2.3.1. Pile–Soil Interaction

The pile embedded in the soil is under lateral reaction of the soil. To investigate the influence of the soil cut-off frequency, this paper adopts the soil model proposed by Anoyatis [26]. This model uses the improved Tajimi soil model, considers the soil as a three-dimensional continuum, and considers the influence of soil vertical displacement on the lateral soil displacement. The proposed dynamic Winkler modulus can reflect the vibration of spring coefficient and damping coefficient around the soil cut-off frequency. The proposed calculation method is shown in the equation below:

$$\begin{array}{c} k^\* = \pi G\_s^\* s \Big[ s + 4 \frac{Y\_1^{"\prime} (s)}{Y\_0^{"\prime} (s)} \Big] \\ s = \frac{1}{2 (\eta\_s) \chi} \sqrt{a\_{cutoff}^2 - \frac{a\_0^2}{1 + 2i\beta\_s}} \end{array} \tag{11}$$

where *G*∗ *<sup>s</sup>* = *Gs*(1 + 2*iβs*), *β<sup>s</sup>* is the soil material damping, valued 0.05 in this paper, *Y* <sup>1</sup>(*s*) and *Y* <sup>0</sup>(*s*) are the first and zero order modified Bessel function of the second kind, *<sup>η</sup><sup>s</sup>* <sup>=</sup> <sup>√</sup><sup>2</sup> <sup>−</sup> *<sup>ν</sup>s*/1 <sup>−</sup> *<sup>ν</sup>s*, where *<sup>υ</sup><sup>s</sup>* is Poisson's ratio, *acutoff* = (*π*/2)(*L*/*D*) −1 , where *L* is the

pile length, and *χ* is the dimensional coefficient, valued according to Poisson's ratio, valued 3 in this paper.

As shown in Figure 3, the dynamic response of the pile embedded in the soil can be divided into four steps:


**Figure 3.** Pile–pile interaction.

2.3.2. Active Pile Displacement

The dynamic equation of the active pile under horizontal dynamic load can be written as:

$$\frac{d^4 w\_i^{d2}}{dz^4} + 4\gamma^4 w\_i^{d2}(z) = 0\tag{12}$$

where *γ* = *<sup>k</sup>*+*iωc*−*mpω*<sup>2</sup> 4*Ep Ip* 1/4 , *k* = Re(*k*∗), *wc* = IM(*k*∗), and *wa*<sup>2</sup> *<sup>i</sup>* means the lateral displacement of the active pile embedded in the soil. The solution of Equation (12) is written below:

$$w\_i^{a2}(z) = e^{(-1+i)\gamma z} A\_{a2} + e^{(-1-i)\gamma z} B\_{a2} + e^{(1-i)\gamma z} C\_{a2} + e^{(1+i)\gamma z} D\_{a2} \tag{13}$$

where *Aa*2, *Ba*2, *Ca*2, *Da*<sup>2</sup> are undetermined coefficients, which can be determined according to boundary conditions.

Similarly, according to differential relations, we can obtain the lateral displacement, rotation angle, bending moment, and shearing force of the active pile embedded in the soil. Written in matrix form:

$$\begin{Bmatrix} w\_i^{a2} \\ \phi\_i^{a2} \\ M\_i^{a2} \\ Q\_i^{a2} \\ 1 \end{Bmatrix} = \left\{ t\_i^{a2} \right\} \begin{Bmatrix} A\_{a2} \\ B\_{a2} \\ C\_{a2} \\ D\_{a2} \\ 1 \end{Bmatrix} \tag{14}$$

where the matrix *ta*2 *i* is shown in Appendix A. The transfer matrix between the pile top and pile tip of each pile section is shown in Equation (15):

$$\begin{Bmatrix} w\_i^{q2}(h\_i) \\ \Phi\_i^{q2}(h\_i) \\ M\_i^{q2}(h\_i) \\ Q\_i^{q2}(h\_i) \\ 1 \end{Bmatrix} = \left\{ t\_i^{q2}(h\_i) \right\} \left\{ t\_i^{q2}(0) \right\}^{-1} \begin{Bmatrix} w\_i^{q2}(0) \\ w\_i^{q2}(0) \\ M\_i^{q2}(0) \\ Q\_i^{q2}(0) \\ 1 \end{Bmatrix} \tag{15}$$

