile body is large, the deformation of the pile body is small, and the horizontal dynamic impedance at the pile top is large.The modulus ratio E p / C 66 has a significant impact on the horizontal dynamic impedance stiffness factor of a single pile in transversely isotropic saturated soil, but its impact on the horizontal dynamic impedance damping factor is relatively small.
Curves of horizontal dynamic impedance stiffness factor varying with frequency Curves of horizontal dynamic impedance damping factor varying with frequency

Figure 2 :
Figure 2: Influence of anisotropic parameter δ 1 on horizontal dynamic impedance of a single pile in transversely isotropic saturated soil.
s v = 0.05 s v = 0.10 s v = 0.15 s v = 0.20(a) Curves of horizontal dynamic impedance stiffness factor varying with frequency f Curves of horizontal dynamic impedance damping factor varying with frequency

Figure 3 :
Figure 3: Influence of liquid-solid coupling coefficient s v on horizontal dynamic impedance of a single pile in transversely isotropic saturated soil.

Figure 4 :
Figure 4: Influence of diameter-length ratio r 0 /H on horizontal dynamic impedance of a single pile in transversely isotropic saturated soil.
E p /C 66 = 1000 E p /C 66 = 1500 E p /C 66 = 2000 E p /C 66 = 2500 (a) Curves of horizontal dynamic impedance stiffness factor varying with frequency C 66 = 1000 E p /C 66 = 1500 E p /C 66 = 2000 E p /C 66 = 2500 (b) Curves of horizontal dynamic impedance damping factor varying with frequency

Figure 5 :
Figure 5: Influence of modulus ratio E p /C 66 on horizontal dynamic impedance of a single pile in transversely isotropic saturated soil.