Dynamic Interaction Factor of Pipe Group Piles Considering the Scattering Effect of Passive Piles

Abstract

Based on the plane–strain assumption, a calculation model of pile–soil–pile vertical coupling vibration response considering the scattering effect of passive piles is established in this paper. Using this model, the vertical displacement expressions of pile core soil and pile surrounding soil, soil displacement attenuation function, and longitudinal complex impedance are obtained. Then, based on the strict pile–soil coupling effect, the displacement of the active pile under vertical load and scattering effect, as well as the displacement of the passive pile under incident waves, are solved separately. A new type of dynamic interaction factor for pipe group piles is derived by introducing scattering effect factors. A numerical example shows that the degenerate solution in this paper is in good agreement with the existing solution, which verifies the rationality of the solution. Considering the scattering effect is helpful in improving the accuracy of vibration response analysis of pile groups. The variation in parameters such as slenderness ratio, pile spacing, and outer diameter has significant effects on the interaction factor, and compared with the solid pile, the influence of parameter change on the pipe pile is smaller.

1. Introduction

In recent years, with the gradual extension of human activities into the ocean, the construction demands of marine engineering have increased sharply. In order to adapt to the complex marine environment, these structures generally use pile groups as their foundation type. Under the action of dynamic loads such as wind and sea waves, the vibration of pile groups’ foundation needs to consider multidimensional factors. It is necessary to consider the pile–soil interaction and the mechanical response analysis of pile–pile coupling, which has always been a hot issue in the fields of soil dynamics, geotechnical engineering, and structure foundation interaction [1,2,3].

Poulos [4] first fully studied pile–soil interaction and proposed the superposition principle, initially proposed the pile–pile dynamic response interaction factor to reflect the pile foundation interaction, and considered the influence of parameter changes such as Poisson’s ratio on the settlement behavior of pile groups. Kaynia et al. [5] solved the dynamic equation of pile groups using the boundary integral method and continuous medium Green’s function and obtained relevant solutions. Mylonakis et al. [6] preliminarily analyzed the vibration impact of the source pile on the receiving pile based on Randolph and Wroth’s shear displacement model and proved that the impact of pile vibration cannot be ignored. On this basis, Mylonakis et al. [7] deduced the pile–pile dynamic interaction factor in uniform geology using the Winkler foundation model and analyzed the influence of soil layer and impedance on the interaction factor. Poulos et al. [8] proposed a double pile interaction coefficient and analyzed the relationship between settlement behavior and interaction coefficient. In addition, Nogami [9], Mylonakis [10], Chen [11], and others carried out pile group vibration research using the Winkler foundation model, taking into account the incident wave generated by the active pile vibration. Bahaa [12], Randolph [13], and others considered various effects of piles and proposed new interaction factors to better reflect the interaction between piles. Mylonakis [10], Wang [14], Wang [15], Zhang [16], and others reasonably layered the foundation, solved the interaction factor of solid pile type in layered soil, and fully analyzed the influence of parameter changes on the interaction factor. Chen et al. [17] and Cao et al. [18] broke down the problem into an extended soil mass and two fictitious piles characterized, respectively, by the Young’s modulus of the soil and that of the difference between the pile and soil and solved the interaction coefficient of double piles in half space based on the model of Muki and Sternberg. Results confirmed the validity of the proposed approach and portrayed the influence of the governing parameters on the pile interaction. Luan et al. [19] considered the infrasound waves generated by passive piles, solved the vibration displacement of each part of the solid pile, and proposed an interaction factor.

Due to the fact that the core soil cannot be ignored in the calculation and analysis of pipe piles, there will inevitably be differences in the solution analysis of the pile group model mentioned above. Liu [20] extended the traditional interaction factor to the type of pipe piles on the basis of the Winkler foundation model, conducted appropriate parameter analysis, and concluded that parameter changes had a greater impact on the interaction factor. Yan [21] and Liu [22] solved the dynamic interaction factor of pipe piles in saturated soil on the basis of the previous work and conducted appropriate parameter change analysis.

In the complex and changeable working environment at sea, small-spacing pile groups are generally used in marine infrastructure construction. When the pile spacing is less than eight times the diameter of pile foundation, the interaction between pile foundation and pile foundation needs to be considered. Generally, the pile foundation with a vibration wave generated by a load is called the active pile, and the pile foundation nearby affected by the vibration wave is called the passive pile. The above study only considers the vibration impact of the active pile on the passive pile, ignoring that the vibration of the passive pile in turn will cause the secondary vibration of the active pile, which will cause large errors. At this time, it is more appropriate to consider the scattering effect of passive piles [19]. Pipe piles have been widely studied and used in engineering in recent years [23,24,25,26], but there is a lack of comparative analysis of pipe pile types and solid pile types in the current pile group vibration research. Therefore, based on the continuum theory and the superposition principle [5,27], this paper extends the solid pile type to the pipe pile type and considers the scattering wave generated by the passive pile vibration, establishes the pile groups calculation model, introduces the scattering effect factor, solves the new interaction factor, and explores the difference in dynamic response characteristics of the pipe pile and the solid pile when considering the scattering effect of passive piles. Based on the assumption of small deformation, this paper only considers the vertical deformation of pipe piles and soil and ignores the lateral deformation. In the subsequent research work, the lateral deformation will be considered for detailed theoretical improvement.