Let {*Ta*<sup>2</sup> *<sup>i</sup>* } <sup>=</sup> *ta*2 *<sup>i</sup>* (*hi*) *ta*<sup>2</sup> *<sup>i</sup>* (0) −<sup>1</sup> and *Ta*<sup>2</sup> = *Ta*<sup>2</sup> *<sup>n</sup>* ··· *<sup>T</sup>a*<sup>2</sup> *<sup>i</sup>* ··· *<sup>T</sup>a*<sup>2</sup> <sup>2</sup> *<sup>T</sup>a*<sup>2</sup> <sup>1</sup> ; the transfer matrix between the active pile tip and top (soil-water surface) can be obtained:

$$\begin{Bmatrix} w^{a2}(H) \\ \Phi^{a2}(H) \\ M^{a2}(H) \\ Q^{a2}(H) \\ 1 \end{Bmatrix} = \{T^{d2}\} \begin{Bmatrix} w^{a2}(0) \\ \Phi^{a2}(0) \\ M^{a2}(0) \\ Q^{a2}(0) \\ 1 \end{Bmatrix} \tag{16}$$

#### 2.3.3. Soil Attenuation Function and Soil Displacement

For the soil S m away from the active pile, its attenuation function and soil displacement can be calculated according to the following equation:

$$M\_p(\mathcal{S}, z) = \psi w^{a2}(z) = \left[\psi(\mathcal{S}, 0^\circ) \cos \theta^2 + \psi(\mathcal{S}, 90^\circ) \sin \theta^2\right] w^{a2}(z) \tag{17}$$

where

$$\psi(S,0^{\circ}) = \sqrt{\frac{R}{S}} \exp\left(\frac{-\beta\omega(R-S)}{V\_{La}}\right) \exp\left(\frac{-i\omega(R-S)}{V\_{La}}\right) \tag{18}$$

$$\left(\boldsymbol{\varphi}(\mathcal{S},\boldsymbol{\theta}0^{\circ})\right) = \sqrt{\frac{R}{S}} \exp\left(\frac{-\beta\omega(R-S)}{V\_{s}}\right) \exp\left(\frac{-i\omega(R-S)}{V\_{s}}\right) \tag{19}$$

where *R* is the radius of the pile, *S* is the distance between the active pile and passive pile, *β* is the soil damping, *VLa* and *Vs* are the shear wave velocity and Lysmer's simulation velocity [30], *VLa* = 3.4 *<sup>π</sup>*(1−*v*)*Vs*, and *<sup>θ</sup>* is the angle of incidence, which will influence the pile–pile interaction factor, as shown in Figure 4.

**Figure 4.** Pile–pile interaction.

The soil displacement around the passive pile after the soil attenuation can then be obtained. In pile groups, *θ* can be different between different piles, which will be considered when calculating the overall impedance of the pile group.

2.3.4. Passive Pile Displacement

The displacement of the passive pile due to the soil displacement can be calculated according to the following equation:

$$\frac{d^4 w\_i^{p2}}{dz^4} + 4\gamma^4 w\_i^{p2}(z) - \frac{(k + i\omega c)}{E\_p I\_p} \mathcal{U}\_\mathbb{P}(S, z) = 0 \tag{20}$$

Substitute Equation (17) into Equation (20), the solution can be obtained:

$$\begin{array}{lcl}w\_{i}^{p2}(z) = & e^{(-1+i)\gamma z}A\_{p2} + e^{(-1-i)\gamma z}B\_{p2} + e^{(1-i)\gamma z}C\_{p2} + e^{(1+i)\gamma z}D\_{p2} + \\ & \frac{(k+i\omega v)\psi z}{E\_{p}I\_{p}}[(1-i)e^{(-1+i)\gamma z}A\_{d2} + (1+i)e^{(-1-i)\gamma z}B\_{d2} + \\ & (-1+i)e^{(1-i)\gamma z}C\_{d2} + (-1-i)e^{(1+i)\gamma z}D\_{d2}\end{array} \tag{21}$$

where *Ap*2, *Bp*2, *Cp*2, *Dp*<sup>2</sup> are undetermined coefficients.

