# Impedance of Inner Saturated Soil to Horizontally Vibrating Large-Diameter Pipe Piles

Authors: Jiang Xie
Journal: Shock and Vibration
 (2021)

---

## Abstract

An analytical solution for the inner soil impedance of saturated soil to a horizontally vibrating large-radius pipe pile was presented. Based on the porous media theory and the assumption that the vertical normal stress is zero, the closed solution of the inner soil impedance of the saturated soil to the movement of the large-diameter pipe pile is obtained. This analytical solution considers the influence of saturated soil parameters on the impedance of the core soil of large-diameter pipe piles. Through numerical examples, the variation law of the inner soil impedance with pile radius, pile length, dimensionless frequency, compression coefficient, effective permeability coefficient, and porosity was analyzed and the pile radius corresponding to effective inner soil impedance is determined.


---

## Body

## 1. Introduction

Pile foundations have been used to transfer structural loads to soil layers, and the soil layers provide reaction force to the piles through the occurrence of small deformations. Pile foundations are often subjected to lateral dynamic loads caused by various factors, such as earthquake, mechanical, wind, and wave loads. When the large-diameter pipe pile is subjected to a horizontal load, the load will be transferred to the pile side and pile inner soil. The pile side and pile inner soil provide the reaction force to the large-diameter pipe pile to bear the horizontal external load.Many mathematical models describing the horizontal impedance have been established. The most commonly used and simplest model is Winkler model, which simplifies the impedance of soil to pile into continuous distributed and independent springs [1–3]. Novak et al. [4] assumed that the vertical strain and vertical displacement were zero and put forward the model of soil impedance at the pile side. Nogami et al. [5] provided a calculation model for pile side soil impedance, which takes into account the vertical change of soil stress gradient. Saitoh and Watanabe [6] studied in detail the influence of flexible sidewalls on the roll impedance of embedded foundations. Anoyatis et al. [7] proposed an analytical expression of the soil reaction force on the side of the pile during dynamic loading of a laterally loaded pile. This analytical solution assumes that the vertical normal stress is zero. All the above are the calculation models of soil impedance at the pile side. Zheng et al. [8, 9] give the calculation models of the inner soil impedance of the pile under vertical and horizontal loads, respectively.Because soil is a multimedium composed of solid and liquid, the above research is based on the assumption that the soil is a single-phase medium, which does not conform to the actual situation. In order to analyze the influence of the inertia, viscosity characteristics of saturated soil, etc. on the bearing characteristics of the soil, Biot [10, 11] established a calculation model using Lagrangian equations. Based on the calculation model of Biot’s theory, Yu et al. [12] and Zheng et al. [13] studied the dynamic characteristics of end-bearing piles and pipe piles embedded in saturated soil under horizontal vibration. Gu et al. [14] studied the composite foundation treatment of pile raft and soft soil foundation. Jiang et al. [15] summarize the research on the axial resistance of rock-socketed piles. In the research of saturated soil and piles, Cao gives an analytical solution for the resistance of saturated soil on the side of the pile based on the normal stress assumption [16].This work addresses the problem of the inner soil impedance of saturated soil to a horizontally vibrating large-diameter pipe pile. The closed solution of the internal soil impedance for the saturated soil to the movement of the large-diameter pipe pile is obtained.


## 2. General Consideration

### 2.1. Proposed Model

The considered internal soil pipe pile system is represented by cylindrical coordinates, as shown in Figure1. It is assumed that the pile-core soil is linear, viscoelastic, isotropic, and homogeneous and is located on a rigid foundation. The pile length is denoted by H, and the inner radius is denoted by R0. There is a harmonic horizontal force Feiωt at the pile head, where t stands for the time variable and F stands for the amplitude of the load.Figure 1
Calculation model of soil impedance in the large-diameter pipe pile.