2. Physical Model Considering the Passive Piles Scattering Effect

2.1. Pile Groups Calculation Model and Basic Assumptions

Figure 1 is the calculation model of pile groups under vertical dynamic load. As shown in the figure, the pipe pile parameter symbol H is the pile foundation length, $r_0^{}$ is the inner diameter, $r_1^{}$ is the outer diameter, $A_{\mathrm{p}}^{}$ is the cross-sectional area of the pile body, S is the pile spacing, ${\rho }_{\mathrm{p}}^{}$ is the pile body density, and $E_{\mathrm{p}}^{}$ is the elastic modulus. The soil parameter symbol $G_{s0}\left(G_{s1}\right)$ is the shear modulus of soil, ${\rho }_{s0}\left({\rho }_{s1}\right)$ is the damping ratio, and ${\delta }_{s0}^{}({\delta }_{s1}^{})$ is the density. The relevant parameters of pile core soil and pile surrounding soil are denoted by subscripts s0 and s1. The vertical harmonic load $P_G^{}{\mathrm{e}}_{}^{i\omega t}$ acts on the top of the pile foundation, where $\omega$ is the excitation circle frequency, $\mathrm{i}=\sqrt{-1}$. The mathematical model established in this paper conforms to the following assumptions:

(1) At the pile–soil interface, the stress and displacement are continuous without relative sliding [28,29];

(2) The pile core soil and the soil around the pile are regarded as homogeneous viscoelastic bodies without anisotropy;

(3) The pile group foundation is placed on rigid bedrock;

(4) The soil around the pile is divided into infinite thin layers without interaction between layers.

Figure 1. Pile group calculation model.

Figure 1. Pile group calculation model.

2.2. Pile Group Model Considering Scattering Effect

Most previous studies only considered the influence of active piles of the solid pile type on passive piles, ignoring the scattered waves generated by passive piles and the dynamic response of the pipe pile type. In fact, the scattered waves generated by passive piles change the original soil stress field, and the influence degree of the pile group’s interaction with pipe pile type is also different; its mechanism is shown in Figure 2. Under the action of vertical dynamic load $P_1$, the active pile generates vertical displacement $W_{1-1}^P$ and incident waves in the soil. When the incident wave propagates S distance in the soil around the pile, the soil displacement is $W_{2-1}^{s1}$, which causes the passive pile to produce displacement $W_{2-1}^P$ and generates scattered waves in the soil. The scattered wave propagates to the position of the active pile, causing its vibration to produce vertical displacement $W_{1-2}^P$. Therefore, the mechanism of pile–pile dynamic interaction considering the scattering effect of passive piles in this paper is:

(1) The derivation of the displacement of the active pile under vertical dynamic load via the analytical method $W_{1-1}^P$.

(2) The active pile vibration produces an incident wave that propagates radially along the soil around the pile, and its displacement can be expressed as:

$$W_{2-1}^{s1}=\mu \left(S\right)W_{1-1}^P$$

(1)

where $\mu \left(S\right)$ is the displacement attenuation function,

$$\mu \left(S\right)=\frac{W_{}^{s1}\left(\mathrm{S}\right)}{W_{}^{s1}\left(r_1\right)}$$

(2)

(3) The incident wave propagates to the passive pile, causing vertical vibration and displacement and generating a scattered wave. The scattered wave propagates to the active pile, causing its secondary vibration.

Figure 2. Action mechanism.

Figure 2. Action mechanism.

3. Solution of Governing Equations and Physical Models

3.1. Soil Impedance

Based on the plane strain assumption, the soil governing equation established in this section conforms to the following assumptions:

(1) It neglects the radial displacement of pile core soil and pile surrounding soil;

(2) It ignores the vertical stress gradient of pile core soil and pile-surrounding soil.

