Analysis of Isolation Effectiveness of Shear Waves by a Row of Hollow Pipe Piles in Saturated Soils

Abstract

Wave numbers about the three types of waves in saturated soils are firstly given in this paper. The lengths of the pipe piles are much larger than their diameters, so the isolation problem about SV waves by discontinuous barriers composed of a row of pipe piles can be simplified as a two-dimensional scattering problem. The expansion method of wave functions is adopted, the stresses and displacements at the boundaries between the pipe piles and adjacent soils are considered as continuous and the inner sides of the pipe piles are free, and then the theoretical solutions are obtained about this two-dimensional scattering problem. Normalized displacements are introduced, which are the displacements behind the barriers caused by both the incident and scattered waves to those only by the incident SV waves, contours and curves of the normalized displacements are drawn, and the influences of wall thickness of pipe piles, modulus ratio of pipe piles to soils, spacing distance between the pipe piles and pipe pile numbers on the isolation effectiveness are analyzed.

Appendices

Appendix A: Total Potential Functions of P$_{1}$ Waves, P $_{2}$ Waves and SV Waves in the Saturated Soils

$$\begin{aligned} \varphi _{1l} \left( {r_l ,\theta _l } \right)= & {} \left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {A_{1m}^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_1 r_l } \right) T_{mn}^+ \left( {k_1 d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {A_{1m}^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_1 r_l } \right) T_{mn}^+ \left( {k_1 d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\sum _{n=0}^{+\infty } {H_n^{\left( 1 \right) } \left( {k_1 r_l } \right) \left( {A_{1n}^l \cos n\theta _l +B_{1n}^l \sin n\theta _l } \right) } \nonumber \\&+\left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {B_{1m}^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_1 r_l } \right) T_{mn}^- \left( {k_1 d_{jl} } \right) \sin n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {B_{1m}^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_1 r_l } \right) T_{mn}^- \left( {k_1 d_{jl} } \right) \sin n\theta _l } } } \nonumber \\ \end{aligned}$$

(A.1)

$$\begin{aligned} \varphi _{2l} \left( {r_l ,\theta _l } \right)= & {} \left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {A_{2m}^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_2 r_l } \right) T_{mn}^+ \left( {k_2 d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {A_{2m}^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_2 r_l } \right) T_{mn}^+ \left( {k_2 d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\sum _{n=0}^{+\infty } {H_n^{\left( 1 \right) } \left( {k_2 r_l } \right) \left( {A_{2n}^l \cos n\theta _l +B_{2n}^l \sin n\theta _l } \right) } \nonumber \\&+\left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {B_{2m}^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_2 r_l } \right) T_{mn}^- \left( {k_2 d_{jl} } \right) \sin n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {B_{2m}^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_2 r_l } \right) T_{mn}^- \left( {k_2 d_{jl} } \right) \sin n\theta _l } } } \nonumber \\ \end{aligned}$$

(A.2)

$$\begin{aligned} \psi _l \left( {r_l ,\theta _l } \right)= & {} \psi _0 \exp \left( {ik_s d_{1l} \cos \beta } \right) \sum _{n=0}^{+\infty } {\varepsilon _n i^{n}J_n \left( {k_s r_l } \right) \cos n\left( {\theta _l -\beta } \right) } \nonumber \\&+\left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {C_m^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_s r_l } \right) T_{mn}^+ \left( {k_s d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {C_m^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_s r_l } \right) T_{mn}^+ \left( {k_s d_{jl} } \right) \cos n\theta _l } } } \nonumber \\&+\sum _{m=0}^{+\infty } {H_n^{\left( 1 \right) } \left( {k_s r_l } \right) \left( {C_n^l \cos n\theta _l +D_n^l \sin n\theta _l } \right) } \nonumber \\&+\left( {1-\delta _{l1} } \right) \sum _{j=1}^{l-1} {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {D_m^j \left( {-1} \right) ^{n}\frac{\varepsilon _n }{2}J_n \left( {k_s r_l } \right) T_{mn}^- \left( {k_s d_{jl} } \right) \sin n\theta _l } } } \nonumber \\&+\left( {1-\delta _{lN} } \right) \sum _{j=l+1}^N {\sum _{m=0}^{+\infty } {\sum _{n=0}^{+\infty } {D_m^j \left( {-1} \right) ^{m}\frac{\varepsilon _n }{2}J_n \left( {k_s r_l } \right) T_{mn}^- \left( {k_s d_{jl} } \right) \sin n\theta _l } } }\nonumber \\ \end{aligned}$$

