Kinematic response of pipe pile embedded in fractional-order viscoelastic unsaturated soil subjected to vertically propagating seismic SH-waves

Abstract

The analytical representation of kinematic interaction between pipe pile and unsaturated soil subjected to vertical SH-waves is revisited using three-dimension continuum modelling. The fractional-order standard line solid (FSLS) model is employed in the governing equation of unsaturated soil to refine the characterization of the flow-independent viscosity of the soil skeleton. The Timoshenko beam theory is utilized to describe the horizontal dynamic behavior of pipe pile. A closed series form solution of horizontal kinematic response for a pipe pile embedded in unsaturated soils under the action of vertical SH-waves is deduced theoretically with boundary conditions of pile-soil system. The solutions of the proposed model are compared with those captured from existing solutions. Finally, the effects of physical parameters in pile-soil system on the horizontal kinematic response of pipe pile under SH-waves are evaluated with analytical examples and parametric study. The results indicate that FSLS model parameters have significant impact on the horizontal kinematic response of the pipe pile, changes in the soil saturation have a relatively slight effect on the horizontal kinematic response of the pipe pile, and the length and radii of pipe pile should be moderate rather than too large or too small to make the pipe pile play a better seismic performance.

Appendices

Appendix A

$B_{11} (\omega )$, $B_{12} (\omega )$, $B_{13} (\omega )$, $B_{21} (\omega )$, $B_{22} (\omega )$, and $B_{23} (\omega )$ in Eqs. (10) ~ (12) in the text.

The complex coefficients $B_{11} (\omega )$, $B_{12} (\omega )$, $B_{13} (\omega )$, $B_{21} (\omega )$, $B_{22} (\omega )$, and $B_{23} (\omega )$ in Eqs. (10) ~ (12) in the text are expressed as

$$\left. \begin{gathered} B_{11} \left( \omega \right) = \frac{{A_{13} \left( \omega \right)A_{22} \left( \omega \right) - A_{23} \left( \omega \right)A_{12} \left( \omega \right)}}{{A_{11} \left( \omega \right)A_{22} \left( \omega \right) - A_{21} \left( \omega \right)A_{12} \left( \omega \right)}} \hfill \\ B_{12} \left( \omega \right) = \frac{{A_{14} A_{22} \left( \omega \right)}}{{A_{11} \left( \omega \right)A_{22} \left( \omega \right) - A_{21} \left( \omega \right)A_{12} \left( \omega \right)}} \hfill \\ B_{13} \left( \omega \right) = \frac{{A_{25} A_{12} \left( \omega \right)}}{{A_{11} \left( \omega \right)A_{22} \left( \omega \right) - A_{21} \left( \omega \right)A_{12} \left( \omega \right)}} \hfill \\ \end{gathered} \right\}$$

(A.1)

$$\left. \begin{gathered} B_{21} \left( \omega \right) = \frac{{A_{13} \left( \omega \right)A_{21} \left( \omega \right) - A_{23} \left( \omega \right)A_{11} \left( \omega \right)}}{{A_{12} \left( \omega \right)A_{21} \left( \omega \right) - A_{22} \left( \omega \right)A_{11} \left( \omega \right)}} \hfill \\ B_{22} \left( \omega \right) = \frac{{A_{14} A_{21} \left( \omega \right)}}{{A_{12} \left( \omega \right)A_{21} \left( \omega \right) - A_{22} \left( \omega \right)A_{11} \left( \omega \right)}} \hfill \\ B_{23} \left( \omega \right) = \frac{{ - A_{25} A_{11} \left( \omega \right)}}{{A_{12} \left( \omega \right)A_{21} \left( \omega \right) - A_{22} \left( \omega \right)A_{11} \left( \omega \right)}} \hfill \\ \end{gathered} \right\}$$