Therefore, the governing equation of soil under vertical dynamic load is:

$$\frac{d^2{W}_{}^{s0}}{d{r}^{}}+\frac{1}{r}\frac{d{W}_{}^{s0}}{dr}+\frac{{\omega }^2{\rho }_{s0}^{}}{G_{s0}^{*}}{W}_{}^{s0}=0$$

(3)

$$\frac{d^2{W}_{}^{s1}}{d{r}^{}}+\frac{1}{r}\frac{d{W}_{}^{s1}}{dr}+\frac{{\omega }^2{\rho }_{s1}^{}}{G_{s1}^{*}}{W}_{}^{s1}=0$$

(4)

where $W_{}^{s0}$ is the vertical displacement of the pile core soil and $W_{}^{s1}$ is the vertical displacement of the soil around the pile. ${\rho }_{s0}^{}$ is the density of soil in the pile core and ${\rho }_{s1}^{}$ is the density of soil around the pile. $G_{s0}^{*}=G_{s0}^{}(1+2i{\delta }_{s0}^{})$, $G_{s1}^{*}=G_{s1}^{}(1+2i{\delta }_{s1}^{})$ is the complex shear modulus of soil mass.

The solutions of Equations (3) and (4) are obtained as follows:

$$W_{}^{s0}=A_0{K}_0^{}(q_0^{}r)+B_0{I}_0^{}(q_0^{}r)$$

(5)

$$W_{}^{s1}=A_1{K}_0^{}(q_1^{}r)+B_1{I}_0^{}(q_1^{}r)$$

(6)

where $q_0^{}=\sqrt{\frac{-{\rho }_{s0}{\omega }^2}{G_{s0}^{*}}}$, $q_1^{}=\sqrt{\frac{-{\rho }_{s1}{\omega }^2}{G_{s1}^{*}}}$. I₀( ) and K₀( ) are modified Bessel functions of type I and type II, respectively, and $A_0$, $B_0$, $A_1$, and $B_1$ are the coefficients to be determined.

Consider the boundary conditions at $r=0$ and infinity and the displacement continuity conditions of the pile–soil interface:

$$W_{}^{s0}\left|{}_{r=0}=\mathrm{Finite} \mathrm{value}\right., W_{}^{s0}\left|{}_{r=r_0^{}}=W_{1-1}^P\right.$$

(7)

$$W_{}^{s1}\left|{}_{r\to \infty }=0\right., W_{}^{s1}\left|{}_{r\to r_1^{}}=W_{1-1}^P\right.$$

(8)

The solution is as follows: $A_0=0$, $B_0=\frac{W_{1-1}^P}{I_0^{}(q_0^{}{r}_0^{})}$, $A_1=\frac{W_{1-1}^P}{K_0^{}(q_1^{}{r}_1)}$, $B_1=0$, namely:

$$W_{}^{s0}=\frac{W_{1-1}^P}{I_0^{}(q_0^{}{r}_0^{})}{I}_0^{}(q_0^{}r)$$

(9)

$$W_{}^{s1}=\frac{W_{1-1}^P}{K_0^{}(q_1^{}{r}_1)}{K}_0^{}(q_1^{}r)$$

(10)

Further, the displacement attenuation function $\mu \left(S\right)$ of the soil at position s away from the active pile can be obtained as follows:

$$\mu \left(S\right)=\frac{K_0^{}(q_1^{}S)}{K_0^{}(q_1^{}{r}_1)}$$

(11)

Based on the relationship between stress and strain, the soil shear stress can be obtained using Formulas (9) and (10):

$${\tau }_{s0}^{}=G_{s0}^{*}\frac{d{W}_{}^{s0}}{dr}=-\frac{G_{s0}^{*}{q}_0^{}{W}_{1-1}^P}{I_0^{}(q_0^{}{r}_0^{})}{I}_1^{}(q_0^{}r)$$

(12)

$${\tau }_{s1}^{}=G_{s1}^{*}\frac{dU}{dr}=-\frac{G_{s1}^{*}{q}_1^{}{W}_{1-1}^P}{K_0^{}(q_1^{}{r}_1)}{K}_1^{}(q_1^{}r)$$

(13)

Thus, the soil resistance $f_{s0}^{}$ and $f_{s1}^{}$ borne by the pile foundation are:

$$f_{s0}^{}=2\pi r_0^{}{\tau }_{s0}^{}\left|{}_{r=r_0^{}}=\right.-{\psi }_{s0}^{}{W}_{1-1}^P$$

(14)

$$f_{s1}^{}=-2\pi r_1^{}{\tau }_{s1}^{}\left|{}_{r=r_1^{}}=\right.{\psi }_{s1}^{}{W}_{1-1}^P$$

(15)

where ${\psi }_{s0}^{}=2\pi r_0\frac{G_{s0}^{*}{q}_0^{}{I}_1^{}(q_0^{}{r}_0)}{I_0^{}(q_0^{}{r}_0)}$, ${\psi }_{s1}^{}=2\pi r_1\frac{G_{s1}^{*}{q}_1^{}{K}_1^{}(q_1^{}{r}_1)}{K_0^{}(q_1^{}{r}_1)}$.