(A.3)

where $T_{nm}^+ \left( \cdot \right)$ and $T_{nm}^- \left( \cdot \right)$ are

$$\begin{aligned} T_{mn}^+ (\cdot )= & {} H_{m+n}^{\left( 1 \right) } (\cdot )+\left( {-1} \right) ^{n}H_{m-n}^{\left( 1 \right) } (\cdot ) \end{aligned}$$

(A.4)

$$\begin{aligned} T_{mn}^- (\cdot )= & {} -H_{m+n}^{\left( 1 \right) } (\cdot )+\left( {-1} \right) ^{n}H_{m-n}^{\left( 1 \right) } (\cdot ) \end{aligned}$$

(A.5)

Appendix B: Components of Stress and Displacements in Saturated Soils

The components of stresses, displacements and water pressures in the saturated soils, soil skeletons and water are listed in radial coordinate system as follows:

• 1. Normal stress $\sigma _r$

$$\begin{aligned} \varphi _1 : \quad \left( {\lambda _c +\gamma _1 \alpha M} \right) \nabla ^{2}\varphi _1 +2\mu \frac{\partial ^{2}\varphi _1 }{\partial r^{2}}=\frac{2\mu }{r^{2}}\Gamma _{11}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.1)

where $\Gamma _{11}^{\left( l \right) } =\left[ {n^{2}+n-\left( {1+{\lambda _c }/{2\mu }+\gamma _1 \alpha M/{2\mu }} \right) k_1^2 r^{2}} \right] Z_n^{\left( l \right) } \left( {k_1 r} \right) -k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right)$.

$$\begin{aligned} \varphi _2 : \quad \left( {\lambda _c +\gamma _2 \alpha M} \right) \nabla ^{2}\varphi _2 +2\mu \frac{\partial ^{2}\varphi _2 }{\partial r^{2}}=\frac{2\mu }{r^{2}}\Gamma _{12}^{\left( l \right) } \left( {r,n} \right) \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.2)

where $\Gamma _{12}^{\left( l \right) } =\left[ {n^{2}+n-\left( {1+{\lambda _c }/{2\mu }+\gamma _2 \alpha M/{2\mu }} \right) k_2^2 r^{2}} \right] Z_n^{\left( l \right) } \left( {k_2 r} \right) -k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right)$.

$$\begin{aligned} \psi : \quad \frac{2\mu }{r^{2}}\left( {r\frac{\partial ^{2}\psi }{\partial r\partial \theta }-\frac{\partial \psi }{\partial \theta }} \right) =\mp \frac{2\mu }{r^{2}}\Gamma _{13}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.3)

where $\Gamma _{13}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_s r} \right) +k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) } \right]$.

• 2. Shear stress $\tau _{r\theta }$

$$\begin{aligned} \varphi _1 : \quad \frac{2\mu }{r^{2}}\left( {r\frac{\partial ^{2}\varphi _1 }{\partial r\partial \theta }-\frac{\partial \varphi _1 }{\partial \theta }} \right) =\mp \frac{2\mu }{r^{2}}\Gamma _{21}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.4)

where $\Gamma _{21}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_1 r} \right) +k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) } \right]$.

$$\begin{aligned} \varphi _2 : \quad \frac{2\mu }{r^{2}}\left( {r\frac{\partial ^{2}\varphi _2 }{\partial r\partial \theta }-\frac{\partial \varphi _2 }{\partial \theta }} \right) =\mp \frac{2\mu }{r^{2}}\Gamma _{22}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.5)

where $\Gamma _{22}^{\left( l \right) } =n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_2 r} \right) +k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) } \right]$.

$$\begin{aligned} \psi : \quad \frac{\mu }{r^{2}}\left( {\frac{\partial ^{2}\psi }{\partial \theta ^{2}}-r^{2}\frac{\partial ^{2}\psi }{\partial r^{2}}+r\frac{\partial \psi }{\partial r}} \right) =\frac{2\mu }{r^{2}}\Gamma _{23}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.6)

where $\Gamma _{23}^{\left( l \right) } =-\left( {n^{2}+n-{k_s^2 r^{2}}/2} \right) Z_n^{\left( l \right) } \left( {k_s r} \right) +k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right)$.