Written in matrix form:

$$\begin{Bmatrix} w\_i^{p2} \\ \Phi\_i^{p2} \\ M\_i^{p2} \\ Q\_i^{p2} \\ 1 \end{Bmatrix} = \left\{ t\_i^{a2} \right\} \begin{Bmatrix} A\_{p2} \\ B\_{p2} \\ C\_{p2} \\ D\_{p2} \\ 1 \end{Bmatrix} + \left\{ t\_i^{p2} \right\} \begin{Bmatrix} A\_{a2} \\ B\_{a2} \\ C\_{a2} \\ D\_{a2} \\ 1 \end{Bmatrix} \tag{22}$$

where the matrix *t p*2 *i* is shown in Appendix A. The transfer matrix between passive pile segment tip and top can be obtained after the following calculation:

$$\begin{Bmatrix} w\_i^{p2}(h\_i) \\ \Phi\_i^{p2}(h\_i) \\ M\_i^{p2}(h\_i) \\ Q\_i^{p2}(h\_i) \\ 1 \end{Bmatrix} = T\_{i1}^{p2} \begin{Bmatrix} w\_i^{p2}(0) \\ \Phi\_i^{p2}(0) \\ M\_i^{p2}(0) \\ Q\_i^{p2}(0) \\ 1 \end{Bmatrix} + T\_{i2}^{p2} \begin{Bmatrix} w\_i^{q2}(0) \\ w\_i^{q2}(0) \\ M\_i^{q2}(0) \\ Q\_i^{p2}(0) \\ 1 \end{Bmatrix} \tag{23}$$

where

where *Tp*<sup>2</sup>

<sup>1</sup> <sup>=</sup> *<sup>T</sup>p*<sup>2</sup>

$$\begin{array}{l} T\_{i1}^{p2} = t\_i^{a2}(h\_i) \left( t\_i^{a2}(0) \right)^{-1} \\ T\_{i2}^{p2} = \left( t\_i^{p2}(h\_i) - t\_i^{a2}(h\_i) \left( t\_i^{a2}(0) \right)^{-1} t\_i^{p2}(0) \right) \left( t\_i^{a2}(0) \right)^{-1} \end{array} \tag{24}$$

The transfer matrix between the passive pile tip and top (soil-water surface) can then be obtained:

$$\begin{Bmatrix} w^{p2}(H) \\ \Phi^{p2}(H) \\ M^{p2}(H) \\ Q^{p2}(H) \\ 1 \end{Bmatrix} = T\_1^{p2} \begin{Bmatrix} w^{p2}(0) \\ \Phi^{p2}(0) \\ M^{p2}(0) \\ Q^{p2}(0) \\ 1 \end{Bmatrix} + T\_2^{p2} \begin{Bmatrix} w^{a2}(0) \\ w^{a2}(0) \\ M^{a2}(0) \\ Q^{a2}(0) \\ 1 \end{Bmatrix} \tag{25}$$
 
$$\begin{Bmatrix} p\_1^{p2} \cdots T\_{i1}^{p2} \cdots T\_{21}^{p2} T\_{11}^{p2} & T\_2^{p2} \cdots T\_{i2}^{p2} \cdots T\_{22}^{p2} T\_{12}^{p2} \\ \end{Bmatrix} \tag{25}$$

2.3.5. Active Pile Displacement Due to Secondary Wave

The passive pile displacement will produce secondary wave, which will influence the active pile displacement. The soil displacement around the active pile can be calculated according to the following equation:

$$\mathcal{U}\_{\mathfrak{a}}(\mathcal{S},z) = \psi w^{p2}(z) = \left[\psi(\mathcal{S},0^{\circ})\cos\theta^{2} + \psi(\mathcal{S},90^{\circ})\sin\theta^{2}\right]w^{p2}(z)\tag{26}$$