### 2.2. Basic Equation of the Inner Saturated Soil

In cylindrical coordinates, the geometric relations of the inner saturated soil of the pipe pile are(1)εR=∂uR∂R,εθ=uRR+1R∂uθ∂θ,εz=∂uz∂z,εz=∂uz∂z,γRθ=1R∂uR∂θ+∂uθ∂R−uθR,γRz=∂uR∂z+∂uz∂R,γθz=∂uθ∂z+1R∂uz∂θ.According to Biot’s theory [12], the motion equations of the inner saturated soil of the large-radius pipe pile are(2)∂σR∂R+∂τRz∂z+1R∂τRθ∂θ+σR−σθR=ρsu¨R+ρfw¨R,(3)∂τRθ∂R+∂τθz∂z+1R∂σθ∂θ+2RτRθ=ρsu¨θ+ρfw¨θ,(4)−∂pf∂R=ρfu¨R+m1w¨R+ρfgkdw˙R,(5)−∂pfR∂θ=ρfu¨θ+m1w¨θ+ρfgkdw˙θ.The constitutive relation of saturated soil inside the large-radius pipe pile can be expressed by the following formula:(6)σij=2μεij+λe−αpfδij,(7)pf=Mζ−αe.uR represents the soil radial displacement, uθ represents the circumferential displacement, and uz represents the displacement. wR is liquid radial displacement d, wθ represents the circumferential displacement, and wz represents the liquid vertical displacement. σR,σθ, and σz are the total stress of the radial, circumferential, and vertical measurements of the soil. us=Gs1+2iζ, λ=2μSνS/1−2νS, e=∂uR/∂R+uR/R+1/R∂uθ/∂θ+∂uz/∂z, α, and M are Biot’s parameters accounting for the compressibility of the two-phase material. ps is the solid density and pf is the liquid density. pf represents the pore pressure.


## 2.1. Proposed Model

The considered internal soil pipe pile system is represented by cylindrical coordinates, as shown in Figure1. It is assumed that the pile-core soil is linear, viscoelastic, isotropic, and homogeneous and is located on a rigid foundation. The pile length is denoted by H, and the inner radius is denoted by R0. There is a harmonic horizontal force Feiωt at the pile head, where t stands for the time variable and F stands for the amplitude of the load.Figure 1
Calculation model of soil impedance in the large-diameter pipe pile.


## 2.2. Basic Equation of the Inner Saturated Soil

In cylindrical coordinates, the geometric relations of the inner saturated soil of the pipe pile are(1)εR=∂uR∂R,εθ=uRR+1R∂uθ∂θ,εz=∂uz∂z,εz=∂uz∂z,γRθ=1R∂uR∂θ+∂uθ∂R−uθR,γRz=∂uR∂z+∂uz∂R,γθz=∂uθ∂z+1R∂uz∂θ.According to Biot’s theory [12], the motion equations of the inner saturated soil of the large-radius pipe pile are(2)∂σR∂R+∂τRz∂z+1R∂τRθ∂θ+σR−σθR=ρsu¨R+ρfw¨R,(3)∂τRθ∂R+∂τθz∂z+1R∂σθ∂θ+2RτRθ=ρsu¨θ+ρfw¨θ,(4)−∂pf∂R=ρfu¨R+m1w¨R+ρfgkdw˙R,(5)−∂pfR∂θ=ρfu¨θ+m1w¨θ+ρfgkdw˙θ.The constitutive relation of saturated soil inside the large-radius pipe pile can be expressed by the following formula:(6)σij=2μεij+λe−αpfδij,(7)pf=Mζ−αe.uR represents the soil radial displacement, uθ represents the circumferential displacement, and uz represents the displacement. wR is liquid radial displacement d, wθ represents the circumferential displacement, and wz represents the liquid vertical displacement. σR,σθ, and σz are the total stress of the radial, circumferential, and vertical measurements of the soil. us=Gs1+2iζ, λ=2μSνS/1−2νS, e=∂uR/∂R+uR/R+1/R∂uθ/∂θ+∂uz/∂z, α, and M are Biot’s parameters accounting for the compressibility of the two-phase material. ps is the solid density and pf is the liquid density. pf represents the pore pressure.