• 3. Radial displacement $u_r$ in the soil skeletons

$$\begin{aligned} \varphi _1 : \quad \frac{\partial \varphi _1 }{\partial r}=\frac{1}{r}\Gamma _{31}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.7)

where $\Gamma _{31}^{\left( l \right) } =k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) -Z_n^{\left( l \right) } \left( {k_1 r} \right)$.

$$\begin{aligned} \varphi _2 : \quad \frac{\partial \varphi _2 }{\partial r}=\frac{1}{r}\Gamma _{32}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.8)

where $\Gamma _{32}^{\left( l \right) } =k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) -Z_n^{\left( l \right) } \left( {k_2 r} \right)$.

$$\begin{aligned} \psi : \quad \frac{1}{r}\frac{\partial \psi }{\partial \theta }=\mp \frac{1}{r}\Gamma _{33}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.9)

where $\Gamma _{33}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_s r} \right)$.

• 4. Circumferential displacement $u_\theta$ in the soil skeletons

$$\begin{aligned} \varphi _1 : \quad \frac{1}{r}\frac{\partial \varphi _1 }{\partial \theta }=\mp \frac{1}{r}\Gamma _{41}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.10)

where $\Gamma _{41}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_1 r} \right)$.

$$\begin{aligned} \varphi _2 : \quad \frac{1}{r}\frac{\partial \varphi _2 }{\partial \theta }=\mp \frac{1}{r}\Gamma _{42}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.11)

where $\Gamma _{42}^{\left( l \right) } =nZ_n^{\left( l \right) } \left( {k_2 r} \right)$.

$$\begin{aligned} \psi : \quad -\frac{\partial \psi }{\partial r}=\frac{1}{r}\Gamma _{43}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.12)

where $\Gamma _{43}^{\left( l \right) } =-\left[ {k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) -nZ_n^{\left( l \right) } \left( {k_s r} \right) } \right]$.

• 5. Radial displacement $w_r$ in the water

$$\begin{aligned} \phi _1 : \quad \frac{\partial \phi _1 }{\partial r}=\frac{1}{r}\Gamma _{51}^{\left( l \right) } \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.13)

where $\Gamma _{51}^{\left( l \right) } =\gamma _1 \Gamma _{31}^{\left( l \right) } =\gamma _1 \left[ {k_1 rZ_{n-1}^{\left( l \right) } \left( {k_1 r} \right) -Z_n^{\left( l \right) } \left( {k_1 r} \right) } \right]$.

$$\begin{aligned} \phi _2 : \quad \frac{\partial \phi _2 }{\partial r}=\frac{1}{r}\Gamma _{52}^{\left( l \right) } \left( {r,n} \right) \left\{ {\begin{array}{l} \cos n\theta \\ \sin n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.14)

where $\Gamma _{52}^{\left( l \right) } =\gamma _2 \Gamma _{32}^{\left( l \right) } =\gamma _2 \left[ {k_2 rZ_{n-1}^{\left( l \right) } \left( {k_2 r} \right) -Z_n^{\left( l \right) } \left( {k_2 r} \right) } \right]$.

$$\begin{aligned} \chi : \quad \frac{1}{r}\frac{\partial \chi }{\partial \theta }=\mp \frac{1}{r}\Gamma _{53}^{\left( l \right) } \left\{ {\begin{array}{l} \sin n\theta \\ \cos n\theta \\ \end{array}} \right\} \end{aligned}$$

(B.15)

where $\Gamma _{53}^{\left( l \right) } =\gamma _s \Gamma _{33}^{\left( l \right) } =\gamma _s nZ_n^{\left( l \right) } \left( {k_s r} \right)$.