3.2. Displacement of the Active Pile under External Load

Based on the elastodynamic analysis, the governing equation of the active pile under external load can be established as follows:

$$E_p^{}{A}_p^{}\frac{d^2{W}_{1-1}^P}{d{z}^2}+\sigma {\omega }^2{W}_{1-1}^P+f_{s0}^{}-f_{s1}^{}=0$$

(16)

where $\sigma ={\rho }_p^{}{A}_p^{}$.

We substitute Formulas (14) and (15) into Formula (16):

$$E_p^{}{A}_p^{}\frac{d^2{W}_{1-1}^P}{d{z}^2}+(\sigma {\omega }^2-{\psi }_{s0}^{}-{\psi }_{s1}^{})W_{1-1}^P=0$$

(17)

The solution is as follows:

$$W_{1-1}^P=A_{1-1}^{}{e}_{}^{\xi z}+B_{1-1}^{}{e}_{}^{-\xi z}$$

(18)

where ${\xi }_{}^2=\frac{{\psi }_{s0}^{}+{\psi }_{s1}^{}-\sigma {\omega }_{}^2}{E_p^{}{A}_p^{}}$. $A_{1-1}^{}$, $B_{1-1}^{}$ are the coefficients to be determined.

Consider the boundary conditions at $z=0$ and $z=H$:

$$\frac{d{W}_{1-1}^P}{dz}\left|{}_{z=0}=\right.-\frac{P_1^{}}{E_p^{}{A}_p^{}}$$

(19)

$$W_{1-1}^P\left|{}_{z=H}=\right.0$$

(20)

where $P_1^{}$ is the external load acting on the active pile.

By introducing the pile foundation displacement expression (18) into the boundary conditions in (19) and (20), we obtain:

$$A_{1-1}^{}=-\frac{P_1^{}}{E_p^{}{A}_p^{}\xi (1+e_{}^{2H\xi })}$$

(21)

$$B_{1-1}^{}=\frac{e_{}^{2H\xi }{P}_1^{}}{E_p^{}{A}_p^{}\xi (1+e_{}^{2H\xi })}$$

(22)

3.3. Passive Piles Displacement Caused by Incident Waves

By introducing Equations (11) and (18) into Equation (1), we can derive that the soil displacement at distance S of the incident wave is:

$$W_{2-1}^{s1}=\frac{K_0^{}(q_1^{}S)}{K_0^{}(q_1^{}{r}_1)}(A_{1-1}^{}{e}_{}^{\xi z}+B_{1-1}^{}{e}_{}^{-\xi z})$$

(23)

The passive pile has dynamic response under the action of the incident wave. At this time, based on the elastodynamic analysis, the passive pile control equation is established:

$$E_p^{}{A}_p^{}\frac{d^2{W}_{2-1}^P}{d{z}^2}+\sigma {\omega }^2{W}_{2-1}^P-{\psi }_{s1}^{}(W_{2-1}^P-W_{2-1}^{s1})-{\psi }_{s0}^{}{W}_{2-1}^P=0$$

(24)

The solution is as follows:

$$W_{2-1}^P=\frac{{\psi }_{s1}^{}\mu (S)}{2\xi E_p^{}{A}_p^{}}z(-A_{1-1}^{}{e}_{}^{\xi z}+B_{1-1}^{}{e}_{}^{-\xi z})+A_{2-1}^{}{e}_{}^{\xi z}+B_{2-1}^{}{e}_{}^{-\xi z}$$

(25)

where $A_{2-1}^{}$ and $B_{2-1}^{}$ are the coefficients to be determined.

Consider the boundary conditions at $z=0$ and $z=H$:

$$\frac{d{W}_{2-1}^P}{dz}\left|{}_{z=0}=\right.0$$

(26)

$$W_{2-1}^P\left|{}_{z=H}=\right.0$$

(27)

Introducing Equation (25) into Equations (26) and (27), the undetermined coefficients of the governing equation are:

$$\begin{array}{l}{A}_{2-1}^{}={\psi }_{s1}^{}\mu (S)\frac{A_{1-1}^{}-B_{1-1}^{}-B_{1-1}^{}\xi H+A_{1-1}^{}\xi H{e}_{}^{2\xi H}}{2E_p^{}{A}_p^{}{\xi }_{}^2(e_{}^{2\xi H}+1)}\\ =-\frac{{\psi }_{s1}^{}\mu (S)P_1^{}(1+e_{}^{2\xi H}+2\xi H{e}_{}^{2\xi H})}{2E_p^2{A}_p^2{\xi }_{}^3{(e_{}^{2\xi H}+1)}_{}^2}\end{array}$$

