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Three-dimensional analytical model for vibrations from a tunnel embedded in an unsaturated half-space

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Abstract

An analytical model for predicting the vibrations from an underground railway tunnel embedded in an unsaturated half-space is proposed. The tunnel lining is modeled as an infinite Flügge cylindrical shell, and the unsaturated soil is modeled as a three-phase medium comprising solid grains and pores containing water and air. By using the transformation properties between the plane wave functions and the cylindrical wave functions, the model is coupled based on the boundary conditions. The developed model is validated by comparison with existing tunnel models, and the effect of saturation on the dynamic response of the tunnel–soil system is demonstrated through a case study.

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Acknowledgements

The study on which this paper is based was supported by the National Natural Science Foundation of China through the Grant No. 51808405.

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Authors and Affiliations

  1. Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai, 201804, China

    Honggui Di, Shunhua Zhou, Huiji Guo, Chao He & Xiaohui Zhang

  2. Shanghai Key Laboratory of Rail Infrastructure Durability and System Safety, Tongji University, Shanghai, 201804, China

    Honggui Di, Shunhua Zhou, Huiji Guo, Chao He & Xiaohui Zhang

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  1. Honggui Di
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Correspondence to Huiji Guo.

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Appendices

Appendix A

$$ \begin{aligned} & a_{11} = \frac{{c_{1} A_{22} }}{{A_{11} A_{22} - A_{12} A_{21} }},\quad a_{12} = \frac{{A_{22} A_{13} - A_{12} A_{23} }}{{A_{11} A_{22} - A_{12} A_{21} }}, \\ & a_{13} = \frac{{A_{22} A_{14} - A_{12} A_{24} }}{{A_{11} A_{22} - A_{12} A_{21} }},\quad a_{21} = - \frac{{c_{1} A_{21} }}{{A_{11} A_{22} - A_{12} A_{21} }}, \\ & a_{22} = \frac{{A_{11} A_{23} - A_{21} A_{13} }}{{A_{11} A_{22} - A_{12} A_{21} }},\quad a_{23} = \frac{{A_{11} A_{24} - A_{21} A_{14} }}{{A_{11} A_{22} - A_{12} A_{21} }}, \\ \end{aligned} $$
$$ \begin{aligned} & A_{11} = \frac{{c_{2} \gamma - n_{0} S_{r} }}{{K_{s} }} + \frac{{n_{0} S_{r} }}{{K_{l} }},\quad A_{12} = \frac{{c_{2} (1 - \gamma ) - n_{0} (1 - S_{r} )}}{{K_{s} }} + \frac{{n_{0} (1 - S_{r} )}}{{K_{g} }}, \\ & A_{13} = n_{0} S_{r} ,\quad A_{14} = n_{0} (1 - S_{r} ),\quad A_{21} = A_{s} - \frac{{S_{r} (1 - S_{r} )}}{{K_{l} }}, \\ & A_{22} = \frac{{S_{r} (1 - S_{r} )}}{{K_{g} }} - A_{s} ,\quad A_{23} = - A_{24} = - S_{r} (1 - S_{r} ), \\ & c_{1} = 1 - n_{0} - \frac{{K_{b} }}{{K_{s} }},\quad c_{2} = 1 - \frac{{K_{b} }}{{K_{s} }},\quad A_{s} = - \alpha_{1} \alpha_{2} \alpha_{3} (1 - S_{w0} )(S_{e} )^{{\frac{{\alpha_{2} + 1}}{{\alpha_{2} }}}} [(S_{e} )^{{ - \frac{1}{{\alpha_{2} }}}} - 1]^{{\frac{{\alpha_{3} - 1}}{{\alpha_{3} }}}} \\ & K_{b} = \lambda + \frac{2}{3}\mu ,\quad S_{e} = \frac{{S_{r} - S_{w0} }}{{1 - S_{w0} }},\quad \mu = \mu_{s} + \frac{2050}{\alpha }\ln \left( {\sqrt {Se^{ - 2} - 1} + Se^{ - 1} } \right)\tan \varphi , \\ & \lambda = \frac{{2\upsilon_{s} \mu }}{{1 - 2\upsilon_{s} }} \\ \end{aligned} $$

Kl and Kg represent the compression modulus of water and air, respectively. α1, α2, α3 represent the fitting parameters of the VG model curve. υs represents Poisson’s ratio of soil. φ and μs represent the internal friction angle and dynamic shear modulus of saturated soil, respectively.