The parameter l in $Z_n^{(l)} (\cdot )$ of Eqs. (B.1)–(B.15) is valued 1–3, and $Z_n^{(l)} (\cdot )$ denotes the different special functions: $Z_n^{\left( 1 \right) } (\cdot )=J_n (\cdot )$, $Z_n^{\left( 2 \right) } (\cdot )=N_n (\cdot )$ and $Z_n^{\left( 3 \right) } (\cdot )=H_n^{\left( 1 \right) } (\cdot )$.

Appendix C: Theoretical Solutions About the Undefined Complex Coefficients

The undefined complex coefficients $A_{1n}^l \sim Q_n^l$ ($1\le l\le N)$ can be finally determined from the linear equation group [Eqs. (C.1)–(C.6)] as follows:

$$\begin{aligned}&\frac{\varepsilon _n }{2}\Gamma _{\kappa 1}^{\left( 1 \right) } \left( {x_1^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) A_{1m}^j T_{mn}^+ \left( {k_1 d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) A_{1m}^j T_{mn}^+ \left( {k_1 d_{jl} } \right) } \right] \nonumber \\&\qquad +\frac{\varepsilon _n }{2}\Gamma _{\kappa 2}^{\left( 1 \right) } \left( {x_2^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) A_{2m}^j T_{mn}^+ \left( {k_2 d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) A_{2m}^j T_{mn}^+ \left( {k_2 d_{jl} } \right) } \right] \nonumber \\&\qquad +\frac{\varepsilon _n }{2}\Gamma _{\kappa 3}^{\left( 1 \right) } \left( {x_s^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) D_m^j T_{mn}^- \left( {k_s d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) D_m^j T_{mn}^- \left( {k_s d_{jl} } \right) } \right] \nonumber \\&\qquad +A_{1n}^l \Gamma _{\kappa 1}^{\left( 3 \right) } \left( {x_1^l } \right) +A_{2n}^l \Gamma _{\kappa 2}^{\left( 3 \right) } \left( {x_2^l } \right) +D_n^l \Gamma _{\kappa 3}^{\left( 3 \right) } \left( {x_s^l } \right) \nonumber \\&\qquad -\hat{{\delta }}_{\kappa 5} E_n^l \mu _l^*\left[ {\Lambda _{\kappa 1}^{\left( 1 \right) } \left( {\tilde{x}_p^l } \right) +p_1 \Lambda _{\kappa 2}^{\left( 1 \right) } \left( {\tilde{x}_s^l } \right) +q_1 \Lambda _{\kappa 2}^{\left( 2 \right) } \left( {\tilde{x}_s^l } \right) } \right] \nonumber \\&\qquad -\hat{{\delta }}_{\kappa 5} F_n^l \mu _l^*\left[ {\Lambda _{\kappa 1}^{\left( 2 \right) } \left( {\tilde{x}_p^l } \right) +p_2 \Lambda _{\kappa 2}^{\left( 1 \right) } \left( {\tilde{x}_s^l } \right) +q_2 \Lambda _{\kappa 2}^{\left( 2 \right) } \left( {\tilde{x}_s^l } \right) } \right] \nonumber \\&\quad =-\psi _0 \exp \left( {ik_s d_{1l} \cos \beta } \right) \varepsilon _n i^{n}\Gamma _{k3}^{\left( 1 \right) } \left( {x_s^l } \right) \sin n\beta \end{aligned}$$

