DYNAMIC RESPONSE ANALYSIS OF SATURATED-UNSATURATED DOUBLE-LAYERED SOIL UNDER RECTANGULAR MOVING LOAD1)

LI Kuikui, ZHAO Jianchang,2), WANG Li'an

College of Civil Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

 基金资助: 1)甘肃省高等学校成果转化基金资助项目(2018D-27)

Abstract

Under the assumption that the foundation is a double-layered half-space of saturated-unsaturated soil, the unified dynamic governing equations of the double-layered foundation are constructed and solved by using the theory of continuum mechanics and multiphase porous media. By using Dirac-delta function and Heaviside step function, the action of rectangular moving load is described as an analytic function of temporal and spatial coordinates. Then the load function is substituted into the ground dynamic governing equation, and the equation is solved with Fourier transform. The results are verified by degradation. The dynamic response of saturated-unsaturated soil double-layer foundation is also analyzed. The results show that when the load moving velocity is smaller than the Rayleigh wave velocity, the peak value of the vertical vibration is very small, and the amplitude increases slightly with the increase of the velocity. However, when the load velocity reaches the Rayleigh wave velocity, the vertical vibration increases sharply, and the vertical displacement exhibits peak value many times. The saturation of unsaturated soil and the thickness of soil layer also affect the amplitude of foundation significantly.

Keywords： reduced-order method; triple Fourier transform; rectangular moving load; saturated foundation; unsaturated foundation; dynamic response

LI Kuikui, ZHAO Jianchang, WANG Li'an. DYNAMIC RESPONSE ANALYSIS OF SATURATED-UNSATURATED DOUBLE-LAYERED SOIL UNDER RECTANGULAR MOVING LOAD1). Mechanics in Engineering, 2022, 44(1): 131-137 DOI:10.6052/1000-0879-21-297

1 问题模型

$\lambda^{\ast }=\dfrac{E^{\ast}\nu^{\ast }}{\left( {1+\nu^{\ast }} \right)\left( {1-2\nu^{\ast }}\right)},\ \mu^{\ast }=\dfrac{E^{\ast }}{2\left( {1+\nu^{\ast }}\right)}$

图1

Fig. 1   Computational model

$\rho =\left( {1-n} \right)\rho_{\rm s} +nS_{\rm r} \rho_{\rm w} +n\left( {1-S_{\rm r} }\right)\rho_{\rm a}$

2 非饱和土控制方程及求解

2.1 运动方程

$\mu \nabla^{2}{u}+\left( {\lambda +\mu } \right)\nabla \varTheta -\alpha \chi \nabla p_{\rm w} -\alpha \left( {1-\chi } \right)\nabla p_{\rm a} = \\ \rho {\ddot{{u}}}+\rho_{\rm w} {\ddot{{w}}}+\rho_{\rm a}{\ddot{{v}}}$
$-p_{w,i} =\rho_{\rm w} \ddot{{u}}_{i} +\dfrac{\rho_{\rm w} }{nS_{\rm r} }\ddot{{w}}_{i}+\dfrac{\rho_{\rm w} g}{k_{\rm w} }\dot{{w}}_{i}$
$-p_{a,i} =\rho_{\rm a} \ddot{{u}}_{i} +\dfrac{\rho_{\rm a} }{n\left( {1-S_{\rm r} }\right)}\ddot{{v}}_{i} +\dfrac{\rho_{\rm a} g}{k_{\rm a} }\dot{{v}}_{i}$

$p=\chi p_{\rm w} +\left( {1-\chi } \right)p_{\rm a}$

2.2 本构方程

$\sigma_{ij} =2\mu \varepsilon_{ij} +\delta_{ij} \lambda \theta -\delta_{ij} \alpha p$

2.3 质量守恒方程

$A_{11} \dot{{p}}_{\rm w} +A_{12} \dot{{p}}_{\rm a} +A_{13} \nabla \cdot\dot{{u}}_{i} +A_{14} \nabla \cdot \dot{{v}}_{i} =0$
$A_{21} \dot{{p}}_{\rm w} +A_{22} \dot{{p}}_{\rm a} +A_{23}\nabla \cdot \dot{{u}}_{i} +A_{24} \nabla \cdot \dot{{w}}_{i} =0$

2.4 非饱和土控制方程求解

$\theta =\nabla \cdot {u},\ \theta_{\rm w} =\nabla \cdot {w},\ \theta_{\rm a} =\nabla \cdot {v}$