Appendix B

$$ \begin{aligned} & \lambda_{c} = \lambda + c_{2} \gamma b_{11} + c_{2} (1 - \gamma )b_{21} ,\quad M = c_{2} \gamma b_{12} + c_{2} (1 - \gamma )b_{22} , \\ & N = c_{2} \gamma b_{13} + c_{2} (1 - \gamma )b_{23} ,\quad b_{11} = \frac{{c_{2} a_{12} }}{{c_{1} }} ,\quad b_{12} = \frac{1}{{n_{0} S_{r} }}a_{12} , \\ & b_{13} = \frac{1}{{n_{0} (1 - S_{r} )}}a_{13} ,\quad b_{21} = \frac{{c_{2} a_{21} }}{{c_{1} }} ,\quad b_{22} = \frac{1}{{n_{0} S_{r} }}a_{22} ,\quad b_{23} = \frac{1}{{n_{0} (1 - S_{r} )}}a_{23} , \\ & \vartheta_{l} = \frac{{\rho_{l} }}{{n_{0} S_{r} }} ,\quad \vartheta_{g} = \frac{{\rho_{g} }}{{n_{0} (1 - S_{r} )}} ,\quad d_{l} = \frac{{\eta_{l} }}{{k_{rl} \kappa }} ,\quad d_{g} = \frac{{\eta_{g} }}{{k_{rg} \kappa }}, \\ & k_{rl} = \sqrt {S_{e} } \left\{ {1 - [1 - (S_{e} )^{{1/\alpha_{2} }} ]^{{\alpha_{2} }} } \right\}^{2} ,\quad k_{rg} = \sqrt {1 - S_{e} } [1 - (S_{e} )^{{1/\alpha_{2} }} ]^{{2\alpha_{2} }} , \\ & p = \gamma p_{l} + (1 - \gamma )p_{g} \\ \end{aligned} $$

Appendix C

$$ \begin{aligned} & D_{1} = \lambda_{c} + 2\mu ,\quad D_{2} { = }M ,\quad D_{3} { = }N ,\quad D_{4} = b_{11} ,\quad D_{5} = b_{12} ,\quad D_{6} = b_{13} ,\quad D_{7} = b_{21}, \\ & D_{8} = b_{22} ,\quad D_{9} = b_{23} ,\quad C_{1} = \omega^{2} \rho ,\quad C_{2} = \omega^{2} \rho_{l} ,\quad C_{3} = \omega^{2} \rho_{g} ,\quad C_{4} = \omega^{2} \rho_{l}, \\ & C_{5} = \omega^{2} \vartheta_{1} - {\text{i}}\omega d_{l} ,\quad C_{6} = 0 ,\quad C_{7} = \omega^{2} \rho_{g} ,\quad C_{8} = 0 ,\quad C_{9} = \omega^{2} \vartheta g - {\text{i}}\omega d_{g}, \\ & Ba = D_{1} D_{5} D_{9} + D_{2} D_{6} D_{7} + D_{3} D_{4} D_{8} - (D_{3} D_{5} D_{7} + D_{1} D_{6} D_{8} + D_{2} D_{4} D_{9} ), \\ & Bb = D_{1} D_{5} C_{9} + D_{1} D_{9} C_{5} + D_{5} D_{9} C_{1} + D_{2} D_{6} C_{7} + D_{2} D_{7} C_{6} + D_{6} D_{7} C_{2} \\ & \quad \quad + \;D_{3} D_{4} C_{8} + D_{3} D_{8} C_{4} + D_{4} D_{8} C_{3} - (D_{3} D_{5} C_{7} + D_{3} D_{7} C_{5} + D_{5} D_{7} C_{3} \\ & \quad \quad + \;D_{1} D_{6} C_{8} + D_{1} D_{8} C_{6} + D_{6} D_{8} C_{1} + D_{2} D_{4} C_{9} + D_{2} D_{9} C_{4} + D_{4} D_{9} C_{2} ), \\ & Bc = D_{1} C_{5} C_{9} + D_{5} C_{1} C_{9} + D_{9} C_{1} C_{5} + D_{2} C_{6} C_{7} + D_{6} C_{2} C_{7} + D_{7} C_{2} C_{6} \\ & \quad \quad + \;D_{3} C_{4} C_{8} + D_{4} C_{3} C_{8} + D_{8} C_{3} C_{4} - (D_{3} C_{5} C_{7} + D_{5} C_{3} C_{7} + D_{7} C_{3} C_{5} \\ & \quad \quad + \;D_{1} C_{6} C_{8} + D_{6} C_{1} C_{8} + D_{8} C_{1} C_{6} + D_{2} C_{4} C_{9} + D_{4} C_{2} C_{9} + D_{9} C_{2} C_{4} ), \\ & Bd = C_{1} C_{5} C_{9} + C_{2} C_{6} C_{7} + C_{3} C_{4} C_{8} - (C_{3} C_{5} C_{7} + C_{1} C_{6} C_{8} + C_{2} C_{4} C_{9} ), \\ & Be = \frac{{C_{1} C_{5} C_{9} - C_{3} C_{5} C_{7} - C_{2} C_{4} C_{9} }}{{\mu C_{5} C_{9} }} \\ \end{aligned} $$