(A.2)

with

$$\left. \begin{gathered} A_{11} \left( \omega \right) = \phi A_{s} + \left[ {\overline{a}\left( \omega \right) - \phi } \right]{{S_{w}^{2} } \mathord{\left/ {\vphantom {{S_{w}^{2} } {K_{s} }}} \right. \kern-0pt} {K_{s} }} + {{\phi S_{w} } \mathord{\left/ {\vphantom {{\phi S_{w} } {K_{w} }}} \right. \kern-0pt} {K_{w} }} \hfill \\ A_{12} \left( \omega \right) = \left[ {\overline{a}\left( \omega \right) - \phi } \right]{{S_{w} S_{g} } \mathord{\left/ {\vphantom {{S_{w} S_{g} } {K_{s} }}} \right. \kern-0pt} {K_{s} }} - \phi A_{s} \hfill \\ A_{13} \left( \omega \right) = \left[ {\overline{a}\left( \omega \right) - \phi } \right]S_{w} , \, A_{14} = \phi S_{w} , \, A_{15} = 0 \hfill \\ \end{gathered} \right\}$$

(A.3)

$$\left. \begin{gathered} A_{21} \left( \omega \right) = \left[ {\overline{a}\left( \omega \right) - \phi } \right]{{S_{w} S_{g} } \mathord{\left/ {\vphantom {{S_{w} S_{g} } {K_{s} }}} \right. \kern-0pt} {K_{s} }} - \phi A_{s} \hfill \\ A_{22} \left( \omega \right) = \phi A_{s} + \left[ {\overline{a}\left( \omega \right) - \phi } \right]{{S_{g}^{2} } \mathord{\left/ {\vphantom {{S_{g}^{2} } {K_{s} }}} \right. \kern-0pt} {K_{s} }} + \phi {{S_{g} } \mathord{\left/ {\vphantom {{S_{g} } {K_{g} }}} \right. \kern-0pt} {K_{g} }} \hfill \\ A_{23} \left( \omega \right) = \left[ {\overline{a}\left( \omega \right) - \phi } \right]S_{g} , \, A_{24} = 0, \, A_{25} = \phi S_{g} \hfill \\ \end{gathered} \right\}$$

(A.4)

$$A_{s} = \alpha_{v} md\left( {1 - S_{res} } \right)S_{e}^{{{{\left( {m + 1} \right)} \mathord{\left/ {\vphantom {{\left( {m + 1} \right)} m}} \right. \kern-0pt} m}}} \left( {S_{e}^{{ - {1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} m}}} - 1} \right)^{{{{\left( {d - 1} \right)} \mathord{\left/ {\vphantom {{\left( {d - 1} \right)} d}} \right. \kern-0pt} d}}}$$

(A.5)

where $K_{w}$ and $K_{g}$ are bulk modulus of liquid and gas, respectively. $\alpha_{v}$, m, and d are material parameters of the V-G model.

Appendix B

$h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$, and $h_{5}$ in Eqs. (20) and (21) in the text.

The coefficients $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$, and $h_{5}$ in Eqs. (20) and (21) in the text are detailed as

$$h_{1} = a_{1} \left( {a_{8} a_{13} - a_{10} a_{12} } \right) + a_{3} \left( {a_{10} a_{11} - a_{7} a_{13} } \right) + a_{5} \left( {a_{7} a_{12} - a_{8} a_{11} } \right)$$

(B.1)

$$\begin{gathered} h_{2} = a_{3} \left( {a_{6} a_{10} - a_{7} a_{14} } \right) + a_{8} \left[ {a_{1} a_{14} + a_{2} a_{13} - a_{6} \left( {a_{5} + a_{11} } \right)} \right] + a_{11} \left( {a_{4} a_{10} - a_{5} a_{9} } \right) \hfill \\ \, + a_{12} \left( {a_{4} a_{5} + a_{6} a_{7} - a_{2} a_{10} } \right) + a_{13} \left[ {a_{1} a_{9} - a_{4} \left( {a_{3} + a_{7} } \right)} \right] \hfill \\ \end{gathered}$$

(B.2)

$$h_{3} = a_{9} \left[ {a_{1} a_{14} + a_{2} a_{13} - a_{6} \left( {a_{5} + a_{11} } \right)} \right] + a_{4} \left[ {a_{6} \left( {a_{10} + a_{12} } \right) - a_{14} \left( {a_{3} + a_{7} } \right) - a_{4} a_{13} } \right] + a_{8} \left( {a_{2} a_{14} - a_{6}^{2} } \right)$$

(B.3)

$$h_{4} = a_{9} \left( {a_{2} a_{14} - a_{6}^{2} } \right) - a_{4}^{2} a_{14}$$

(B.4)

$$h_{5} = {{\left[ {a_{4}^{2} a_{14} - a_{9} \left( {a_{2} a_{14} - a_{6}^{2} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {a_{4}^{2} a_{14} - a_{9} \left( {a_{2} a_{14} - a_{6}^{2} } \right)} \right]} {\left( {a_{9} a_{14} a_{15} } \right)}}} \right. \kern-0pt} {\left( {a_{9} a_{14} a_{15} } \right)}}$$

(B.5)

Appendix C

h₁₁ ~ h₂₃ in Eq. (60) in the text.

