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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (7): 957-966.  DOI: 10.11908/j.issn.0253-374x.2019.07.007 0

### 引用本文

GAO Guangyun, YAO Shaofeng, SUN Yuming, YANG Chengbin. Unsaturated Ground Vibration Induced by High-speed Train Loads Based on 2.5D Finite Element Method[J]. Journal of Tongji University (Natural Science), 2019, 47(7): 957-966. DOI: 10.11908/j.issn.0253-374x.2019.07.007

### 文章历史

2.5维有限元分析高铁荷载诱发非饱和土地面振动

1. 同济大学 土木工程学院，上海 200092;
2. 同济大学 岩土及地下工程教育部重点实验室，上海 200092;
3. 上海应用技术大学 城市建设与安全工程学院，上海 201418;
4. 合肥工业大学 资源与环境工程学院，安徽 合肥 230009

Unsaturated Ground Vibration Induced by High-speed Train Loads Based on 2.5D Finite Element Method
GAO Guangyun 1,2, YAO Shaofeng 1,2, SUN Yuming 3, YANG Chengbin 4
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China;
3. College of Urban Construction and Safety Engineering, Shanghai Institute of Technology, Shanghai 201418, China;
4. School of Resources and Environmental Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: A two-and-a-half-dimension finite element method (2.5D FEM) was established to investigate the surface vibration of the unsaturated ground subjected to moving loads caused by high-speed trains. The track structure was simplified as an Euler beam resting on an unsaturated porous half-space. The Galerkin method was used and the governor equations of unsaturated soil of 2.5D in frequency-wavenumber domain was derived by applying the Fourier transform with respect to time and the load moving direction. The influences of train speed and water saturation of unsaturated ground on ground vibration and excess pore water pressure are analyzed. Results show that, at the track center, the displacement amplitude largely decreases when the water saturation decreases from 100% (fully saturated) to 99% (nearly saturated); for a given speed, the ground vibration displacement of unsaturated ground attenuates faster with time than the saturated ground. At 8 m away from the track center, the displacement amplitude of the unsaturated ground is larger than that of the saturated ground when the train speed is below 250 km·h-1; as the speed increasing, the duration time of the unsaturated ground vibration displacement becomes shorter as the speed increases, while of the saturated ground it becomes longer. The displacement amplitude at 200 km·h-1 is larger than the other speed near the track (less than 5m) and attenuates rapidly at an equal rate. The rebound phenomenon of acceleration may occur at some train speed, and the location is strongly related to the train speed. The excess pore water pressure is mainly distributes within 4.5 m below the ground surface and the maximum amplitude is located at 1.5~2.0 m depth and decreases significantly as the water saturation decreases.
Key words: high-speed train    unsaturated subgrade    2.5D finite element method(FEM)    ground vibration    excess pore pressure

1 计算理论与模型 1.1 路基控制方程的2.5D有限元格式

 $\left( {1 - n} \right)\frac{{\partial {\rho _{\rm{s}}}}}{{\partial t}} - {\rho _{\rm{s}}}\frac{{\partial n}}{{\partial t}} + {\rho _{\rm{s}}}\left( {1 - n} \right)\nabla \cdot \dot u = 0$ (1)

 $\frac{1}{\rho_{\mathrm{s}}} \frac{\partial p_{\mathrm{c}}}{\partial t}=\frac{\alpha-n}{K_{\mathrm{g}}} \frac{\partial p_{\mathrm{c}}}{\partial t}-(1-\alpha) \frac{\partial \varepsilon_{\mathrm{s}}}{\partial t}$ (2)

 $\frac{\partial n}{\partial t}=\frac{\alpha-n}{K_{g}} \frac{\partial\left(S_{r} p^{w}+\left(1-S_{r}\right) p^{a}\right)}{\partial t}+(\alpha-n) \nabla \cdot \dot{u}$ (3)

 ${S_{\rm{r}}}{\rho _{\rm{w}}}\frac{{\partial n}}{{\partial t}} + n{S_{\rm{r}}}\frac{{\partial {\rho _{\rm{w}}}}}{{\partial t}} + \eta {\rho _{\rm{w}}}\frac{{\partial {S_{\rm{r}}}}}{{\partial t}} + {\rho _{\rm{w}}}n{S_{\rm{r}}}\nabla \cdot {\dot u^{\rm{w}}} = 0$ (4)

