Modeling of ground vibrations from a tunnel in layered unsaturated soil with spatial variability

Abstract

This paper presents a method for modeling the ground vibrations from a railway tunnel in layered unsaturated soil, considering the uncertainty and spatial variability of soil parameters. A deterministic ground vibration prediction model, which is an improved Euler beam model, is developed to evaluate the vibrations for a tunnel in layered unsaturated soil. Furthermore, the spatial variability of soil parameters is simulated by random fields using the Monte Carlo theory and the middle point method of Cholesky decomposition. By coupling the random fields of soil parameters to the deterministic vibration prediction model, the effect of uncertainty and spatial variability of soil on the ground vibrations is demonstrated through a case study. It is found that the variability of soil parameters has little influence on the spatial distribution regularities of ground vibrations, but it has a significant effect on the dynamic response amplitude and the critical velocity of the system.

Appendices

Appendix A

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

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

Appendix B

$$\begin{gathered} D_{u}^{(i)} = \left[ {\begin{array}{*{20}c} {ik_{z} } & {ik_{z} } & {ik_{z} } & {ik_{sx}^{(i)} } \\ {ik_{p1x}^{(i)} } & {ik_{p2x}^{(i)} } & {ik_{p3x}^{(i)} } & { - ik_{z}^{(i)} } \\ {\mu_{l1}^{(i)} ik_{p1x}^{(i)} } & {\mu_{l2}^{(i)} ik_{p2x}^{(i)} } & {\mu_{l3}^{(i)} ik_{p3x}^{(i)} } & { - \mu_{ls}^{(i)} ik_{z} } \\ {\mu_{g1}^{(i)} ik_{p1x}^{(i)} } & {\mu_{g2}^{(i)} ik_{p2x}^{(i)} } & {\mu_{g3} ik_{p3x}^{(i)} } & { - \mu_{gs}^{(i)} ik_{z} } \\ \end{array} } \right] \hfill \\ D_{d}^{(i)} = \left[ {\begin{array}{*{20}c} {ik_{z} } & {ik_{z} } & {ik_{z} } & { - ik_{sx}^{(i)} } \\ { - ik_{p1x}^{(i)} } & { - ik_{p2x}^{(i)} } & { - ik_{p3x}^{(i)} } & { - ik_{z}^{(i)} } \\ { - \mu_{l1}^{(i)} ik_{p1x}^{(i)} } & { - \mu_{l2}^{(i)} ik_{p2x}^{(i)} } & { - \mu_{l3}^{(i)} ik_{p3x}^{(i)} } & { - \mu_{ls}^{(i)} ik_{z} } \\ { - \mu_{g1}^{(i)} ik_{p1x}^{(i)} } & { - \mu_{g2}^{(i)} ik_{p2x}^{(i)} } & { - \mu_{g3}^{(i)} ik_{p3x}^{(i)} } & { - \mu_{gs}^{(i)} ik_{z} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$

$$\begin{gathered} S_{u}^{(i)} = \left[ {\begin{array}{*{20}c} { - \;2\mu^{(i)} k_{z} k_{p1x}^{(i)} } & { - \;2\mu^{(i)} k_{z} k_{p2x}^{(i)} } & { - \;2\mu^{(i)} k_{z} k_{p3x}^{(i)} } & { - \;\mu^{(i)} k_{sx}^{2(i)} + \mu^{(i)} k_{z}^{2} } \\ { - \;T_{1}^{(i)} } & { - \;T_{2}^{(i)} } & { - \;T_{3}^{(i)} } & {2\mu^{(i)} k_{z} k_{sx}^{(i)} } \\ {A_{p1}^{l(i)} k_{p1}^{2(i)} } & {A_{p2}^{l(i)} k_{p2}^{2(i)} } & {A_{p3}^{l(i)} k_{p3}^{2(i)} } & 0 \\ {A_{p1}^{g(i)} k_{p1}^{2(i)} } & {A_{p2}^{g(i)} k_{p2}^{2(i)} } & {A_{p3}^{g(i)} k_{p3}^{2(i)} } & 0 \\ \end{array} } \right] \hfill \\ S_{d}^{(i)} = \left[ {\begin{array}{*{20}c} {2\mu^{(i)} k_{z} k_{p1x}^{(i)} } & {2\mu^{(i)} k_{z} k_{p2x}^{(i)} } & {2\mu^{(i)} k_{z} k_{p3x}^{(i)} } & { - \;\mu^{(i)} k_{sx}^{2(i)} + \mu^{(i)} k_{z}^{2} } \\ { - \;T_{1}^{(i)} } & { - \;T_{2}^{(i)} } & { - \;T_{3}^{(i)} } & { - \;2\mu^{(i)} k_{z} k_{sx}^{(i)} } \\ {A_{p1}^{l(i)} k_{p1}^{2(i)} } & {A_{p2}^{l(i)} k_{p2}^{2(i)} } & {A_{p3}^{l(i)} k_{p3}^{2(i)} } & 0 \\ {A_{p1}^{g(i)} k_{p1}^{2(i)} } & {A_{p2}^{g(i)} k_{p2}^{2(i)} } & {A_{p3}^{g(i)} k_{p3}^{2(i)} } & 0 \\ \end{array} } \right] \hfill \\ \end{gathered}$$