## 3. Model Development

Assume thatσz = 0, and the following expression can be obtained with equation (6):(8)σr=μsησ2εr+εθ−2εθ−α1−2νs1−νspf,σθ=μsησ2εr+εθ−2εr−α1−2νs1−νspf,where ησ=2/1−νs.Considering the harmonic soil response of the formuR=uRR,θ,z,ωeiωt, uθ=uθR,θ,z,ωeiωt, wR=wRR,θ,z,ωeiωt, and wθ=wθR,θ,z,ωeiωt, substituting equations (1), (6), and (8) into equations (2) and (3) yields(9)ηs2∂∂R1R∂RuR∂R+∂uθ∂θ−1R2∂∂θ∂Ruθ∂R−∂ur∂θ+∂2uR∂z2+κ1ωVs2uR−κ2∂pf∂R=0,(10)∂∂r1R∂Ruθ∂R−∂uR∂θ+ηs21R2∂∂θ∂RuR∂R+∂uθ∂θ+∂2uθ∂z2+κ1ωVs2uθ−κ2∂pfR∂θ=0,where κ1=1+ω2ρf2/ρsϑ, κ2=α/2μs1−2νs/1−νs+ρfω2/μsϑ, ηs2=2−νs/1−νs, and Vs=μs/ρs.In addition, assuming that the vertical displacement of the fluid is zero, the following expression can be obtained from equation (7):(11)ω2ρfϑ−α1−2νs1−νs∂uR∂R+uRR+1R∂uθ∂θ=1ϑ∇2pf+α2λνs1−νs+1Mpf.The following potential functionsϕ and ψ are introduced:(12a)ur=∂Φ∂R+1R∂Ψ∂θ,(12b)uθ=1R∂Φ∂θ−∂Ψ∂R.Substituting equations (12a) and (12b) into equations (9), (10), and (11) yields(13)ηs2∇2Φ+∂2Φ∂z2+κ1ωVs2Φ=κ2pf,(14)∇2Ψ+∂2Ψ∂z2+κ1ωVs2Ψ=0,(15)∇2pf+ϑα2λνs1−νs+ϑMpf=ω2ρf−αϑ1−2νs1−νs∇2Φ,where ∇2=∂2/∂R2+1/R∂/∂R+1/R∂2/∂θ2. Substituting equation (13) into equation (15) yields(16)∇4Φ+κ1ηs2ωVs2−κ2ω2ρfηs2+κ2αϑηs21−2νs1−νs+α2ϑλνs1−νs+ϑM∇2Φ+ϑα2ληs2νs1−νs+ϑMηs2+1ηs2∇2∂2Φ∂z2+κ1ωVs2ϑα2ληs2νs1−νs+ϑMηs2Φ=0.The general solutions of equations (14) and (17) can be obtained using the variable separation method:(17)Φ11=A11Inζ11R+B11Knζ11RC11sinnθ+D11cosnθE1cosaz+F1sinaz,(18)Φ12=A12Inζ12R+B12Knζ12RC12sinnθ+D12cosnθE1cosaz+F1sinaz,(19)Ψ=A2Inζ2R+B2Knζ2RC11sinnθ+D2cosnθE2cosaz+F2sinaz,where Φ=Φ11+Φ12, d11=κ1/ηs2ω/Vs2+ϑ/M−a2/ηs2+κ2αϑ/ηs21−2νs/1−νs−κ2ω2ρf/ηs2+ϑα2/λνs/1−νs, d12=ϑ/ηs2κ1ω/Vs2α2/λνs/1−νs+1/M−a2α2/λνs/1−νs−a2/M, ζ112=−d11+d112−4d12/2, ζ122=−d11+d112−4d12/2, and ζ2=a2−κ1ω/Vs2, with n being the positive real number and In () and Kn () being the modified Bessel functions of the n-th order of the first and second kind, respectively.To ensure a bounded response atR = 0, the constants B11, B12, and B2 associated with the modified Bessel functions Kn () must disappear. Considering the direction of pile loading to be