 $\frac{{{\rm{d}}{\rho _{\rm{w}}}}}{{{\rho _{\rm{w}}}}} = \frac{{{\rm{d}}{p^{\rm{w}}}}}{{{K_{\rm{w}}}}}$ (5)

 ${S_{\rm{r}}} = {S_{\rm{r}}}\left( s \right) = {S_{\rm{r}}}\left( {{p^{\rm{a}}} - {p^{\rm{w}}}} \right)$ (6)

 ${A_{11}}{\dot p^{\rm{w}}} + {A_{12}}{\dot p^{\rm{a}}} + {A_{13}}\nabla \dot u + {A_{14}}\nabla {\dot u^{\rm{w}}} = 0$ (7)

 $\begin{array}{*{20}{c}} {\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}\frac{{\partial n}}{{\partial t}} + n\left( {1 - {S_{\rm{r}}}} \right)\frac{{\partial {\rho _{\rm{a}}}}}{{\partial t}} - n{\rho _{\rm{a}}}\frac{{\partial {S_{\rm{r}}}}}{{\partial t}} + }\\ {n\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}\nabla \cdot {{\dot u}^{\rm{a}}} = 0} \end{array}$ (8)

 $\frac{{{\rm{d}}{\rho _{\rm{a}}}}}{{{\rho _{\rm{a}}}}} = \frac{{{\rm{d}}{p^{\rm{a}}}}}{{{p^{\rm{a}}}}}$ (9)

 ${A_{21}}{\dot p^{\rm{w}}} + {A_{22}}{\dot p^{\rm{a}}} + {A_{23}}\nabla \dot u + {A_{24}}\nabla {\dot u^{\rm{a}}} = 0$ (10)
 ${A_{21}} = \frac{{\left( {\alpha - n} \right){S_{\rm{r}}}\left( {1 - {S_{\rm{r}}}} \right)}}{{{K_{\rm{g}}}}} + {A_{{\rm{ss}}}}\left( {n + \frac{{\left( {\alpha - n} \right)\left( {1 - {S_{\rm{r}}}} \right)s}}{{{K_{\rm{g}}}}}} \right),$
 $\begin{array}{*{20}{c}} {{A_{22}} = \frac{{\left( {\alpha - n} \right){{\left( {1 - {S_{\rm{r}}}} \right)}^2}}}{{{K_{\rm{g}}}}} + \frac{{n\left( {1 - {S_{\rm{r}}}} \right)}}{{{p^{\rm{a}}}}} - }\\ {{A_{{\rm{ss}}}}\left( {n + \frac{{\left( {\alpha - n} \right)\left( {1 - {S_{\rm{r}}}} \right)s}}{{{K_{\rm{g}}}}}} \right),} \end{array}$
 ${A_{23}} = \left( {1 - {S_{\rm{r}}}} \right)\left( {\alpha - n} \right),{A_{24}} = n\left( {1 - {S_{\rm{r}}}} \right).$

 $n{S_{\rm{r}}}\left( {\dot u_i^{\rm{w}} - {{\dot u}_i}} \right) = - \frac{{{k_{\rm{w}}}}}{{{\rho _{\rm{w}}}g}}\left( {p_{,i}^{\rm{w}} + {\rho _{\rm{w}}}\ddot u_i^{\rm{w}}} \right)$ (11)
 $n\left( {1 - {S_{\rm{r}}}} \right)\left( {\dot u_i^{\rm{a}} - {{\dot u}_i}} \right) = - \frac{{{k_{\rm{a}}}}}{{{\rho _{\rm{a}}}g}}\left( {p_{,i}^{\rm{a}} + {\rho _{\rm{a}}}\ddot u_i^{\rm{a}}} \right)$ (12)

 $\tilde u_i^{\rm{w}} = \left( {{F_{\rm{w}}}{{\tilde u}_i} - \tilde p_{,i}^{\rm{w}}} \right)/\left( {{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}} \right)$ (13)
 $\tilde u_i^{\rm{a}} = \left( {{F_{\rm{a}}}{{\tilde u}_i} - \tilde p_{,i}^{\rm{a}}} \right)/\left( {{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}} \right)$ (14)