(C.1)

$$\begin{aligned}&\frac{\varepsilon _n }{2}\Gamma _{\kappa 1}^{\left( 1 \right) } \left( {x_1^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) B_{1m}^j T_{mn}^- \left( {k_1 d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) B_{1m}^j T_{mn}^- \left( {k_1 d_{jl} } \right) } \right] \nonumber \\&\qquad +\frac{\varepsilon _n }{2}\Gamma _{\kappa 2}^{\left( 1 \right) } \left( {x_2^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) B_{2m}^j T_{mn}^- \left( {k_2 d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) B_{2m}^j T_{mn}^- \left( {k_2 d_{jl} } \right) } \right] \nonumber \\&\qquad -\frac{\varepsilon _n }{2}\Gamma _{\kappa 3}^{\left( 1 \right) } \left( {x_s^l } \right) \sum _{m=0}^{+\infty } \left[ \sum _{j=1}^{l-1} \left( {-1} \right) ^{n}\left( {1-\delta _{l1} } \right) C_m^j T_{mn}^+ \left( {k_s d_{jl} } \right) \right. \nonumber \\&\qquad \left. +\sum _{j=l+1}^N {\left( {-1} \right) ^{m}\left( {1-\delta _{lN} } \right) C_m^j T_{mn}^+ \left( {k_s d_{jl} } \right) } \right] \nonumber \\&\qquad +B_{1n}^l \Gamma _{\kappa 1}^{\left( 3 \right) } \left( {x_1^l } \right) +B_{2n}^l \Gamma _{\kappa 2}^{\left( 3 \right) } \left( {x_2^l } \right) -C_n^l \Gamma _{\kappa 3}^{\left( 3 \right) } \left( {x_s^l } \right) \nonumber \\&\qquad -\hat{{\delta }}_{\kappa 5}{\mathop {E}\limits ^{\frown }}_n^l \mu _l^*\left[ {\Lambda _{\kappa 1}^{\left( 1 \right) } \left( {\tilde{x}_p^l } \right) +p_1 \Lambda _{\kappa 2}^{\left( 1 \right) } \left( {\tilde{x}_s^l } \right) +q_1 \Lambda _{\kappa 2}^{\left( 2 \right) } \left( {\tilde{x}_s^l } \right) } \right] \nonumber \\&\qquad -\hat{{\delta }}_{\kappa 5}{\mathop {F}\limits ^{\frown }}_n^l \mu _l^*\left[ {\Lambda _{\kappa 1}^{\left( 2 \right) } \left( {\tilde{x}_p^l } \right) +p_2 \Lambda _{\kappa 2}^{\left( 1 \right) } \left( {\tilde{x}_s^l } \right) +q_2 \Lambda _{\kappa 2}^{\left( 2 \right) } \left( {\tilde{x}_s^l } \right) } \right] \nonumber \\&\quad =-\psi _0 \exp \left( {ik_s d_{1l} \cos \beta } \right) \varepsilon _n i^{n}\Gamma _{k3}^{\left( 1 \right) } \left( {x_s^l } \right) \cos n\beta \end{aligned}$$

(C.2)

$$\begin{aligned}&M_n^l =p_1 E_n^l +p_2 I_n^l \end{aligned}$$

(C.3)

$$\begin{aligned}&Q_n^l =q_1 E_n^l +q_2 I_n^l \end{aligned}$$

(C.4)

$$\begin{aligned}&L_n^l =-\left( {p_1 F_n^l +p_2 K_n^l } \right) \end{aligned}$$

(C.5)

$$\begin{aligned}&P_n^l =-\left( {q_1 F_n^l +q_2 K_n^l } \right) \end{aligned}$$

(C.6)

where

$$\begin{aligned} p_1= & {} \frac{\Lambda _{21}^{\left( 1 \right) } \left( {\tilde{y}_p^l } \right) \Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) +\Lambda _{11}^{\left( 1 \right) } \left( {\tilde{y}_p^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) }{\Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) -\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) } \end{aligned}$$

(C.7)

$$\begin{aligned} p_2= & {} \frac{\Lambda _{21}^{\left( 2 \right) } \left( {\tilde{y}_p^l } \right) \Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) +\Lambda _{11}^{\left( 2 \right) } \left( {\tilde{y}_p^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) }{\Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) -\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) } \end{aligned}$$

(C.8)

$$\begin{aligned} q_1= & {} \frac{\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{21}^{\left( 1 \right) } \left( {\tilde{y}_p^l } \right) +\Lambda _{11}^{\left( 1 \right) } \left( {\tilde{y}_p^l } \right) \Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) }{\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) -\Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) } \end{aligned}$$