(28)

$$\begin{array}{l}{B}_{2-1}^{}={\psi }_{s1}^{}\mu (S)\frac{B_{1-1}^{}{e}_{}^{2\xi H}+A_{1-1}^{}\xi H{e}_{}^{2\xi H}-B_{1-1}^{}\xi H-A_{1-1}^{}{e}_{}^{2\xi H}}{2E_p^{}{A}_p^{}{\xi }_{}^2(e_{}^{2\xi H}+1)}\\ =\frac{{\psi }_{s1}^{}\mu (S)P_1^{}{e}_{}^{2\xi H}(1+e_{}^{2\xi H}-2\xi H)}{2E_p^2{A}_p^2{\xi }_{}^3{(e_{}^{2\xi H}+1)}_{}^2}\end{array}$$

(29)

3.4. Secondary Displacement of Active Piles Caused by Scattered Waves

Considering the scattered wave generated by the passive pile, the soil displacement at position S from the passive pile can be obtained by introducing Equations (11) and (25) into Equation (1) as:

$$W_{1-2}^{s1}=\mu \left(S\right)W_{2-1}^P=\mu \left(S\right)[\frac{{\psi }_{s1}^{}\mu \left(S\right)}{2\xi E_p^{}{A}_p^{}{}_{}^{}}z(-A_{1-1}^{}{e}_{}^{\xi z}+B_{1-1}^{}{e}_{}^{-\xi z})+A_{2-1}^{}{e}_{}^{\xi z}+B_{2-1}^{}{e}_{}^{-\xi z}]$$

(30)

The active pile receives the scattered wave and causes a dynamic response. Based on the elastodynamic analysis,, the control equation of the active pile is established:

$$E_p^{}{A}_p^{}\frac{d^2{W}_{1-2}^P}{d{z}^2}+\sigma {\omega }^2{W}_{1-2}^P-{\psi }_{s1}^{}(W_{1-2}^P-W_{1-2}^{s1})-{\psi }_{s0}^{}{W}_{1-2}^P=0$$

(31)

The solution is:

$$\begin{array}{l}{W}_{1-2}^P=z[\frac{{\psi }_{s1}^2\mu {\left(S\right)}_{}^2}{8{\xi }_{}^2{E}_p^2{A}_p^2}{A}_{1-1}^{}z-\frac{{\psi }_{s1}^{}\mu \left(S\right)}{2\xi E_p^{}{A}_p^{}}{A}_{2-1}^{}-\frac{{\psi }_{s1}^2\mu {\left(S\right)}_{}^2}{8{\xi }_{}^3{E}_p^2{A}_p^2}{A}_{1-1}^{}]e_{}^{\xi z}+z[\frac{{\psi }_{s1}^2\mu {\left(S\right)}_{}^2}{8{\xi }_{}^2{E}_p^2{A}_p^2}{B}_{1-1}^{}z\\ +\frac{{\psi }_{s1}^{}\mu \left(S\right)}{2\xi E_p^{}{A}_p^{}}{B}_{2-1}^{}+\frac{{\psi }_{s1}^2\mu {\left(S\right)}_{}^2}{8{\xi }_{}^3{E}_p^2{A}_p^2}{B}_{1-1}^{}]e_{}^{-\xi z}+A_{1-2}^{}{e}_{}^{\xi z}+B_{1-2}^{}{e}_{}^{-\xi z}\end{array}$$

(32)

where $A_{1-2}^{}$ and $B_{1-2}^{}$ are the coefficients to be determined.

At this time, only the scattering wave is considered. According to the superposition principle, the boundary conditions are:

$$\frac{d{W}_{1-2}^P}{dz}\left|{}_{z=0}=\right.0$$

(33)

$$W_{1-2}^P\left|{}_{z=H}=\right.0$$

(34)

Introducing Equation (32) into Equations (33) and (34), the undetermined coefficients of the governing equation are:

$$A_{1-2}^{}=-\frac{V_1^{}+e_{}^{\xi H}{V}_2^{}\xi }{\xi +\xi e_{}^{2\xi H}}$$

(35)

$$B_{1-2}^{}=\frac{e_{}^{\xi H}(e_{}^{\xi H}{V}_1^{}-V_2^{}\xi )}{\xi +\xi e_{}^{2\xi H}}$$

(36)

where:

$$V_1^{}=\frac{\mu {(S)}_{}^2{\psi }_{s1}^2(B_{1-1}^{}-A_{1-1}^{})}{8A_p^2{E}_p^2{\xi }_{}^3}+\frac{\mu (S){\psi }_{s1}^{}(B_{2-1}^{}-A_{2-1}^{})}{2A_p^{}{E}_p^{}\xi }=\frac{\mu {(S)}_{}^2{\psi }_{s1}^2{P}_1^{}}{8A_p^3{E}_p^3{\xi }_{}^4}+\frac{\mu (S){\psi }_{s1}^{}{P}_1^{}}{2A_p^2{E}_p^2{\xi }_{}^3}$$