$\left. \!\!{\begin{array}{l} \tilde{{\tilde{{\tilde{{f}}}}}}\left( {\xi,\eta,z,s}\right)\!=\!\displaystyle\int_{-\infty }^{+\infty } {\displaystyle\int_{-\infty }^{+\infty }{\displaystyle\int_{-\infty }^{+\infty } {f\left( {x,y,z,t} \right)} } } \\ {\rm e}^{-{i}\left( {\xi x+\eta y+st} \right)}{\rm d}x{\rm d}y{\rm d}t \\ f\left( {x,y,z,t} \right)\!=\!\dfrac{1}{8\pi^{3}}\displaystyle\int_{-\infty }^{+\infty } {\displaystyle\int_{-\infty }^{+\infty } {\displaystyle\int_{-\infty }^{+\infty } {\tilde{{\tilde{{\tilde{{f}}}}}}\left( {\xi,\eta,z,s} \right)} } } \\ {\rm e}^{{i}\left( {\xi x+\eta y+st} \right)}{\rm d}\xi {\rm d}\eta {\rm d}s \\ \end{array}}\!\! \right\}\quad$

\begin{align} \left[ {\tilde{{\tilde{{\tilde{{u}}}}}}_{x} \tilde{{\tilde{{\tilde{{u}}}}}}_{y}\ \tilde{{\tilde{{\tilde{{u}}}}}}_{z} \tilde{{\tilde{{\tilde{{p}}}}}}_{\rm w} \tilde{{\tilde{{\tilde{{p}}}}}}_{\rm a} }\right]^{\rm T}= \\ \quad{\varTheta }\left[ {D_{1} {\rm e}^{\sqrt {\gamma_{1} } \cdot z}\ D_{2} {\rm e}^{\sqrt {\gamma_{2} } \cdot z}\ \cdots\ D_{5} {\rm e}^{\sqrt{\gamma_{5} } \cdot z}} \right]^{\rm T}+ \\ \quad{\varTheta }\left[ {D_{6}{\rm e}^{-\sqrt {\gamma_{1} } \cdot z}\ D_{7} {\rm e}^{-\sqrt {\gamma_{2} } \cdot z}\ \cdots\ D_{10} {\rm e}^{-\sqrt {\gamma_{5} } \cdot z}} \right]^{\rm T}\qquad \end{align}
$\left. {\begin{array}{l} \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{z} ={i}\lambda \left( {\xi \tilde{{\tilde{{\tilde{{u}}}}}}_{x} +\eta \tilde{{\tilde{{\tilde{{u}}}}}}_{y} } \right)+\left( {\lambda +2\mu } \right)\tilde{{\tilde{{\tilde{{u}}}}}}_{z}^{\prime }-\\ \alpha \chi \tilde{{\tilde{{\tilde{{p}}}}}}_{\rm w} -\alpha \left( {1-\chi } \right)\tilde{{\tilde{{\tilde{{p}}}}}}_{\rm a} \\ \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{xz} =\mu \left( {i\xi \tilde{{\tilde{{\tilde{{u}}}}}}_{z} +\tilde{{\tilde{{\tilde{{u}}}}}}_{x} ^{\prime }} \right),\ \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{yz} =\mu \left( {i\eta \tilde{{\tilde{{\tilde{{u}}}}}}_{z} +\tilde{{\tilde{{\tilde{{u}}}}}}_{y}^{\prime }} \right) \\ \end{array}} \right\}$

3 饱和土控制方程及求解

3.1 饱和土控制方程

(1)运动方程

$\left. {\begin{array}{l} \mu^{{\ast}}\nabla^{2}{u}^{{\ast}}+\left( {\lambda^{{\ast}}+\mu^{{\ast}}} \right)\nabla \theta^{{\ast}}-\nabla p^{\ast }=\\ \rho^{\ast }{\ddot{{u}}}^{{\ast}}+\rho _{\rm w} {w}^{{\ast}} \\ -p^{\ast }=\rho_{\rm w} \ddot{{u}}_{i}^{\ast } +\dfrac{\rho_{\rm w} }{n}\ddot{{w}}_{i}^{\ast } +\dfrac{\rho_{\rm w} g}{k_{\rm w}^{\ast } }\dot{{w}}_{i}^{\ast } \\ \end{array}} \right\}$

(2)本构方程

$\sigma_{ij}^{\ast } =2\mu^{\ast }\varepsilon_{ij}^{\ast } +\lambda^{\ast }\delta_{ij} \theta^{\ast }-\delta_{ij} \alpha p^{\ast }$

(3)渗流连续方程

$-p^{\ast }=\alpha M\nabla \cdot {\dot{{u}}}^{\ast }+M\nabla \cdot {\dot{{w}}}^{\ast }$