 $\begin{array}{*{20}{c}} {\left( {{A_{13}} + \frac{{{A_{14}}{F_{\rm{w}}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}}} \right){{\tilde u}_{i,i}} - \frac{{{A_{14}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}}\tilde p_{,ii}^{\rm{w}} + }\\ {{A_{11}}{{\tilde p}^{\rm{w}}} + {A_{12}}{{\tilde p}^{\rm{a}}} = 0} \end{array}$ (15)
 $\begin{array}{*{20}{c}} {\left( {{A_{23}} + \frac{{{A_{24}}{F_{\rm{a}}}}}{{{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}}}} \right){{\tilde u}_{i,i}} - \frac{{{A_{24}}}}{{{F_a} - {\rho _a}{\omega ^2}}}\tilde p_{,ii}^{\rm{a}} + }\\ {{A_{21}}{{\tilde p}^{\rm{w}}} + {A_{22}}{{\tilde p}^{\rm{a}}} = 0} \end{array}$ (16)

 $\begin{array}{*{20}{c}} {\mu {u_{i,jj}} + \left( {\lambda + \mu } \right){u_{i,jj}} - {S_r}p_{,i}^{\rm{w}} - \left( {1 - {S_r}} \right)p_{,i}^{\rm{a}} = }\\ {{{\bar \rho }_{\rm{s}}}{{\ddot u}_i} + {{\bar \rho }_{\rm{w}}}\ddot u_i^{\rm{w}} + {{\bar \rho }_{\rm{a}}}\ddot u_i^{\rm{a}}} \end{array}$ (17)

 $\begin{array}{l} \bar \mu {{\tilde u}_{i,jj}} + \left( {\bar \lambda + \bar \mu } \right){{\tilde u}_{i,ji}} - {S_{\rm{r}}}\tilde p_{,i}^{\rm{w}} - \left( {1 - {S_{\rm{r}}}} \right)\tilde p_{,i}^{\rm{a}} + \\ {\omega ^2}\left[ {\left( {1 - n} \right){\rho _{\rm{s}}} + \frac{{n{S_{\rm{r}}}{\rho _{\rm{w}}}{F_{\rm{w}}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}} + \frac{{n\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}{F_{\rm{a}}}}}{{{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}}}} \right]{{\tilde u}_i} - \\ \frac{{{\omega ^2}n{S_{\rm{r}}}{\rho _{\rm{w}}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}}\tilde p_{,i}^{\rm{w}} - \frac{{{\omega ^2}n\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}}}{{{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}}}\tilde p_{,i}^{\rm{a}} = 0 \end{array}$ (18)

 $\sigma_{i j}^{\mathrm{s}} n_{j}=f_{i}, p=p^{\prime},-k_{\mathrm{d}} p_{, j}^{\prime} n_{j}=\rho_{\mathrm{f}} g v_{\mathrm{n}}=q$

 $\begin{array}{l} \int {\left[ {\delta \varepsilon _i^ * {{\tilde \sigma '}_{ij}} - \delta u_i^ * {\omega ^2}\left[ {\left( {1 - n} \right){\rho _{\rm{s}}} + \frac{{n{S_{\rm{r}}}{\rho _{\rm{w}}}{F_{\rm{w}}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}} + } \right.} \right.} \\ \left. {\left. {\frac{{n\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}{F_{\rm{a}}}}}{{{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}}}} \right]{{\tilde u}_i}} \right]{\rm{d}}V + \int {\left( {\delta u_i^ * \frac{{{\omega ^2}n{S_{\rm{r}}}{\rho _{\rm{w}}}}}{{{F_{\rm{w}}} - {\rho _{\rm{w}}}{\omega ^2}}}\tilde p_{,i}^{\rm{w}} - } \right.} \\ \left. {\delta \varepsilon _i^ * {\delta _{ij}}{S_{\rm{r}}}\tilde p_{,i}^{\rm{w}}} \right){\rm{d}}V + \int {\left( {\delta u_i^ * \frac{{{\omega ^2}n\left( {1 - {S_{\rm{r}}}} \right){\rho _{\rm{a}}}}}{{{F_{\rm{a}}} - {\rho _{\rm{a}}}{\omega ^2}}}\tilde p_{,i}^{\rm{a}} - } \right.} \\ \left. {\delta \varepsilon _i^ * {\delta _{ij}}\left( {1 - {S_{\rm{r}}}} \right)\tilde p_{,i}^{\rm{a}}} \right){\rm{d}}V = \int {\delta u_i^ * {f_i}{\rm{d}}S} \end{array}$ (19)