(37)

$$\begin{array}{cc}\hfill & V_2^{}=H[\frac{\mu {\left(S\right)}_{}^2{\psi }_{s1}^2}{8A_p^2{E}_p^2{\xi }_{}^2}{A}_{1-1}^{}H-\frac{\mu \left(S\right){\psi }_{s1}^{}}{2A_p^{}{E}_p^{}\xi }{A}_{2-1}^{}-\frac{\mu {\left(S\right)}_{}^2{\psi }_{s1}^2}{8A_p^2{E}_p^2{\xi }_{}^3}{A}_{1-1}^{}]e_{}^{\xi H}\hfill \\ \hfill & +H[\frac{\mu {\left(S\right)}_{}^2{\psi }_{s1}^2}{8A_p^2{E}_p^2{\xi }_{}^2}{B}_{1-1}^{}H+\frac{\mu \left(S\right){\psi }_{s1}^{}}{2A_p^{}{E}_p^{}\xi }{B}_{2-1}^{}+\frac{\mu {\left(S\right)}_{}^2{\psi }_{s1}^2}{8A_p^2{E}_p^2{\xi }_{}^3}{B}_{1-1}^{}]e_{}^{-\xi H}\hfill \\ \hfill & =\frac{\mu {\left(S\right)}_{}^2{\psi }_{s1}^2{P}_1^{}H{e}_{}^{\xi H}(3+3e_{}^{2\xi H}+2\xi H{e}_{}^{2\xi H}-2\xi H)}{4A_p^3{E}_p^3{\xi }_{}^4{(1+e_{}^{2\xi H})}_{}^2}\hfill \end{array}$$

(38)

3.5. A New Interaction Factor Considering Scattering Effect of Passive Piles

Mylonakis et al. [10] and Dobry et al. [30] deduced the traditional interaction factor in the previous study, which only considered the incident wave generated by the vibration of the active pile and ignored the scattering effect of the passive pile. The expression is:

$$\alpha =\frac{W_{2-1}^P}{W_{1-1}^P}$$

(39)

The model only quantifies the unidirectional effect of active pile vibration on passive piles. However, in the pile groups, the interaction between piles is mutual. There is not only the impact of the active pile on the passive pile, but also the impact of the passive pile on the active pile. Therefore, for pile groups with small spacing, the influence of scattered waves generated by passive piles on active piles cannot be ignored. Therefore, the scattering effect factor $\gamma$ is introduced in this paper, and the scattering effect of passive piles is included in the interaction analysis between pile groups. Its mathematical expression is:

$$\gamma =\frac{W_{1-2}^P}{W_{1-1}^P}$$

(40)

The scattering effect factor is introduced into Equation (39), and the new double pile interaction factor is obtained:

$$\alpha =\frac{W_{2-1}^P}{W_{1-1}^P-W_{1-2}^P}=\frac{W_{2-1}^P}{(1-\gamma )W_{1-1}^P}$$

(41)

4. Example Analysis

The parameter system for the example analysis is shown in

Table 1. Example parameters.

$E_p^{}/E_s^{}=6.5\times {10}^3$	$H/d=40$	$S/d=2$	${\rho }_p^{}/{\rho }_s^{}=1.25$
${\nu }_s^{}=0.4$	${\delta }_s^{}=0.05$	$r_0^{}=0.3$	$r_1^{}=0.6$

:

4.1. Comparative Analysis

The obtained solution is reduced to a solid pile to verify the rationality of this derivation and to explore the difference between the traditional model and the model considering the scattering effect of the passive piles. The obtained dynamic interaction factors of pile groups are compared with the solutions of Mylonakis et al. [10]. It can be seen from Figure 3 that the degenerated solutions in this paper are in good agreement with the existing solutions. It can be seen from the figure that in the case of small spacing, the scattering effect has a certain impact on the pile–pile dynamic interaction factor. In the range of a₀ < 0.75, the real part of the solution in this paper is greater than the Mylonakis et al. [10] solution; in the range of a₀ > 0.75, the real part of the solution in this paper is smaller than that of the traditional result. For the imaginary part of the interaction factor, the above situation also exists, and the corresponding frequency is a₀ = 0.5. Therefore, considering the scattering effect in pile group analysis can improve the accuracy.