$$\begin{gathered} E_{u}^{(i)} = \left[ {\begin{array}{*{20}c} {e^{{ixk_{p1x}^{(i)} }} } & 0 & 0 & 0 \\ 0 & {e^{{ixk_{p2x}^{(i)} }} } & 0 & 0 \\ 0 & 0 & {e^{{ixk_{p3x}^{(i)} }} } & 0 \\ 0 & 0 & 0 & {e^{{ixk_{sx}^{(i)} }} } \\ \end{array} } \right] \hfill \\ E_{d}^{(i)} = \left[ {\begin{array}{*{20}c} {e^{{ - ixk_{p1x}^{(i)} }} } & 0 & 0 & 0 \\ 0 & {e^{{ - ixk_{p2x}^{(i)} }} } & 0 & 0 \\ 0 & 0 & {e^{{ - ixk_{p3x}^{(i)} }} } & 0 \\ 0 & 0 & 0 & {e^{{ - ixk_{sx}^{(i)} }} } \\ \end{array} } \right] \hfill \\ A_{u}^{(i)} = \left[ {\begin{array}{*{20}c} {A_{u1}^{(i)} } & {A_{u2}^{(i)} } & {A_{u3}^{(i)} } & {A_{u4}^{(i)} } \\ \end{array} } \right]^{T} \hfill \\ A_{d}^{(i)} = \left[ {\begin{array}{*{20}c} {A_{d1}^{(i)} } & {A_{d2}^{(i)} } & {A_{d3}^{(i)} } & {A_{d4}^{(i)} } \\ \end{array} } \right]^{T} \hfill \\ \end{gathered}$$

$$\left\{ \begin{gathered} A_{p1}^{l} = (b_{11} + b_{12} \mu_{l1} + b_{13} \mu_{g1} )(k_{z}^{2} + k_{p1x}^{2} ) \hfill \\ A_{p2}^{l} = (b_{11} + b_{12} \mu_{l2} + b_{13} \mu_{g2} )(k_{z}^{2} + k_{p2x}^{2} ) \hfill \\ A_{p3}^{l} = (b_{11} + b_{12} \mu_{l3} + b_{13} \mu_{g3} )(k_{z}^{2} + k_{p3x}^{2} ) \hfill \\ A_{p1}^{g} = (b_{21} + b_{22} \mu_{l1} + b_{23} \mu_{g1} )(k_{z}^{2} + k_{p1x}^{2} ) \hfill \\ A_{p2}^{g} = (b_{21} + b_{22} \mu_{l2} + b_{23} \mu_{g2} )(k_{z}^{2} + k_{p2x}^{2} ) \hfill \\ A_{p3}^{g} = (b_{21} + b_{22} \mu_{l3} + b_{23} \mu_{g3} )(k_{z}^{2} + k_{p3x}^{2} ) \hfill \\ T_{1} = (\lambda + 2\mu )k_{p1x}^{2} + \lambda k_{z}^{2} + ak_{p1}^{2} (\gamma A_{p1}^{l} - \gamma A_{p1}^{g} + A_{p1}^{g} ) \hfill \\ T_{2} = (\lambda + 2\mu )k_{p2x}^{2} + \lambda k_{z}^{2} + ak_{p2}^{2} (\gamma A_{p2}^{l} - \gamma A_{p2}^{g} + A_{p2}^{g} ) \hfill \\ T_{3} = (\lambda + 2\mu )k_{p3x}^{2} + \lambda k_{z}^{2} + ak_{p3}^{2} (\gamma A_{p3}^{l} - \gamma A_{p3}^{g} + A_{p3}^{g} ) \hfill \\ \end{gathered} \right.$$