 ${u_i} = \sum\limits_{i = 1}^4 {{N_j}\left( {\eta ,\xi } \right)u_{ij}^{\rm{e}}} ,p = \sum\limits_{i = 1}^4 {{N_j}\left( {\eta ,\xi } \right)p_i^{\rm{e}}}$

 $\begin{array}{*{20}{c}} {\left( {{{\mathit{\boldsymbol{K'}}}_{{\rm{up}}}} - {\mathit{\boldsymbol{M}}_{{\rm{up}}}}} \right)\mathit{\boldsymbol{\tilde {\bar u}}} + \left( {{{\mathit{\boldsymbol{Q'}}}_{{\rm{up}}}} - {\mathit{\boldsymbol{Q}}_{{\rm{up}}}}} \right){{\mathit{\boldsymbol{\tilde {\bar p}}}}^{\rm{w}}} + }\\ {\left( {{{\mathit{\boldsymbol{G'}}}_{{\rm{up}}}} - {\mathit{\boldsymbol{G}}_{{\rm{up}}}}} \right){{\mathit{\boldsymbol{\tilde {\bar p}}}}^{\rm{g}}} = \mathit{\boldsymbol{\tilde {\bar f}}}_{{\rm{up}}}^{\rm{s}}} \end{array}$ (20)

 $\begin{gathered} {{\mathit{\boldsymbol{K'}}}_{{\text{up}}}} = \sum\limits_{\text{e}} {\iint {{{\left( {{\mathit{\boldsymbol{B}}^ * }\mathit{\boldsymbol{N}}} \right)}^{\text{T}}}\mathit{\boldsymbol{D}}\left( {\mathit{\boldsymbol{BN}}} \right)\left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }} ;{\mathit{\boldsymbol{M}}_{{\text{up}}}} = \hfill \\ \;\;\;\;\;\;{\omega ^2}\left[ {\left( {1 - n} \right){\rho _{\text{s}}} + n{S_{\text{r}}}{\rho _{\text{w}}}\frac{{{F_{\text{w}}}}}{{{F_{\text{w}}} - {\rho _{\text{w}}}{\omega ^2}}} + } \right. \hfill \\ \;\;\;\;\;\;\left. {n\left( {1 - {S_{\text{r}}}} \right){\rho _{\text{w}}}\frac{{{F_{\text{a}}}}}{{{F_{\text{a}}} - {\rho _{\text{a}}}{\omega ^2}}}} \right]\sum\limits_{\text{e}} {\iint {\mathit{\boldsymbol{N}}{\mathit{\boldsymbol{N}}^{\text{T}}}\left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }} ; \hfill \\ \end{gathered}$
 ${\mathit{\boldsymbol{Q'}}_{up}} = \frac{{{\omega ^2}n{S_{\text{r}}}{\rho _{\text{w}}}}}{{{F_{\text{w}}} - {\rho _{\text{w}}}{\omega ^2}}}\sum\limits_{\text{e}} {\iint {{\mathit{\boldsymbol{N}}^{\text{T}}}\overline {\mathit{\boldsymbol{BN}}} \left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }} ;$
 ${\mathit{\boldsymbol{Q}}_{up}} = \alpha {S_{\text{r}}}\sum\limits_{\text{e}} {\iint {{{\left( {{\mathit{\boldsymbol{B}}^*}\mathit{\boldsymbol{N}}} \right)}^{\text{T}}}}} \mathit{\boldsymbol{m}}\mathit{\boldsymbol{\overline N}} |J|{\text{d}}\eta {\text{d}}\xi ;$
 ${\mathit{\boldsymbol{G'}}_{{\text{up}}}} = \frac{{{\omega ^2}n\left( {1 - {S_r}} \right){\rho _{\text{a}}}}}{{{F_{\text{a}}} - {\rho _{\text{a}}}{\omega ^2}}}\sum\limits_{\text{e}} {\iint {{N^{\text{T}}}\overline {\mathit{\boldsymbol{BN}}} \left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }} ;$
 ${\mathit{\boldsymbol{G}}_{{\text{up}}}} = \alpha \left( {1 - {S_{\text{r}}}} \right)\sum\limits_{\text{e}} {\iint {{{\left( {{\mathit{\boldsymbol{B}}^ * }\mathit{\boldsymbol{N}}} \right)}^{\text{T}}}\mathit{\boldsymbol{m\bar N}}\left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }} ;$
 $\mathit{\boldsymbol{\tilde {\bar f}}}_{up}^{\text{s}} = \sum\limits_{\text{e}} {\iint {{\mathit{\boldsymbol{N}}^T}\mathit{\boldsymbol{\tilde {\bar f}}}\left| \mathit{\boldsymbol{J}} \right|{\text{d}}\eta {\text{d}}\xi }}$