3.2 饱和土控制方程求解

$\left[ {\tilde{{\tilde{{\tilde{{u}}}}}}_{x}^{\ast } \ \tilde{{\tilde{{\tilde{{u}}}}}}_{y}^{\ast }\ \tilde{{\tilde{{\tilde{{u}}}}}}_{z}^{\ast }\ \tilde{{\tilde{{\tilde{{p}}}}}}^{\ast }} \right]^{\rm T}={\varXi}\Big[ D_{11} {\rm e}^{\sqrt {\gamma_{1}^{\ast } } \cdot z}\ D_{12} {\rm e}^{\sqrt {\gamma_{2}^{\ast } } \cdot z} \quad D_{13} {\rm e}^{\sqrt {\gamma_{3}^{\ast } } \cdot z}\ D_{14} {\rm e}^{\sqrt {\gamma_{4}^{\ast } } \cdot z} \Big]^{\rm T}$
$\left. {\begin{array}{l} \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{z}^{\ast } ={i}\lambda^{\ast }\left({\xi \tilde{{\tilde{{\tilde{{u}}}}}}_{x}^{\ast } +\eta \tilde{{\tilde{{\tilde{{u}}}}}}_{y}^{\ast } } \right)+\left( {\lambda^{\ast}+2\mu^{\ast }} \right)\tilde{{\tilde{{\tilde{{u}}}}}}_{z}^{\ast\prime }-\tilde{{\tilde{{\tilde{{p}}}}}}^{\ast } \\ \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{xz}^{\ast } =\mu^{\ast }\left({i\xi \tilde{{\tilde{{\tilde{{u}}}}}}_{z}^{\ast }+\tilde{{\tilde{{\tilde{{u}}}}}}_{x}^{\ast \prime }}\right),\ \tilde{{\tilde{{\tilde{{\sigma }}}}}}_{yz}^{\ast } =\mu \left({i\eta \tilde{{\tilde{{\tilde{{u}}}}}}_{z}^{\ast }+\tilde{{\tilde{{\tilde{{u}}}}}}_{y}^{\ast \prime }} \right) \\ \end{array}} \right\}\qquad$

4 边界问题

4.1 矩形移动荷载的数学描述

$Q=H\left( {a-\left| {x-ct} \right|} \right)\cdot H\left( {b-\left| y\right|} \right)\cdot q_{0}$

$\tilde{{\tilde{{\tilde{{Q}}}}}}=4\pi \left[ {\delta \left( {s-\xi c} \right)+\delta \left( {s+\xi c} \right)} \right]\cdot \\ \dfrac{\sin \left( {a\xi } \right)\sin \left( {b\eta } \right)}{\xi \eta }\cdot q_{0}$

4.2 边界条件

$\left. {\begin{array}{l} \sigma_{z} \left( {x,y,0,t} \right)=Q\left( {x,y,t} \right) \\ \sigma_{xz} \left( {x,y,0,t} \right)=0,\sigma_{yz} \left( {x,y,0,t}\right)=0 \\ p_{\rm w} \left( {x,y,0,t} \right)=0,p_{\rm a} \left( {x,y,0,t} \right)=0 \\ \end{array}} \right\}$

$\left. {\begin{array}{l} u_{x} =u_{x}^{\ast },\ u_{y} =u_{y}^{\ast },\ u_{z} =u_{z}^{\ast}\\ \sigma _{z} =\sigma_{z}^{\ast },\ \sigma_{xz} =\sigma_{xz}^{\ast },\ \sigma_{yz} =\sigma_{yz}^{\ast }\\\chi p_{\rm w} +\left( {1-\chi }\right)p_{\rm a} =p^{\ast},\ \dfrac{\partial p_{\rm w} }{\partial z}=0,\ \dfrac{\partial p_{\rm a} }{\partial z}=0 \\ \end{array}} \right\}$

4.3 边界方程求解

${\varPhi }\cdot \left[ {D_{1},D_{2},\cdots D_{14} } \right]^{\rm T}=\left[ {\tilde{{\tilde{{\tilde{{Q}}}}}},0,\cdots, 0} \right]^{\rm T}$

$\left[ {D_{1},D_{2},\cdots D_{14} } \right]^{\rm T}={\varPhi }^{-1}\left[ {\tilde{{\tilde{{\tilde{{Q}}}}}},0,\cdots, 0} \right]^{\rm T}$

5 验证与分析

图2

Fig. 2   Calculation of degradation and comparison of results ($q_{0}=60$ kN/m$^{2}$, $a=$2 m, $b=1$ m)

图3

Fig. 3   Spatial distribution of vertical ground displacement ($z=0$)

图4

Fig. 4   Variation of peak surface displacement along upper unsaturated soil saturation and soil thickness

图5

Fig. 5   Curve of relationship between peak value of vertical vibration and velocity