The coefficients h₁₁ ~ h₂₃ in Eq. (60) in the text are detailed as

$$h_{11} = \frac{{\left( {F_{13} F_{24} - F_{14} F_{23} } \right)\left( {F_{21} F_{34} - F_{24} F_{31} } \right) - \left( {F_{11} F_{24} - F_{14} F_{21} } \right)\left( {F_{23} F_{34} - F_{24} F_{33} } \right)}}{{\left( {F_{12} F_{24} - F_{14} F_{22} } \right)\left( {F_{23} F_{34} - F_{24} F_{33} } \right) - \left( {F_{13} F_{24} - F_{14} F_{23} } \right)\left( {F_{22} F_{34} - F_{24} F_{32} } \right)}}$$

(C.1)

$$h_{12} = \frac{{\left( {F_{12} F_{24} - F_{14} F_{22} } \right)\left( {F_{21} F_{34} - F_{24} F_{31} } \right) - \left( {F_{11} F_{24} - F_{14} F_{21} } \right)\left( {F_{22} F_{34} - F_{24} F_{32} } \right)}}{{\left( {F_{13} F_{24} - F_{14} F_{23} } \right)\left( {F_{22} F_{34} - F_{24} F_{32} } \right) - \left( {F_{12} F_{24} - F_{14} F_{22} } \right)\left( {F_{23} F_{34} - F_{24} F_{33} } \right)}}$$

(C.2)

$$h_{13} = - \frac{{F_{11} + F_{12} h_{11} + F_{13} h_{12} }}{{F_{14} }}$$

(C.3)

$$h_{21} = \frac{{\left( {G_{13} G_{24} - G_{14} G_{23} } \right)\left( {G_{21} G_{34} - G_{24} G_{31} } \right) - \left( {G_{11} G_{24} - G_{14} G_{21} } \right)\left( {G_{23} G_{34} - G_{24} G_{33} } \right)}}{{\left( {G_{12} G_{24} - G_{14} G_{22} } \right)\left( {G_{23} G_{34} - G_{24} G_{33} } \right) - \left( {G_{13} G_{24} - G_{14} G_{23} } \right)\left( {G_{22} G_{34} - G_{24} G_{32} } \right)}}$$

(C.4)

$$h_{22} = \frac{{\left( {G_{12} G_{24} - G_{14} G_{22} } \right)\left( {G_{21} G_{34} - G_{24} G_{31} } \right) - \left( {G_{11} G_{24} - G_{14} G_{21} } \right)\left( {G_{22} G_{34} - G_{24} G_{32} } \right)}}{{\left( {G_{13} G_{24} - G_{14} G_{23} } \right)\left( {G_{22} G_{34} - G_{24} G_{32} } \right) - \left( {G_{12} G_{24} - G_{14} G_{22} } \right)\left( {G_{23} G_{34} - G_{24} G_{33} } \right)}}$$

(C.5)

$$h_{23} = - \frac{{G_{11} + G_{12} h_{21} + G_{13} h_{22} }}{{G_{14} }}$$

(C.6)