The dynamic equation can be written as:

$$4\frac{d^4\overline{w}\_i^2}{dz^4} + 4\gamma^4\overline{w}\_i^2(z) - \frac{(k + i\omega c)}{E\_p I\_p} \mathcal{U}\_a(\mathcal{S}, z) = 0\tag{27}$$

where *wa*<sup>2</sup> is the lateral displacement of the active pile under the influence of the passive pile. The solution to Equation (27) is:

$$\begin{array}{l} \overline{w}\_{i}^{\mathfrak{q}2}(z) = \begin{array}{l} e^{(-1+i)\gamma z} \overline{A}\_{d2} + e^{(-1-i)\gamma z} \overline{B}\_{d2} + e^{(1-i)\gamma z} \overline{C}\_{d2} + e^{(1+i)\gamma z} \overline{D}\_{d2} + \\ \frac{(k+i\omega c)\psi z}{16E\_{p}l\_{p}\gamma^{3}} [(1-i)e^{(-1+i)\gamma z} A\_{p2} + (1+i)e^{(-1-i)\gamma z} B\_{p2} + \\ (-1+i)e^{(1-i)\gamma z} \mathcal{C}\_{p2} + (-1-i)e^{(1+i)\gamma z} D\_{p2}] + \\ \frac{(k+i\omega c)\psi z^{2}}{512E\_{p}l\_{p}\gamma^{6}} [(1-i)^{2}e^{(-1+i)\gamma z} A\_{d2} + (1+i)^{2}e^{(-1-i)\gamma z} B\_{d2} + \\ (-1+i)^{2}e^{(1-i)\gamma z} \mathcal{C}\_{d2} + (-1-i)^{2}e^{(1+i)\gamma z} D\_{d2}] \end{array} \tag{28}$$

where *Aa*2, *Ba*2, *Ca*2, *Da*<sup>2</sup> are undetermined coefficients. Written in matrix form:

$$\begin{aligned} \left\{ \begin{array}{c} \overline{w}\_{i}^{2} \\ \overline{\theta}\_{i}^{2} \\ \overline{A}\_{i}^{2} \\ \overline{C}\_{i}^{2} \\ 1 \end{array} \right\} &= \left\{ t\_{i}^{\text{el}} \right\} \left\{ \begin{array}{c} \overline{A}\_{a2} \\ \overline{B}\_{a2} \\ \overline{C}\_{a2} \\ 1 \end{array} \right\} + \frac{1}{16\gamma^{3}} \left\{ t\_{i}^{\text{el}} \right\} \begin{Bmatrix} A\_{p2} \\ B\_{p2} \\ C\_{p2} \\ D\_{p2} \\ 1 \end{Bmatrix} + \\ &+ \left\{ \begin{array}{c} (1-i)^{2} & 0 & 0 & 0 \\ 0 & (1+i)^{2} & 0 & 0 \\ 0 & 0 & (-1+i)^{2} & 0 \\ 0 & 0 & 0 & (-1-i)^{2} \end{array} \right\} \left\{ t\_{i}^{p,2} \right\} \end{aligned} \tag{29}$$

where the matrix *t a*2 *i* is shown in Appendix A. Let:

$$\begin{aligned} \left\{ \begin{matrix} t\_i^1 \\ t\_i^2 \end{matrix} \right\} &= \frac{\left\{ t\_i^2 \right\}}{16\gamma^3} \left\{ t\_i^{p^2} \right\} \\ \left\{ t\_i^3 \right\} &= \left\{ \overline{t}\_i^{d^2} \right\} + \frac{3i}{1024\gamma^7} \left\{ \begin{matrix} (1-i)^2 & 0 & 0 & 0 \\ 0 & (1+i)^2 & 0 & 0 \\ 0 & 0 & (-1+i)^2 & 0 \\ 0 & 0 & 0 & (-1-i)^2 \end{matrix} \right\} \left\{ t\_i^{p^2} \right\} \end{aligned} \tag{30}$$

Equation (29) can be simplified:

$$
\begin{Bmatrix} \overline{w}\_i^{p2} \\ \overline{\phi}\_i^{p2} \\ \overline{M}\_i^{p2} \\ \overline{Q}\_i^{p2} \\ 1 \end{Bmatrix} = \left\{ t\_i^1 \right\} \begin{Bmatrix} \overline{A}\_{d2} \\ \overline{B}\_{d2} \\ \overline{C}\_{d2} \\ 1 \end{Bmatrix} + \left\{ t\_i^2 \right\} \begin{Bmatrix} A\_{p2} \\ B\_{p2} \\ C\_{p2} \\ D\_{p2} \\ 1 \end{Bmatrix} + \left\{ t\_i^3 \right\} \begin{Bmatrix} A\_{d2} \\ B\_{d2} \\ C\_{d2} \\ D\_{d2} \\ 1 \end{Bmatrix} \tag{31}
$$