along θ = 0, the radial and tangential displacement components must satisfy the condition of ur|θ=π/2=uθ|θ=0=0. Thus, C11 = C12 = D2 = 0 and n = 1. The conditions of no soil displacement at the bottom of the soil layer and no stress on the soil surface are enforced, E1=E2=0 and cosaH=0, yielding(20)am=2m−1π2H,m=1,2,3,.Hence,(21)Φm=cosθB11mI1ζ11mR+B12mI1ζ12mRsinamz,(22)Ψm=B2msinθI1ζ2mRsinamz.Substituting equations (21) and (22) into (12) yields(23)ur=cosθB11mζ11mI0ζ11mR−I1ζ11mRR+B12mζ12mI0ζ12mR−I1ζ12mRR+I1ζ2mRRB2msinamz,(24)uθ=−sinθB11mI1ζ11mRR+B12mI1ζ12mRR+B2mζ12mI0ζ2mR−I1ζ2mRRsinamz.The inner wall of pipe pile-soil interface (R=R0) is assumed to be fully permeable, thus giving pf|R=R0=0. It can be obtained by equation (13) and equation (21):(25)B12m=−κ1ω/Vs2−a2m+ηs2ζ211mκ1ω/Vs2−a2m+ηs2ζ212mI1ζ11mRI0ζ12mRB11m.Assuming that the soil is in complete contact with the pile, we can get the following expression:(26)Wz,ω=∑m=0∞Wmωcosamz.According to equations (23)–(26), we get(27)B11m=R0Wδ12+δ22δ12δ21−δ11δ22,(28)B12m=R0Wδ12+δ22δ12δ21−δ11δ22δ0,(29)B2m=R0Wδ11+δ21δ12δ21−δ11δ22,where δ0=−κ1ω/Vs2−am2+ηs2ζ11m2/κ1ω/Vs2−am2+ηs2ζ12m2I1s11m/I1s12m, δ11=s11mI0s11m−I1s11m+δ0s12mI0s12m−I1s12m, δ12=I1s2m, δ21=I1s11m+δ0I1s12m, δ22=s2mI0s2m−I1s2m, s11m=R0ζ11m, s12m=R0ζ12m, and s2m=R0ζ2m.At the soil-inner wall of the pipe pile interface, the impedance amplitude of saturated soil inside the pilepz,w resulting from the pile motion is expressed as [2, 5](30)pz,w=−∫02πσRR0cosθ−τRθR0sinθR0dθ,where σRR0=σRR0,θ,z,ω and τRθR0=τRθR0,θ,z,ω are the radial normal and shear stresses, respectively. Substituting equations (1), (17), (23), (24), and (27)–(29) into (30), we can obtain the impedance amplitude as follows:(31)pz,w=πμ∑i=0nRmωUmω=πμ∑i=0nRmωWmωsinamz,where Rm (ω) is the inner soil impedance factor associated with the m-th soil mode. The expression for Rm (ω) is shown as follows:(32)Rmω=−δ12+δ22δ51−δ54+δ52−δ55+δ53δ11+δ21δ12δ21−δ11δ22,where δ51=ησ2s211mI1s11m, δ52=ησ2s212mI1s12m, δ53=s22mI1s2m,δ54=R02α/μs1−2νs/1−νsk1/k2ω/Vs+ηs2ζ211m/k2−am2/κ2I1s11m, and δ54=R02α/μs1−2νs/1−νsk1/k2ω/Vs2+ηs2ζ212m/k2−am2/κ2I0s12m.