with

$F_{11} = \beta_{1n} K_{2} \left( {\beta_{1n} r_{1} } \right)$, $F_{12} = \beta_{2n} K_{2} \left( {\beta_{2n} r_{1} } \right)$, $F_{13} = \beta_{3n} K_{2} \left( {\beta_{3n} r_{1} } \right)$, $F_{14} = - \beta_{4n} K_{2} \left( {\beta_{4n} r_{1} } \right)$, $F_{21} = \chi_{w1} \beta_{1n} \left[ {K_{2} \left( {\beta_{1n} r_{1} } \right) + K_{0} \left( {\beta_{1n} r_{1} } \right)} \right]$, $F_{22} = \chi_{w2} \beta_{2n} \left[ {K_{2} \left( {\beta_{2n} r_{1} } \right) + K_{0} \left( {\beta_{2n} r_{1} } \right)} \right]$, $F_{23} = \chi_{w3} \beta_{3n} \left[ {K_{2} \left( {\beta_{3n} r_{1} } \right) + K_{0} \left( {\beta_{3n} r_{1} } \right)} \right]$, $F_{24} = - \chi_{w4} \beta_{4n} \left[ {K_{2} \left( {\beta_{4n} r_{1} } \right) - K_{0} \left( {\beta_{4n} r_{1} } \right)} \right]$, $F_{31} = \chi_{g1} \beta_{1n} \left[ {K_{2} \left( {\beta_{1n} r_{1} } \right) + K_{0} \left( {\beta_{1n} r_{1} } \right)} \right]$, $F_{32} = \chi_{g2} \beta_{2n} \left[ {K_{2} \left( {\beta_{2n} r_{1} } \right) + K_{0} \left( {\beta_{2n} r_{1} } \right)} \right]$, $F_{33} = \chi_{g3} \beta_{3n} \left[ {K_{2} \left( {\beta_{3n} r_{1} } \right) + K_{0} \left( {\beta_{3n} r_{1} } \right)} \right]$, $F_{34} = - \chi_{g4} \beta_{4n} \left[ {K_{2} \left( {\beta_{4n} r_{1} } \right) - K_{0} \left( {\beta_{4n} r_{1} } \right)} \right]$, $G_{11} = \beta_{1n} I_{2} \left( {\beta_{1n} r_{2} } \right)$, $G_{12} = \beta_{2n} I_{2} \left( {\beta_{2n} r_{2} } \right)$, $G_{13} = \beta_{3n} I_{2} \left( {\beta_{3n} r_{2} } \right)$, $G_{14} = - \beta_{4n} I_{2} \left( {\beta_{4n} r_{2} } \right)$, $G_{21} = \chi_{w1} \beta_{1n} \left[ {I_{2} \left( {\beta_{1n} r_{2} } \right) + I_{0} \left( {\beta_{1n} r_{2} } \right)} \right]$, $G_{22} = \chi_{w2} \beta_{2n} \left[ {I_{2} \left( {\beta_{2n} r_{2} } \right) + I_{0} \left( {\beta_{2n} r_{2} } \right)} \right]$, $G_{23} = \chi_{w3} \beta_{3n} \left[ {I_{2} \left( {\beta_{3n} r_{2} } \right) + I_{0} \left( {\beta_{3n} r_{2} } \right)} \right]$, $G_{24} = - \chi_{w4} \beta_{4n} \left[ {I_{2} \left( {\beta_{4n} r_{2} } \right) - I_{0} \left( {\beta_{4n} r_{2} } \right)} \right]$, $G_{31} = \chi_{g1} \beta_{1n} \left[ {I_{2} \left( {\beta_{1n} r_{2} } \right) + I_{0} \left( {\beta_{1n} r_{2} } \right)} \right]$, $G_{32} = \chi_{g2} \beta_{2n} \left[ {I_{2} \left( {\beta_{2n} r_{2} } \right) + I_{0} \left( {\beta_{2n} r_{2} } \right)} \right]$, $G_{33} = \chi_{g3} \beta_{3n} \left[ {I_{2} \left( {\beta_{3n} r_{2} } \right) + I_{0} \left( {\beta_{3n} r_{2} } \right)} \right]$, $G_{34} = - \chi_{g4} \beta_{4n} \left[ {I_{2} \left( {\beta_{4n} r_{2} } \right) - I_{0} \left( {\beta_{4n} r_{2} } \right)} \right]$.

Appendix D

$\varsigma_{1n}$, $\varsigma_{2n}$, $\varsigma_{3n}$, $\varsigma_{4n}$, $\varsigma_{5n}$, $\varsigma_{6n}$, $\varsigma_{7n}$, and $\varsigma_{8n}$ in Eqs. (61) ~ (68) in the text.