 $\left( {{\mathit{\boldsymbol{H}}_{{\text{md}}}}} \right)\mathit{\boldsymbol{\tilde {\bar u}}} + \left( {{\mathit{\boldsymbol{Q}}_{{\text{md}}}} + {{\mathit{\boldsymbol{Q'}}}_{{\text{md}}}}} \right){\mathit{\boldsymbol{\tilde {\bar p}}}^w} + \left( {{{\mathit{\boldsymbol{G'}}}_{{\text{md}}}}} \right){\mathit{\boldsymbol{\tilde {\bar p}}}^{\text{g}}} = {\mathit{\boldsymbol{\tilde {\bar f}}}^{\text{w}}}$ (21)
 $\left( {{\mathit{\boldsymbol{H}}_{{\text{dw}}}}} \right)\mathit{\boldsymbol{\tilde {\bar u}}} + \left( {{{\mathit{\boldsymbol{Q'}}}_{{\text{dw}}}}} \right){\mathit{\boldsymbol{\tilde {\bar p}}}^w} + \left( {{\mathit{\boldsymbol{G}}_{{\text{dw}}}} + {{\mathit{\boldsymbol{G'}}}_{{\text{dw}}}}} \right){\mathit{\boldsymbol{\tilde {\bar p}}}^{\text{g}}} = {\mathit{\boldsymbol{\tilde {\bar f}}}^{\text{g}}}$ (22)
 ${\mathit{\boldsymbol{H}}_{{\text{md}}}} = \left( {{A_{13}} + \frac{{{A_{14}}{F_{\text{w}}}}}{{{F_{\text{w}}} - {\rho _{\text{w}}}{\omega ^2}}}} \right)\sum\limits_{\text{e}} {\iint {{{\mathit{\boldsymbol{\overline N}} }^{\text{T}}}}} {\mathit{\boldsymbol{m}}^{\text{T}}}\mathit{\boldsymbol{BN}}|\mathit{\boldsymbol{J}}|{\text{d}}\eta {\text{d}}\xi ;$
 ${\mathit{\boldsymbol{H}}_{{\text{dw}}}} = \left( {{A_{23}} + \frac{{{A_{24}}{F_{\text{a}}}}}{{{F_{\text{a}}} - {\rho _{\text{a}}}{\omega ^2}}}} \right)\sum\limits_{\text{e}} {\iint {{{\mathit{\boldsymbol{\overline N}} }^{\text{T}}}}} {\mathit{\boldsymbol{m}}^{\text{T}}}\mathit{\boldsymbol{BN}}|\mathit{\boldsymbol{J}}|{\text{d}}\eta {\text{d}}\xi ;$
 $\boldsymbol{Q}_{\mathrm{md}}=\frac{A_{14}}{F_{\mathrm{w}}-\rho_{\mathrm{w}} \omega^{2}} \sum\limits_{\mathrm{e}} \iint\left(\boldsymbol{B}_{\mathrm{s}}^{*} \overline{\boldsymbol{N}}\right)^{\mathrm{T}}\left(\boldsymbol{B}_{\mathrm{s}} \overline{\boldsymbol{N}}\right)|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi;$
 $Q_{\mathrm{md}}^{\prime}=A_{11} \sum\limits_{\mathrm{e}} \iint \overline{\boldsymbol{N}}^{\mathrm{T}} \overline{\boldsymbol{N}}|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi;$
 $\boldsymbol{G}_{\mathrm{dw}}=\frac{A_{24}}{F_{\mathrm{a}}-\rho_{\mathrm{a}} \omega^{2}} \sum\limits_{\mathrm{e}} \iint\left(\boldsymbol{B}_{\mathrm{s}}^{*} \overline{\boldsymbol{N}}\right)^{\mathrm{T}}\left(\boldsymbol{B}_{\mathrm{s}} \overline{\boldsymbol{N}}\right)|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi;$
 $\boldsymbol{Q}_{\mathrm{dw}}^{\prime}=A_{21} \sum\limits_{\mathrm{e}} \iint \overline{\boldsymbol{N}}^{\mathrm{T}} \overline{\boldsymbol{N}}|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi;$
 $\boldsymbol{G}_{\mathrm{md}}^{\prime}=A_{12} \sum\limits_{\mathrm{e}} \iint \overline{\boldsymbol{N}}^{\mathrm{T}} \overline{\boldsymbol{N}}|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi$
 $\boldsymbol{G}_{\mathrm{dw}}^{\prime}=A_{22} \sum\limits_{\bf{e}} \iint \overline{\boldsymbol{N}}^{\mathrm{T}} \overline{\boldsymbol{N}}|\boldsymbol{J}| \mathrm{d} \eta \mathrm{d} \xi;$
 $\mathit{\boldsymbol{\tilde {\bar f}}}_{{\text{up}}}^{\text{w}} = - \sum\limits_{\text{e}} {\iint {\frac{{{F_{\text{w}}}{\rho _{\text{w}}}g{{\mathit{\boldsymbol{\tilde v}}}_{\text{n}}}}}{{{k_{\text{w}}}}}}} |\mathit{\boldsymbol{J}}|{\text{d}}\eta {\text{d}}\xi ;$
 $\mathit{\boldsymbol{\tilde {\bar f}}}_{{\text{up}}}^{\text{g}} = - \sum\limits_{\text{e}} {\iint {\frac{{{F_{\text{g}}}{\rho _{\text{g}}}g{{\mathit{\boldsymbol{\tilde v}}}_{\text{n}}}}}{{{k_{\text{g}}}}}}} |\mathit{\boldsymbol{J}}|{\text{d}}\eta {\text{d}}\xi .$