Appendix D

$$ \begin{aligned} & Bp = \frac{{3Ba*Bc - Bb^{2} }}{{3Ba^{2} }} ,\quad Bq = \frac{{27Ba^{2} Bd - 9Ba*Bb*Bc + 2Bb^{3} }}{{27Ba^{3} }} ,\quad Bw = \frac{{ - 1 + \sqrt 3 {\text{i}}}}{2}, \\ & k_{1} = \sqrt[3]{{ - \frac{Bq}{2} + \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }} + \sqrt[3]{{ - \frac{Bq}{2} - \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }}, \\ & k_{2} = Bw\sqrt[3]{{ - \frac{Bq}{2} + \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }} + Bw^{2} \sqrt[3]{{ - \frac{Bq}{2} - \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }}, \\ & k_{3} = Bw^{2} \sqrt[3]{{ - \frac{Bq}{2} + \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }} + Bw\sqrt[3]{{ - \frac{Bq}{2} - \sqrt {\left( {\frac{Bq}{2}} \right)^{2} + \left( {\frac{Bp}{3}} \right)^{3} } }}, \\ & k_{p1}^{2} = k_{1} - \frac{Bb}{3Ba} ,\quad k_{p2}^{2} = k_{2} - \frac{Bb}{3Ba} ,\quad k_{p3}^{2} = k_{3} - \frac{Bb}{3Ba}, \\ & k_{s}^{2} = \frac{{C_{1} C_{5} C_{9} - C_{3} C_{5} C_{7} - C_{2} C_{4} C_{9} }}{{\mu C_{5} C_{9} }} \\ \end{aligned} $$

Appendix E

$$ \begin{aligned} & Bx1 = D1*( - 1*k_{p1}^{2} ) + C1,\quad Bx2 = D2*( - 1*k_{p1}^{2} ) + C2, \\ & Bx3 = D3*( - 1*k_{p1}^{2} ) + C3,\quad Bx4 = D4*( - 1*k_{p1}^{2} ) + C4, \\ & Bx5 = D5*( - 1*k_{p1}^{2} ) + C5,\quad Bx6 = D6*( - 1*k_{p1}^{2} ) + C6, \\ & By1 = D1*( - 1*k_{p2}^{2} ) + C1,\quad By2 = D2*( - 1*k_{p2}^{2} ) + C2, \\ & By3 = D3*( - 1*k_{p2}^{2} ) + C3,\quad By4 = D4*( - 1*k_{p2}^{2} ) + C4, \\ & By5 = D5*( - 1*k_{p2}^{2} ) + C5,\quad By6 = D6*( - 1*k_{p2}^{2} ) + C6, \\ & Bz1 = D1*( - 1*k_{p3}^{2} ) + C1,\quad Bz2 = D2*( - 1*k_{p3}^{2} ) + C2, \\ & Bz3 = D3*( - 1*k_{p3}^{2} ) + C3,\quad Bz4 = D4*( - 1*k_{p3}^{2} ) + C4, \\ & Bz5 = D5*( - 1*k_{p3}^{2} ) + C5,\quad Bz6 = D6*( - 1*k_{p3}^{2} ) + C6, \\ & \mu_{1l} = \frac{Bx3*Bx4 - Bx1*Bx6}{Bx2*Bx6 - Bx3*Bx5},\quad \mu_{2l} = \frac{By3*By4 - By1*By6}{By2*By6 - By3*By5}, \\ & \mu_{3l} = \frac{Bz3*Bz4 - Bz1*Bz6}{Bz2*Bz6 - Bz3*Bz5},\quad \mu_{1g} = \frac{Bx2*Bx4 - Bx1*Bx5}{Bx3*Bx5 - Bx2*Bx6}, \\ & \mu_{2g} = \frac{By2*By4 - By1*By5}{By3*By5 - By2*By6},\quad \mu_{3g} = \frac{Bz2*Bz4 - Bz1*Bz5}{Bz3*Bz5 - Bz2*Bz6}, \\ & \mu_{tl} = - \frac{C4}{C5},\quad \mu_{tg} = - \frac{C7}{C9} \\ \end{aligned} $$