The coefficients $\varsigma_{1n}$, $\varsigma_{2n}$, $\varsigma_{3n}$, $\varsigma_{4n}$, $\varsigma_{5n}$, $\varsigma_{6n}$, $\varsigma_{7n}$, and $\varsigma_{8n}$ in Eqs. (61) ~ (68) in the text are detailed as

$$\varsigma_{1n} = \frac{1}{2}\left\{ \begin{gathered} - \beta_{1n} \left[ {K_{2} \left( {\beta_{1n} r} \right) + K_{0} \left( {\beta_{1n} r} \right)} \right] - h_{11} \beta_{2n} \left[ {K_{2} \left( {\beta_{2n} r} \right) + K_{0} \left( {\beta_{2n} r} \right)} \right] \hfill \\ - h_{12} \beta_{3n} \left[ {K_{2} \left( {\beta_{3n} r} \right) + K_{0} \left( {\beta_{3n} r} \right)} \right] + h_{13} \beta_{4n} \left[ {K_{2} \left( {\beta_{4n} r} \right) - K_{0} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.1)

$$\varsigma_{2n} = \frac{1}{2}\left\{ \begin{gathered} - \beta_{1n} \left[ {K_{2} \left( {\beta_{1n} r} \right) - K_{0} \left( {\beta_{1n} r} \right)} \right] - h_{11} \beta_{2n} \left[ {K_{2} \left( {\beta_{2n} r} \right) - K_{0} \left( {\beta_{2n} r} \right)} \right] \hfill \\ - h_{12} \beta_{3n} \left[ {K_{2} \left( {\beta_{3n} r} \right) - K_{0} \left( {\beta_{3n} r} \right)} \right] + h_{13} \beta_{4n} \left[ {K_{2} \left( {\beta_{4n} r} \right) + K_{0} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.2)

$$\varsigma_{3n} = \frac{1}{2}\left\{ \begin{gathered} \beta_{1n} \left[ {I_{2} \left( {\beta_{1n} r} \right) + I_{0} \left( {\beta_{1n} r} \right)} \right] + h_{21} \beta_{2n} \left[ {I_{2} \left( {\beta_{2n} r} \right) + I_{0} \left( {\beta_{2n} r} \right)} \right] \hfill \\ + h_{22} \beta_{3n} \left[ {I_{2} \left( {\beta_{3n} r} \right) + I_{0} \left( {\beta_{3n} r} \right)} \right] - h_{23} \beta_{4n} \left[ {I_{2} \left( {\beta_{4n} r} \right) - I_{0} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.3)

$$\varsigma_{4n} = \frac{1}{2}\left\{ \begin{gathered} \beta_{1n} \left[ {I_{2} \left( {\beta_{1n} r} \right) - I_{0} \left( {\beta_{1n} r} \right)} \right] + h_{21} \beta_{2n} \left[ {I_{2} \left( {\beta_{2n} r} \right) - I_{0} \left( {\beta_{2n} r} \right)} \right] \hfill \\ + h_{22} \beta_{3n} \left[ {I_{2} \left( {\beta_{3n} r} \right) - I_{0} \left( {\beta_{3n} r} \right)} \right] - h_{23} \beta_{4n} \left[ {I_{2} \left( {\beta_{4n} r} \right) + I_{0} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.4)

$$\begin{gathered} \varsigma_{5n} = m_{1} \beta_{1n}^{2} K_{1} \left( {\beta_{1n} r} \right) + m_{2} h_{11} \beta_{2n}^{2} K_{1} \left( {\beta_{2n} r} \right) + m_{3} h_{12} \beta_{3n}^{2} K_{1} \left( {\beta_{3n} r} \right) + m_{4} h_{13} \beta_{4n}^{2} K_{1} \left( {\beta_{4n} r} \right) \hfill \\ \, + m_{4} \left[ {\beta_{1n}^{2} K_{3} \left( {\beta_{1n} r} \right) + h_{11} \beta_{2n}^{2} K_{3} \left( {\beta_{2n} r} \right) + h_{12} \beta_{3n}^{2} K_{3} \left( {\beta_{3n} r} \right) - h_{13} \beta_{4n}^{2} K_{3} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered}$$