## 4. Results and Discussion

In this section, in order to verify the correctness of the model proposed in this paper and analyze the influence of saturated soil parameters on the impedance of pile-core saturated soil, the selected material parameters includeH = 50, a1 = 0.5, α = 0.9, ρs = 2200 kg/m3, ρf = 1000 kg/m3, Gs = 2.5 MPa, ξ = 0.01, vs = 0.4, kd = 1 × 10–8 m/s, Kf = 33 MPa, Ks = 370 MPa, and a1 = r1ω/Vs. The impedance of pile-core saturated soil varies with saturated soil parameters as shown in Figures 2– 5.Figure 2
Variation in the inner soil impedance with the inner radius with different porosity ranges.
(a)(b)Figure 3
Inner soil impedance coefficient changes with the inner diameter for different effective permeability coefficients.
(a)(b)Figure 4
Variation of the inner soil impedance coefficient with the inner diameter under different compressibility rates.
(a)(b)Figure 5
Variation of inner soil impedance with the inner radius under different dimensionless frequencies.
(a)(b)Some common trends are observed in Figures2–5. From Figure 2, it can be seen that when the inner radius of the pipe pile is less than 0.5 m, the real part of the inner soil impedance factor increases rapidly with the increase in the inner radius of the pipe pile. When the inner radius of the pipe pile is greater than 0.5 m and less than 1.5 m, the inner soil impedance factor increases slowly with the increase in the inner radius of the pipe pile. When the inner radius of the pipe pile is greater than 1.5 m, a tends to be stable. Similarly, in the same inner radius range as the real part of the inner soil impedance factor, the imaginary part of the inner soil impedance factor decreases with increase in the pile inner radius.Under different porosity rangesϕ, the stiffness and damping changes of the inner soil dynamic response are as shown in Figure 2. As the porosity of saturated soil increases, the stiffness of the internal soil decreases. As the porosity of saturated soil increases, the damping of saturated soil increases.With the change in the inner diameter of the pipe pile, the effect of the effective permeability coefficient on the impedance of the inner soil will be as shown in Figure3. With the increase in the effective permeability coefficient of the saturated soil, the stiffness of the inner soil decreases. With the increase in the effective permeability coefficient of saturated soil, the imaginary part of the impedance coefficient increases.Figure4 shows the influence of the compression coefficient on the impedance factor in the soil as the inner radius of the pipe pile changes. As the effective permeability coefficient of saturated soil increases, the stiffness of the internal soil also increases. As the effective permeability coefficient of saturated soil increases, the imaginary part of the impedance coefficient decreases.In the low-frequency range, Figure5 shows the effect of the dimensionless frequency on the inner soil impedance factor. When the effective permeability of saturated soil increases, the damping of saturated soil decreases. When the pile inner radius is less than 2.2 m, the inner soil impedance factor increases with the increase in dimensionless frequency. When the pile inner radius is greater than 2.2 m, the inner soil impedance factor decreases with the increase in dimensionless frequency.Figure6 shows the effect of the pile length on the inner soil impedance factor. With the increase in the pile length, the stiffness of the inner soil decreases. With the increase in the pile length, the imaginary part of the impedance coefficient increases.Figure 6
Variation of the internal soil impedance coefficient with the inner radius under different pile lengths.
(a)(b)


## 5. Conclusions

This paper presented an innovative solution to study the impedance of inner soil to a horizontally vibrating pipe pile. Some interesting conclusions arise from the application of the solution to typical problems:(1)
The effective inner soil impedance corresponding to the pile radius is 0.5 m, and with the increase in the pile radius, the stiffness of the inner soil increases. With the increase in the pile length, the damping of the saturated soil decreases. When the inner radius of the pipe pile is greater than 1.5 m,a tends to be stable.(2)
The influence of porosity and effective permeability coefficient on the inner soil impedance factor is analyzed. With the increase in the porosity and effective permeability coefficient of the saturated soil, the stiffness of the inner soil decreases and the damping of the saturated soil increases.(3)
The influence of pile length on the inner soil impedance factor is analyzed. With the increase in the pile length, the stiffness of the inner soil decreases. With the increase in the pile length, the damping of the saturated soil increases.



---

*Source: 1845109-2021-09-22.xml*