Appendix F

$$ \begin{aligned} & A_{p1}^{l} = - (b_{11} + b_{12} \mu_{1l} + b_{13} \mu_{1g} ), \\ & A_{p2}^{l} = - (b_{11} + b_{12} \mu_{2l} + b_{13} \mu_{2g} ), \\ & A_{p3}^{l} = - (b_{11} + b_{12} \mu_{3l} + b_{13} \mu_{3g} ), \\ & A_{p1}^{g} = - (b_{21} + b_{22} \mu_{1l} + b_{23} \mu_{1g} ), \\ & A_{p2}^{g} = - (b_{21} + b_{22} \mu_{2l} + b_{23} \mu_{2g} ), \\ & A_{p3}^{g} = - (b_{21} + b_{22} \mu_{3l} + b_{23} \mu_{3g} ) \\ \end{aligned} $$

Appendix G

$$ \begin{aligned} & E1 = - a\gamma (b_{11} + b_{12} \mu_{1l} + b_{13} \mu_{1g} ), \\ & E2 = - a\gamma (b_{11} + b_{12} \mu_{2l} + b_{13} \mu_{2g} ), \\ & E3 = - a\gamma (b_{11} + b_{12} \mu_{3l} + b_{13} \mu_{3g} ), \\ & E4 = - a(1 - \gamma )(b_{21} + b_{22} \mu_{1l} + b_{23} \mu_{1g} ), \\ & E5 = - a(1 - \gamma )(b_{21} + b_{22} \mu_{2l} + b_{23} \mu_{2g} ), \\ & E6 = - a(1 - \gamma )(b_{21} + b_{22} \mu_{3l} + b_{23} \mu_{3g} ) \\ \end{aligned} $$

Appendix H

$$ \begin{aligned} & \varvec{S} = \frac{ - Eh}{{r_{1} (1 - \upsilon^{2} )}}\left[ {\begin{array}{*{20}c} {s_{11} } &\quad {s_{12} } &\quad {s_{13} } \\ {s_{21} } &\quad {s_{22} } &\quad {s_{23} } \\ {s_{31} } &\quad {s_{32} } &\quad {s_{33} } \\ \end{array} } \right], \\ & s_{11} { = }\frac{{\rho_{t} r_{1} (1 - \upsilon^{2} )}}{E}\omega^{2} - r_{1} \xi^{2} - \frac{(1 - \upsilon )}{{2r_{1} }}\left( {1 + \frac{{h^{2} }}{{12r_{1}^{2} }}} \right)m^{2} , \\ & s_{12} = - \frac{(1 + \upsilon )}{2}\xi m ,\quad s_{13} = - \upsilon i\xi + \frac{{h^{2} }}{12}({\text{i}}\xi )^{3} + \frac{{h^{2} }}{{12r_{1}^{2} }}\frac{(1 - \upsilon )}{2}{\text{i}}\xi m^{2} , \\ & s_{21} = - \frac{(1 + \upsilon )}{2}\xi m, \\ & s_{22} = \frac{{\rho_{t} r{}_{1}(1 - \upsilon^{2} )}}{E}\omega^{2} - \frac{{r_{1} (1 - \upsilon )}}{2}\left( {1 + \frac{{h^{2} }}{{4r_{1}^{2} }}} \right)\xi^{2} - \frac{1}{{r_{1} }}m^{2} , \\ & s_{23} = - \frac{\text{i}}{{r_{1} }}m - {\text{i}}\frac{{h^{2} }}{12}\frac{(3 - \upsilon )}{{2r_{1} }}\xi^{2} m ,\quad s_{31} = \upsilon {\text{i}}\xi - \frac{{h^{2} }}{12}({\text{i}}\xi )^{3} - \frac{{h^{2} }}{{12r_{1}^{2} }}\frac{(1 - \upsilon )}{2}{\text{i}}\xi m^{2} , \\ & s_{32} = \frac{\text{i}}{{r_{1} }}m + {\text{i}}\frac{{h^{2} }}{12}\frac{(3 - \upsilon )}{{2r_{1} }}\xi^{2} m, \\ & s_{33} = \frac{{\rho_{t} r_{1} (1 - \upsilon^{2} )}}{E}\omega^{2} - \frac{{h^{2} }}{12}\left( {r_{1} \xi^{4} + \frac{2}{{r_{1} }}\xi^{2} m^{2} + \frac{1}{{r_{1}^{3} }}m^{4} } \right) - \frac{1}{{r_{1} }} + \frac{{h^{2} }}{{12r_{1}^{3} }}(2m^{2} - 1) \\ \end{aligned} $$