(D.5)

$$\begin{gathered} \varsigma_{6n} = m_{1} \beta_{1n}^{2} I_{1} \left( {\beta_{1n} r} \right) + m_{2} h_{21} \beta_{2n}^{2} I_{1} \left( {\beta_{2n} r} \right) + m_{3} h_{22} \beta_{3n}^{2} I_{1} \left( {\beta_{3n} r} \right) + m_{4} h_{23} \beta_{4n}^{2} I_{1} \left( {\beta_{4n} r} \right) \hfill \\ \, + m_{4} \left[ {\beta_{1n}^{2} I_{3} \left( {\beta_{1n} r} \right) + h_{21} \beta_{2n}^{2} I_{3} \left( {\beta_{2n} r} \right) + h_{22} \beta_{3n}^{2} I_{3} \left( {\beta_{3n} r} \right) - h_{23} \beta_{4n}^{2} I_{3} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered}$$

(D.6)

$$\varsigma_{7n} = \frac{{\overline{\mu }\left( \omega \right)}}{2}\left\{ \begin{gathered} \beta_{1n}^{2} \left[ {K_{3} \left( {\beta_{1n} r} \right) - K_{1} \left( {\beta_{1n} r} \right)} \right] + h_{11} \beta_{2n}^{2} \left[ {K_{3} \left( {\beta_{2n} r} \right) - K_{1} \left( {\beta_{2n} r} \right)} \right] \hfill \\ + h_{12} \beta_{3n}^{2} \left[ {K_{3} \left( {\beta_{3n} r} \right) - K_{1} \left( {\beta_{3n} r} \right)} \right] - h_{13} \beta_{4n}^{2} \left[ {K_{3} \left( {\beta_{4n} r} \right) + K_{1} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.7)

$$\varsigma_{8n} = \frac{{\overline{\mu }\left( \omega \right)}}{2}\left\{ \begin{gathered} \beta_{1n}^{2} \left[ {I_{3} \left( {\beta_{1n} r} \right) - I_{1} \left( {\beta_{1n} r} \right)} \right] + h_{21} \beta_{2n}^{2} \left[ {I_{3} \left( {\beta_{2n} r} \right) - I_{1} \left( {\beta_{2n} r} \right)} \right] \hfill \\ + h_{22} \beta_{3n}^{2} \left[ {I_{3} \left( {\beta_{3n} r} \right) - I_{1} \left( {\beta_{3n} r} \right)} \right] - h_{23} \beta_{4n}^{2} \left[ {I_{3} \left( {\beta_{4n} r} \right) + I_{1} \left( {\beta_{4n} r} \right)} \right] \hfill \\ \end{gathered} \right\}$$

(D.8)

where$m_{1} = \overline{\lambda }\left( \omega \right) + S_{w} C_{21} \left( \omega \right) + S_{g} C_{31} \left( \omega \right) + \chi_{w1} \left[ {S_{w} C_{22} \left( \omega \right) + S_{g} C_{32} \left( \omega \right)} \right] + \chi_{g1} \left[ {S_{w} C_{23} \left( \omega \right) + S_{g} C_{33} \left( \omega \right)} \right] + {{3\overline{\mu }\left( \omega \right)} \mathord{\left/ {\vphantom {{3\overline{\mu }\left( \omega \right)} 2}} \right. \kern-0pt} 2}$,$m_{2} = \overline{\lambda }\left( \omega \right) + S_{w} C_{21} \left( \omega \right) + S_{g} C_{31} \left( \omega \right) + \chi_{w2} \left[ {S_{w} C_{22} \left( \omega \right) + S_{g} C_{32} \left( \omega \right)} \right] + \chi_{g2} \left[ {S_{w} C_{23} \left( \omega \right) + S_{g} C_{33} \left( \omega \right)} \right] + {{3\overline{\mu }\left( \omega \right)} \mathord{\left/ {\vphantom {{3\overline{\mu }\left( \omega \right)} 2}} \right. \kern-0pt} 2}$,$m_{3} = \overline{\lambda }\left( \omega \right) + S_{w} C_{21} \left( \omega \right) + S_{g} C_{31} \left( \omega \right) + \chi_{w3} \left[ {S_{w} C_{22} \left( \omega \right) + S_{g} C_{32} \left( \omega \right)} \right] + \chi_{g3} \left[ {S_{w} C_{23} \left( \omega \right) + S_{g} C_{33} \left( \omega \right)} \right] + {{3\overline{\mu }\left( \omega \right)} \mathord{\left/ {\vphantom {{3\overline{\mu }\left( \omega \right)} 2}} \right. \kern-0pt} 2}$,$m_{4} = {{\overline{\mu }\left( \omega \right)} \mathord{\left/ {\vphantom {{\overline{\mu }\left( \omega \right)} 2}} \right. \kern-0pt} 2}$.