The transfer matrix between active pile segment tip and top can then be obtained:

$$\begin{Bmatrix} \overline{w}\_i^{p2}(h\_i) \\ \overline{\Phi}\_i^{a2}(h\_i) \\ \overline{M}\_i^{a2}(h\_i) \\ \overline{Q}\_i^{a2}(h\_i) \\ 1 \end{Bmatrix} = T\_{i1} \begin{Bmatrix} \overline{w}\_i^{p2}(0) \\ \overline{\Phi}\_i^{a2}(0) \\ \overline{M}\_i^{a2}(0) \\ \overline{Q}\_i^{a2}(0) \\ 1 \end{Bmatrix} + T\_{i2} \begin{Bmatrix} w\_i^{p2}(0) \\ w\_i^{p2}(0) \\ M\_i^{p2}(0) \\ Q\_i^{p2}(0) \\ 1 \end{Bmatrix} + T\_{i3} \begin{Bmatrix} w\_i^{a2}(0) \\ w\_i^{a2}(0) \\ M\_i^{a2}(0) \\ Q\_i^{a2}(0) \\ 1 \end{Bmatrix} \tag{32}$$

where

$$\begin{array}{ll} T\_{i1} &=& t\_i^1(h\_i) \left(t\_i^1(0)\right)^{-1} \\ T\_{i2} &=& \left(t\_i^2(h\_i) - t\_i^1(h\_i) \left(t\_i^1(0)\right)^{-1} t\_i^2(0)\right) \left(t\_i^{a2}(0)\right)^{-1} \\ T\_{i3} &=& -\left(t\_i^2(h\_i) - t\_i^1(h\_i) \left(t\_i^1(0)\right)^{-1} t\_i^2(0)\right) \left(t\_i^{a2}(0)\right)^{-1} t\_i^{p2}(0) \left(t\_i^{a2}(0)\right)^{-1} + \\ & \left(t\_i^3(h) - t\_i^1(h) \left(t\_i^1(0)\right)^{-1} t\_i^3(0)\right) \left(t\_i^{a2}(0)\right)^{-1} \end{array} \tag{33}$$

The transfer matrix between the active pile tip and top (soil–water surface) due to the secondary wave can then be obtained:

$$\begin{Bmatrix} \overline{w}^{\text{d}^{2}}(H) \\ \overline{\phi}^{\text{d}^{2}}(H) \\ \overline{M}^{\text{d}^{2}}(H) \\ \overline{\mathbb{Q}}^{\text{d}^{2}}(H) \\ 1 \end{Bmatrix} = T\_{1} \begin{Bmatrix} \overline{w}^{\text{d}^{2}}(0) \\ \overline{\phi}^{\text{d}^{2}}(0) \\ \overline{M}^{\text{d}^{2}}(0) \\ 1 \end{Bmatrix} + T\_{2} \begin{Bmatrix} w^{\text{p}^{2}}(0) \\ w^{\text{p}^{2}}(0) \\ M^{\text{p}^{2}}(0) \\ 1 \end{Bmatrix} + T\_{3} \begin{Bmatrix} w^{\text{d}^{2}}(0) \\ w^{\text{d}^{2}}(0) \\ M^{\text{d}^{2}}(0) \\ 1 \end{Bmatrix} \tag{34}$$

where *T*<sup>1</sup> = *Tn*<sup>1</sup> ··· *Ti*<sup>1</sup> ··· *T*21*T*11, *T*<sup>2</sup> = *Tn*<sup>2</sup> ··· *Ti*<sup>2</sup> ··· *T*22*T*12, *T*<sup>3</sup> = *Tn*<sup>3</sup> ··· *Ti*<sup>3</sup> ··· *T*23*T*13.