r1, h, E, υ, and ρt represent the radius, thickness, Young’s modulus, Poisson’s ratio and density of the shell, respectively. qr, qθ, and qz represent the net stresses along the r-, θ-, and z-directions of the central surface of the lining shell, respectively.

Appendix I

$$ I_{mj} = i^{ - m} \left\{ {\begin{array}{*{20}l} {e^{{im\beta_{s} }} }, \hfill \\ {e^{{im\beta_{s} }} }, \hfill \\ {e^{{im\beta_{p1} }} }, \hfill \\ {e^{{im\beta_{p2} }} }, \hfill \\ {e^{{im\beta_{p3} }} }, \hfill \\ \end{array} \begin{array}{*{20}l} {\beta_{s} = \arcsin (k_{y} /k_{sr} ),} \hfill \\ {\beta_{s} = \arcsin (k_{y} /k_{sr} ),} \hfill \\ {\beta_{p1} = \arcsin (k_{y} /k_{p1r} ),} \hfill \\ {\beta_{p2} = \arcsin (k_{y} /k_{p2r} ),} \hfill \\ {\beta_{p3} = \arcsin (k_{y} /k_{p3r} ),} \hfill \\ \end{array} \begin{array}{*{20}l} {h_{j} = k_{sx} }, \hfill \\ {h_{j} = k_{sx} }, \hfill \\ {h_{j} = k_{p1x} }, \hfill \\ {h_{j} = k_{p2x} }, \hfill \\ {h_{j} = k_{p3x} }, \hfill \\ \end{array} \begin{array}{*{20}l} {j = 1} \hfill \\ {j = 2} \hfill \\ {j = 3} \hfill \\ {j = 4} \hfill \\ {j = 5} \hfill \\ \end{array} } \right. $$

Appendix J

$$ \left[ {K_{{jj^{\prime}}} } \right] = - 2\left[ {\begin{array}{*{20}c} {\hat{\bar{\tilde{U}}}_{1}^{d} } &\quad {\hat{\bar{\tilde{U}}}_{2}^{d} } &\quad {\hat{\bar{\tilde{U}}}_{3}^{d} } &\quad {\hat{\bar{\tilde{U}}}_{4}^{d} } &\quad {\hat{\bar{\tilde{U}}}_{5}^{d} } \\ 0 &\quad 0 &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p1}}^{d} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p2}}^{d} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p3}}^{d} } \\ 0 &\quad 0 &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p1}}^{d} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p2}}^{d} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p3}}^{d} } \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{*{20}c} {\hat{\bar{\tilde{U}}}_{1}^{u} } &\quad {\hat{\bar{\tilde{U}}}_{2}^{u} } &\quad {\hat{\bar{\tilde{U}}}_{3}^{u} } &\quad {\hat{\bar{\tilde{U}}}_{4}^{u} } &\quad {\hat{\bar{\tilde{U}}}_{5}^{u} } \\ 0 &\quad 0 &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p1}}^{u} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p2}}^{u} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{l{\kern 1pt} p3}}^{u} } \\ 0 &\quad 0 &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p1}}^{u} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p2}}^{u} } &\quad {\hat{\bar{\tilde{\vartheta }}}_{{g{\kern 1pt} p3}}^{u} } \\ \end{array} } \right] $$

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Di, H., Zhou, S., Guo, H. et al. Three-dimensional analytical model for vibrations from a tunnel embedded in an unsaturated half-space. Acta Mech 232, 1543–1562 (2021). https://doi.org/10.1007/s00707-020-02892-4

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