6 结论

(1) 随着上层非饱和土厚度的增大,地表振动位移也随之增大,层厚处于0$\sim$0.5 m时,地表振动位移发生剧烈波动。

(2) 地基上层土体为中度饱和时,层状土的影响将减小。

(3) 当荷载移动速度小于瑞利波速时,竖向振动峰值很小,振幅随速度的增大发生小幅增涨,但当荷载速度达到瑞利波速时,竖向振动发生激增;随着速度进一步增大,竖向位移多次出现峰点。

附录

\begin{align} \varPhi_{1j} ={i}\lambda \left( {\xi \phi_{1j} +\eta \phi_{2j} } \right)+r_{k}\left( {\lambda +2\mu } \right)\phi_{3j} -\alpha \chi \phi_{4j} - \\ \alpha\left( {1-\chi } \right)\cdot\phi_{5j}\ (j=1,2,\ldots, 14; k=1,2,\ldots, 10;\\ j=11,12,\ldots, 14, \varPhi_{1j}=0); \end{align}

$\varPhi_{2j} =\mu \left( {i\xi \phi_{3j} +r_{k} \phi_{1j} } \right)\ (j=1,2,\ldots, 14; k=1,2,\ldots, 10;\\ j=11,12,\ldots, 14, \varPhi_{ij} =0);$

$\varPhi_{3j} =\mu \left( {i\eta \phi_{3j} +r_{k} \phi_{2j} }\right)\ (j=1,2,\ldots, 14; k=1,2,\ldots, 10;\\ j=11,12,\ldots, 14, \varPhi_{ij} =0);$

$\varPhi_{ij} =\phi_{ij}\ (i=4,5; j=1,2,\ldots, 14; j=11,12,\ldots, 14,\\ \varPhi_{ij} =0);$

\begin{align} \varPhi_{ij} =\left\{ {\begin{array}{ll} \phi_{i-5,j} {\rm e}^{r_{k} H} & \left( {j=1,2,..., 10;k=1,2,..., 10} \right) \\ -\phi_{i-5,j}^{\ast } {\rm e}^{r_{k}^{\ast } H}& \left( {j=11,12,..., 14;k=1,2,..., 4} \right) \\ \end{array}} \right. \\ (i=6,7,8); \end{align}

$\varPhi_{ij} =\left\{ \begin{array}{l} \phi_{i-8,j} {\rm e}^{r_{k} H}\\ \left( {i=9,10,11;j=1,2,..., 10;k=1,2,..., 10} \right) \\ \left.\begin{array}{l} -\Big[ {i}\lambda^{\ast }\left( {\xi \phi_{i-8,j-10}^{\ast } +\eta \phi_{i-7,j-10}^{\ast } } \right)+\\ r_{k}^{\ast } \left( {\lambda^{\ast }+2\mu^{\ast }} \right)\phi_{i-6,j-10}^{\ast } - \\ \phi_{i-5,j-10}^{\ast }\Big]{\rm e}^{r_{k}^{\ast } H}\\ \left( {i=9} \right) \\ -\mu^{\ast }\Big( i\xi \phi_{i-7,j-10}^{\ast } +\\ r_{k}^{\ast } \phi_{i-9,j-10}^{\ast } \Big){\rm e}^{r_{k}^{\ast } H}\ \left( {i=10} \right) -\mu^{\ast }\Big( i\eta \phi_{i-8,j-10}^{\ast } +\\ r_{k}^{\ast } \phi_{i-9,j-10}^{\ast } \Big){\rm e}^{r_{k}^{\ast } H}\ \left( {i=11} \right) \end{array}\right\} \\ \left( {j=11,12,..., 14;k=1,2,..., 4} \right) \end{array} \right.;$

$\varPhi_{ij} =\left\{ {\begin{array}{l} \left[ {\chi \phi_{i-8,j} +\left( {1-\chi } \right)\phi_{i-7,j} } \right]{\rm e}^{r_{k} H}\\ \left( {j=1,2,..., 10;k=1,2,..., 10} \right) \\ -\phi_{i-8,j-10}^{\ast } \cdot {\rm e}^{r_{k}^{\ast } H}\\ \left( {j=11,12,..., 14;k=1,2,..., 4} \right) \\ \end{array}} \right.\ (i=12);$

\begin{align} \varPhi_{ij} =r_{k} \phi_{i-9,j} \cdot {\rm e}^{r_{k}H}\ (i=13,14; j=1,2,\ldots, 14;\\ k=1,2,\ldots, 10; \mbox{当} j=11,12,\ldots, 14\mbox{时}, \varPhi_{ij}=0)\\ (r_{1}=-r_{2}, r_{3}=-r_{4}, r_{5}=-r_{6}, r_{7}=-r_{8}, r_{9}=-r_{10}). \end{align}

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