(C.9)

$$\begin{aligned} q_2= & {} \frac{\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{21}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) +\Lambda _{11}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) }{\Lambda _{12}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) -\Lambda _{12}^{\left( 2 \right) } \left( {\tilde{y}_s^l } \right) \Lambda _{22}^{\left( 1 \right) } \left( {\tilde{y}_s^l } \right) } \end{aligned}$$

(C.10)

where $x_1^l =k_1 b_l$, $x_2^l =k_2 b_l$, $x_s^l =k_s b_l$, $\tilde{x}_p^l =\tilde{k}_p b_l$, $\tilde{x}_s^l =\tilde{k}_s^l b_l$, $\tilde{y}_p^l =\tilde{k}_p a_l$ and $\tilde{y}_s^l =\tilde{k}_s a_l$; $\hat{{\delta }}_{ij}$ is opposite to $\delta _{ij}$: $\hat{{\delta }}_{ij} =1(i\ne j)$ and $\hat{{\delta }}_{ij} =0(i=j)$; $\mu _l^*={\tilde{\mu }_l }/\mu$, which is the modulus ratio of the $l\hbox {th}$ pipe pile to the solid skeleton of saturated soils, and $\mu _l^*=1$ when $\kappa =3,4$; the parameter of $\kappa$ in valued 1–5 in $\Gamma _{\kappa 1}^{\left( l \right) } \sim \Gamma _{\kappa 3}^{\left( l \right) }$ and 1–4 in $\Lambda _{\kappa 1}^{\left( l \right) } \sim \Lambda _{\kappa 2}^{\left( l \right) }$, the expressions of $\Gamma _{\kappa 1}^{\left( l \right) }$ are given in Eq. (B.1)–(B.15), and the expressions of $\Lambda _{11}^{\left( l \right) } \sim \Lambda _{42}^{\left( l \right) }$ are given as follows:

$$\begin{aligned} \Lambda _{11}^{\left( l \right) }= & {} \left[ {n^{2}+n-\left( {1+\lambda /{2\mu }} \right) \tilde{k}_p^2 r^{2}} \right] Z_n^{\left( l \right) } \left( {\tilde{k}_p r} \right) -\tilde{k}_p rZ_{n-1}^{\left( l \right) } \left( {\tilde{k}_p r} \right) , \\ \Lambda _{12}^{\left( l \right) }= & {} n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {k_s r} \right) +k_s rZ_{n-1}^{\left( l \right) } \left( {k_s r} \right) } \right] , \\ \Lambda _{21}^{\left( l \right) }= & {} n\left[ {-\left( {1+n} \right) Z_n^{\left( l \right) } \left( {\tilde{k}_p r} \right) +\tilde{k}_p rZ_{n-1}^{\left( l \right) } \left( {\tilde{k}_p r} \right) } \right] , \\ \Lambda _{22}^{\left( l \right) }= & {} -\left( {n^{2}+n-{\tilde{k}_s^2 r^{2}}/2} \right) Z_n^{\left( l \right) } \left( {\tilde{k}_s r} \right) +\tilde{k}_s rZ_{n-1}^{\left( l \right) } \left( {\tilde{k}_s r} \right) , \\ \Lambda _{31}^{\left( l \right) }= & {} \tilde{k}_p rZ_{n-1}^{\left( l \right) } \left( {\tilde{k}_p r} \right) -Z_n^{\left( l \right) } \left( {\tilde{k}_p r} \right) , \\ \Lambda _{32}^{\left( l \right) }= & {} nZ_n^{\left( l \right) } \left( {\tilde{k}_s r} \right) , \\ \Lambda _{41}^{\left( l \right) }= & {} nZ_n^{\left( l \right) } \left( {\tilde{k}_p r} \right) , \\ \Lambda _{42}^{\left( l \right) }= & {} -\left[ {\tilde{k}_s rZ_{n-1}^{\left( l \right) } \left( {\tilde{k}_s r} \right) -nZ_n^{\left( l \right) } \left( {\tilde{k}_s r} \right) } \right] . \end{aligned}$$