4.2. Parameter Analysis

Figure 4 shows the difference between the change in interaction factors of pipe pile type and solid pile type caused by the change in slenderness ratio when considering the scattering effect of passive piles in 1 × 2 pile groups, where $r_0^{}=0.2$. Within the frequency range shown in the figure, the interaction factors of pipe piles and solid piles show certain oscillation characteristics. Due to the existence of pile core soil, the acting force of pipe piles consumes part of the energy, the oscillation amplitude is small, and the interaction between pile groups is smaller. As the slenderness ratio increases, the oscillation frequency decreases gradually. In the range of a₀ < 0.6, the real part of the interaction factor increases gradually; in the range of a₀ < 1.1, the amplitude of the imaginary part of the interaction factor increases gradually. This is because in the low-frequency range, the longer the pile foundation, the more difficult it is for the load to be transmitted to the bedrock, the greater the impact on the soil is around the pile, and the more obvious the interaction between piles. In the range of a₀ > 1.2, the amplitude level of the interaction factor of solid pile type gradually increases with the increase in slenderness ratio, but when the slenderness ratio is greater than 60, the variation amplitude decreases, because the influence of pile top load on the soil around the pile is mainly concentrated in the upper soil, and when the pile length reaches a certain value, the influence of vibration on adjacent piles does not increase significantly. At the same time, in the above change trend, the change range of pile group interaction of pipe pile type is small, because the existence and length of pile core soil increase, the force generated on the pipe pile increases, and the energy consumed increases, indicating that compared with the solid pile type, the change in slenderness ratio has less interference with the pile groups interaction of pipe pile type.

Figure 5 shows the difference between the change in interaction factors of pipe pile type and solid pile type caused by the change in pile spacing when considering the scattering effect of passive piles in 1 × 2 pile groups. It can be seen from the figure that the interaction factors of pipe piles and solid piles show certain oscillation characteristics. Due to the existence of pile core soil, the acting force of pipe piles consumes part of the energy, the oscillation amplitude is small, and the interaction between pile groups is smaller. With the increase in pile spacing, the oscillation amplitude level decreases gradually, and the change in pipe pile is small. The oscillation frequency decreases gradually, and the difference between the real part and the virtual part is small. This is because when a₀ < 1.5, the shear wave generated by the pile foundation vibration attenuates rapidly, and its influence on the soil around the pile is small. Therefore, the greater the pile spacing, the less obvious the interaction between pile groups. At the same time, in the above change trend, the change range of pile groups interaction of pipe pile type is small because the incident wave and scattered wave generated by the vibration in the soil around the pile weaken with the increase in pile spacing, and the force of pile core soil on the pipe pile is more prominent, indicating that compared with the solid pile type, the change in pile spacing has less interference on the pile groups interaction of pipe pile type.

Figure 6 shows the difference between the change in interaction factor of pipe pile type and solid pile type caused by the change in outer diameter when considering the scattering effect of passive piles in 1 × 2 pile groups. It can be seen from the figure that the interaction factors of pipe pile and solid pile show certain oscillation characteristics. Due to the existence of pile core soil, the acting force of pipe pile consumes part of energy, the oscillation amplitude is small, and the interaction between pile groups is smaller. With the increase in outer diameter, the oscillation amplitude increases gradually; the oscillation frequency increases gradually because the pile wall becomes thicker and its stiffness stronger, and the vibration effect of the pile foundation is strengthened. At the same time, in the above trends, the change range of the pile group interaction of pipe pile type is small, because the pile wall of the pipe pile is thin, and there is the effect of pile core soil on the pile, indicating that compared with the solid pile type, the change in outer diameter has less interference on the interaction of the pipe pile.

Figure 7 shows the difference between the change in interaction factors of pipe pile type and solid pile type caused by the change in pile–soil modulus ratio when considering the scattering effect of passive piles in 1 × 2 pile groups. Within the frequency range shown in the figure, the interaction factors of pipe piles and solid piles show certain oscillation characteristics. Due to the existence of pile core soil, the acting force of pipe piles consumes part of the energy, the oscillation amplitude is small, and the interaction between pile groups is smaller. With the increase in pile–soil modulus ratio, the oscillation frequency decreases gradually, and the change range of the pipe pile is larger. In the range of a₀ > 1.0, the amplitude level of the interaction factor gradually increases because the modulus of soil around the pile decreases and its constraint effect on the pile foundation decreases, so the interaction between pile groups is enhanced. In the amplitude change, the change amplitude of the pipe pile is not obvious because the existing pile core soil has a binding effect on the pile foundation, so the interaction between pile groups is not obvious, indicating that compared with the solid pile type, the change in pile–soil modulus ratio has less interference on the interaction between pile groups of pipe pile type, but has greater interference on the oscillation frequency.

The following is the difference analysis of the change in interaction factors between pipe pile type and solid pile type caused by the change in length diameter ratio and pile spacing when considering the scattering effect of passive piles in 2 × 2 pile groups. Figure 8 shows the designed 2 × 2 pile groups model. Unless otherwise specified, the model parameters are the same as those of the 1 × 2 pile groups.