 $\boldsymbol{K U}=\boldsymbol{R}$ (23)
 $\begin{array}{l} \mathit{\boldsymbol{K}} = \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{K'}}}_{{\rm{up}}}} - {\mathit{\boldsymbol{M}}_{{\rm{up}}}}}&{{{\mathit{\boldsymbol{Q'}}}_{{\rm{up}}}} - {\mathit{\boldsymbol{Q}}_{{\rm{up}}}}}&{{{\mathit{\boldsymbol{G'}}}_{{\rm{up}}}} + {\mathit{\boldsymbol{G}}_{{\rm{up}}}}}\\ {{\mathit{\boldsymbol{H}}_{{\rm{md}}}}}&{{\mathit{\boldsymbol{Q}}_{{\rm{md}}}} + {{\mathit{\boldsymbol{Q'}}}_{{\rm{md}}}}}&{{{\mathit{\boldsymbol{G'}}}_{{\rm{md}}}}}\\ {{\mathit{\boldsymbol{H}}_{{\rm{dw}}}}}&{{{\mathit{\boldsymbol{Q'}}}_{{\rm{dw}}}}}&{{{\mathit{\boldsymbol{G'}}}_{{\rm{dw}}}} + {\mathit{\boldsymbol{G}}_{{\rm{dw}}}}} \end{array}} \right],\mathit{\boldsymbol{U}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\tilde {\bar u}}}}\\ {{{\mathit{\boldsymbol{\tilde {\bar p}}}}^{\rm{w}}}}\\ {{{\mathit{\boldsymbol{\tilde {\bar p}}}}^{\rm{g}}}} \end{array}} \right],\\ \mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\tilde {\bar f}}}_{{\rm{up}}}^{\rm{s}}}\\ {{{\mathit{\boldsymbol{\tilde {\bar f}}}}^{\rm{w}}}}\\ {{{\mathit{\boldsymbol{\tilde {\bar f}}}}^{\rm{g}}}} \end{array}} \right]. \end{array}$
1.2 轨道模型