Appendix E

The nomenclature of symbols in the text.

The nomenclature of symbols in the text is listed below:

Roman

A_p:

Cross-section area of pipe pile

A_u:

Kinematic amplification factor of pile head

$\overline{a}(\omega )$ :

Complex moduli of unsaturated soil

E₀ :

Relaxation modulus

E_p :

Young’s modulus of pipe pile

F_p:

Shear force of pipe pile

f :

Frequency

f₁ :

Characteristic frequency of soil

f_s1 :

Horizontal resistance of outer soil to pipe pile

f_s2 :

Horizontal resistance of inner soil to pipe pile

G_p :

Shear modulus of pipe pile

H :

Pile length

I₀() :

First kind of zero-order modified Bessel function

I_p:

Moment of inertia for pipe pile

I_u:

Horizontal kinematic interaction factor of pile-soil system

i :

Imaginary unit

K :

Modified shear factor of pipe pile

K₀() :

Second kind of zero-order modified Bessel function

K_b0 :

Relaxation bulk moduli of soil skeleton

K_g :

Bulk modulus of gas

K_s :

Bulk moduli of soil particles

K_w :

Bulk modulus of liquid

k :

Intrinsic permeability

k_rw :

Relative permeability of liquid

k_rg :

Relative permeability of gas

M_p:

Bending moment of pipe pile

m, α_v, d :

Material parameters of V-G model

P :

Vertical load due to the mass of superstructure

p_g:

Gas pressure

p_w:

Liquid pressure

r :

Fractional-order index of FSLS model

r₁ :

Outer radius of pipe pile

r₂ :

Inner radius of pipe pile

S_e :

Effective liquid saturation

S_g :

Gas saturation

S_res:

Liquid saturation at residual state

S_w :

Liquid saturation

t :

Time

U_F :

Displacement of free-field soil relative to bedrock

U_sh:

Horizontal displacement of SH-wave

u_F :

Free-field displacement of soil

u_i :

Displacement component of soil skeleton

u_p :

Horizontal displacement amplitude of pipe pile

u_sh:

Horizontal displacement amplitude of SH-wave

V_s:

Shear velocity of soil

v_i :

Relative displacement component of gas

w_i :

Relative displacement component of liquid

Greek

$\overline{\varepsilon }(\omega )$ :

Strain after Fourier transformation

$\hat{\varepsilon }_{s}$ :

Volumetric strain of soil skeleton

$\hat{\varepsilon }_{w}$ :

Relative volumetric strains of liquid

$\hat{\varepsilon }_{g}$ :

Relative volumetric strains of gas

θ_p :

Rotation angle of pipe pile

$\overline{\lambda }(\omega )$ :

Complex moduli of unsaturated soil

μ :

Shear modulus of soil

μ₀ :

Relaxation shear moduli of soil skeleton

$\overline{\mu }(\omega )$ :

Complex moduli of unsaturated soil

μ_g:

Dynamic viscosity of gas

μ_w:

Dynamic viscosity of liquid

ν_p:

Poisson’s ratio of pipe pile

ρ :

Soil density

ρ_g :

Absolute mass density of the gas

ρ_p :

Density of pipe pile

ρ_s :

Absolute mass density of the solid

ρ_w :

Absolute mass density of the liquid

$\overline{\sigma }(\omega )$ :

Stress after Fourier transformation

$\hat{\sigma }_{r}$ :

Normal stress of unsaturated soil

$\hat{\sigma }_{r\theta }$ :

Shear stress of unsaturated soil

τ_ε :

Strain relaxation time

τ_σ :

Stress relaxation time

φ_g:

Scalar potential of gas

φ_s:

Scalar potential of solid

φ_w:

Scalar potential of liquid

ϕ :

Porosity of unsaturated soil

ψ_g:

Vector potential of gas

ψ_s:

Vector potential of solid

ψ_w:

Vector potential of liquid

ω :

Angular frequency

∇² :

Laplace operator