Figure 9 shows the difference between the change in interaction factors of pipe pile type and solid pile type caused by the change in slenderness ratio when considering the scattering effect of passive piles in 2 × 2 pile groups. In the frequency range shown in the figure, the interaction factors of pipe pile and solid pile show certain oscillation characteristics, in which the oscillation amplitude of the pipe pile is larger in the low-frequency range, and the difference between the two amplitudes is smaller in the high-frequency range. As the slenderness ratio increases, the oscillation frequency decreases gradually. In the range of a₀ < 1.0, the amplitude of the imaginary part of the interaction factor increases gradually. This is because in the low-frequency range, the longer the pile foundation is, the more difficult it is for the load to be transmitted to the bedrock, the greater the impact on the soil around the pile, and the more obvious the interaction between piles. In the range of a₀ > 1.0, the amplitude level of the interaction factor gradually increases, but when the slenderness ratio is greater than 60, the change amplitude decreases because the influence of the pile top load on the soil around the pile is mainly concentrated in the upper soil, and when the length of the pile reaches a certain value, the influence of the vibration on the adjacent pile does not increase significantly. At the same time, in the above change trend, the change range of pile groups interaction of pipe pile type is small because the size and length of the pile core soil increase, the force generated on the pipe pile increases, and the energy consumed increases, indicating that compared with the solid pile type, the change in slenderness ratio has less interference on the pile groups interaction of pipe pile type. Among the amplitude changes caused by the slenderness ratio between the 1 × 2 and 2 × 2 pile groups, the amplitude difference between the pipe pile and the solid pile of pile group 2 × 2 is smaller because the pile–pile connection is closer.

Figure 10 shows the difference between the change in interaction factors of pipe pile type and solid pile type caused by the change in pile spacing when considering the scattering effect of passive piles in 2 × 2 pile groups. In the frequency range shown in the figure, the interaction factors of pipe pile and solid pile show certain oscillation characteristics. With the increase in pile spacing, the oscillation amplitude of the interaction factor gradually decreases, indicating that the interaction between piles is gradually weakening. The oscillation frequency of the interaction factor decreases gradually. This is because when a₀ < 1.5, the shear wave generated by the pile foundation vibration attenuates rapidly, and its influence on the soil around the pile is small. Therefore, the greater the pile spacing, the less obvious the interaction between pile groups. At the same time, in the above change trend, the change range of pile groups interaction of pipe pile type is small because the incident wave and scattered wave generated by the vibration in the soil around the pile weaken with the increase in pile spacing and the force of pile core soil on the pipe pile is more prominent, indicating that compared with the solid pile type, the change in pile spacing has less interference on the pile groups interaction of pipe pile type. Among the amplitude changes caused by the pile spacing between the 1 × 2 and 2 × 2 pile groups, the amplitude difference between the pipe pile and the solid pile of pile group 2 × 2 is smaller because the pile–pile connection is closer.

5. Conclusions

Based on elastodynamic analysis, a calculation model of pile groups under vertical dynamic load is established in this paper. Considering the scattered wave generated by passive piles’ vibration, the diffraction effect factor is proposed, and the traditional interaction factor is modified. In this paper, the degenerate solution is compared with the existing theoretical solution to verify the rationality of the solution in this paper. Via the analysis of calculation examples, the differences between the changes in interaction factors of 1 × 2, 2 × 2 pipe piles and solid piles caused by the changes in slenderness ratio, pile spacing, and other changes are discussed, respectively. The conclusions are as follows:

(1) As the slenderness ratio increases, the oscillation frequency decreases gradually. In the low-frequency range, the amplitude level of the interaction factor gradually increases. In the high-frequency range, the amplitude of the interaction factor of solid pile type increases, and the change amplitude of the pipe pile in the above comparison is small. The change in slenderness ratio has less interference with the interaction between pipe piles.

(2) As the pile spacing increases, the oscillation amplitude level gradually decreases, indicating that the interaction between piles is gradually weakening. The oscillation frequency also gradually decreases, and the variation amplitude of the pipe pile in the above comparison is small. The change in pile spacing has less interference with the interaction between pipe piles.

(3) As the outer diameter increases, the oscillation amplitude increases gradually, indicating that the larger the outer diameter, the more obvious the interaction. The oscillation frequency increases gradually, and the variation amplitude of the pipe pile in the above comparison is small. The change in outer diameter has less interference with the interaction between pipe piles.

(4) As the pile–soil modulus ratio increases, the oscillation frequency decreases gradually. In the high-frequency range, the amplitude level of the interaction factor gradually increases, and the amplitude change in the pipe pile is small in the comparison. The change in pile–soil modulus ratio has less interference with the interaction between pipe piles, but greater interference with oscillation frequency.

In engineering practice, the soil plug effect is inevitable in pipe piles, which needs to be considered in order to improve the solution results of the dynamic response analysis of pile groups. At the same time, most natural soil is layered, and due to sedimentation, there is radial heterogeneity, so it is necessary to carry out the dynamic response analysis of pile groups in layered soil and anisotropic soil, which will have greater practical significance for engineering practice.