 $\left( {EI\xi _x^4 - m{\omega ^4}} \right)u_r^{xt} = f_{IT}^{xt}\left( {{\xi _x},\omega } \right) + p_0^{xt}\left( {{\xi _x},\omega } \right)$ (24)

 图 1 轨道和地基的2.5D有限元模型及黏弹性边界 Fig.1 2.5D FEM of track-ground and the viso-elastic boundary
2 模型验证与计算参数

 图 2 不同车速下轨道中心处地面振动时程曲线实测值与模拟值 Fig.2 Time history curve of ground vertical displacement at track center for both test data and simulations with different train speeds

3 不同车速下非饱和路基地面振动位移时程分析

3.1 轨道中心处分析

 图 3 不同速度和饱和度下轨道中心处地面竖向位移时程图 Fig.3 Time history curve of ground vertical displacement at track center at different saturations and train speeds

3.2 轨道中心8 m远处时程分析

 图 4 不同车速和饱和度下距轨道中心8 m处地面振动位移时程曲线 Fig.4 Time history curve of ground vertical acceleration at 8m away from track center at different saturations and train speeds
4 不同车速下非饱和土路基地面振动衰减特性

 图 5 不同车速下饱和度为90%路基地面振动位移幅值和加速度幅值随轨道中心距离衰减曲线 Fig.5 Attenuation curves with distance from track center of ground vertical displacement and acceleration of Sr=90% at different train speeds
5 不同车速下非饱和土路基超静孔隙水压力

 图 6 不同速度和饱和度下轨道中心下超静孔隙水压力随深度变化曲线 Fig.6 Excess pore water pressure distribution with depth under track center at different saturations and train speeds
6 结论

(1) 在轨道中心处：列车速度不超过250 km·h-1时，非饱和土路基地面位移振动幅值明显小于饱和路基，路基从完全饱和到非饱和状态轨道中心处地面振动位移显著减小，完全饱和是振动最不利状态；在车速小于等于250 km·h-1时，随着车速增大，饱和与非饱和路基地面位移幅值差值呈减小趋势；车速为350 km·h-1时位移幅值几近相等.相比于饱和路基，非饱和路基地面振动随时间衰减更快，持时更短.

(2) 距离轨道中心8 m远处：车速不超过250 km·h-1时非饱和路基振动位移幅值明显大于饱和路基，车速高于300 km·h-1后随车速提高饱和与非饱和路基地面振动位移幅值增大且两者逐渐趋于接近，在350 km·h-1时饱和路基地面振动位移幅值有超过非饱和土路基的趋势.在同一速度下，距离轨道中心8 m远处饱和路基路面振动持时大于非饱和路基；车速高于250 km·h-1后随车速增大非饱和土振动持续时间变短，而饱和土地面振动持时变长.

(3) 近轨道处(距离小于5 m)位移幅值低速大于高速且均以大小相当的速率快速衰减；远轨道处(距离超过5 m)200 km·h-1车速下衰减最快，其余车速下地面振动位移振幅大小相当且几乎不衰减.同一位置处低速下地面振动加速度级大于高速下的加速度级，且距轨道中心2 m以内均快速衰减.地面振动加速度级在某些列车速度下的衰减会出现反弹增大现象，反弹增大现象的出现与否及其位置与车速密切相关.

(4) 轨道中心下饱和与非饱和路基超静孔隙水压力分布深度约为0~4.5 m，在轨道中心下1.5~2.0 m之间达最大值，在3.5~4 m深度内急剧减小；同一车速下土体由饱和变为非饱和时超静孔隙水压力峰值显著减小，对于饱和度较低的路基，高铁荷载产生的超静孔隙水压力可以略去不计.