饱和土中非连续屏障对弹性波的隔离
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摘要
单相弹性土体中的振动隔离问题很早就被注意并加以研究,但饱和土体中的振动隔离问题少有文献报道。随着现代工业的迅速发展和城市规模的日益扩大,机器基础、交通荷载、爆破和打桩施工等活动的场所也从弹性地基扩展到饱和土地基,使得振动对周围环境的影响日益显著。因此,研究如何阻止或减小饱和土体中振动的传播,使其对周围地区的不利影响降至最低,不仅可为深入揭示饱和土中屏障和振动波作用机理提供有力的手段,同时在工程上也有重要的应用价值。
本文首先在Biot波动理论的基础上,考虑饱和土体中快纵波、慢纵波和横波的作用,利用圆柱波势函数推导出单个圆柱体对倾斜入射弹性波的散射方程,并给出了背向散射谱和总散射截面积的计算公式。数值计算结果表明,饱和土渗透性对背向散射谱影响较大;随着倾斜入射角的增大,总散射截面积在减小。
其次,分别采用刚性和弹性排桩作为屏障,对全空间饱和土中排桩对平面纵波和横波的隔离问题进行了研究。使用波函数展开法,利用一组圆柱坐标系统和Graff加法转换定理,将散射波势函数的表达式变换到同一个坐标系。通过施加问题的连续条件和平衡条件,利用三角函数的正交性以及不同阶三角函数线性无关的性质,得到问题的解。结果表明随着饱和土体渗透性降低,排桩对横波的隔离效果越好;排桩隔离效果在饱和土与相应的弹性土中差异较大。
然后对排桩后饱和土体竖向(与桩轴线方向一致)位移衰减进行了研究,使用坐标转换和加法转换定理,通过施加桩土界面处的应力和位移连续的边界条件,求得弹性波倾斜入射时散射波势函数的待定系数,进而得到排桩后相应位置处的土体位移。计算结果表明桩土弹性模量比变化对竖向位移衰减有较大影响,但饱和土渗透性对其影响较小。
最后采用不考虑固-液惯性耦合效应的饱和土波动方程,将桩视为欧拉-伯努力梁,根据求出的饱和半空间中瑞利波的膨胀和旋转势函数表达式,将散射瑞利波势函数表达式用一组圆柱坐标系统进行转换,通过施加桩顶和桩底的边界条件以及桩的动力平衡条件,使用复数域内的最小二乘法得到问题的近似解。数值分析结果表明在饱和土中瑞利波作用下,土体渗透性对排桩隔离效果的影响与横波作用时情况不同,排桩的存在可以显著减小入射瑞利波引起的孔隙水压力。
Vibration isolation of structures from ground transmitted waves generated by machines, traffic, blasting or pile driving has received considerable attention in single phase soil, while little work has been done in poroelastic soil. With the rapid development of modern industry and urban expansion, the vibration in poroelastic soil has increasingly important influences on the surroundings. It is important to investigate how to impede or reduce the propagation of elastic waves in poroelastic soil, which not only provides a useful way for revealing the action mechanism of barrier and waves in poroelastic soil, but also has important application value in engineering.
Firstly, based on Biot's poroelastic theory, the scattering of an elastic wave obliquely incident on a cylinder is studied and the scattering equation is given using cylindrical wave functions. The expressions for the backscattering spectrum and total scattering cross-section are subsequently calculated. The numerical results show that the influence of soil permeability on the backscattering spectrum is remarkable, and the total scattering cross-section decreases with an increase of obliquely incident angle.
Secondly, the isolation of plane P and S waves by using a row of rigid or elastic piles as a barrier is investigated in a homogeneous unbounded space. Utilizing wave function expansion method, all of the potential functions of scattered waves are transformed into one coordinate system with the aid of Graff's addition theorem. Taking into account the orthogonality and linear independence of trigonometric functions, closed-form expressions for scattering coefficients are developed by imposing continuity conditions of pile-soil interfaces and equilibrium conditions of piles. The results indicate that the isolation effectiveness for plane S waves will become better with the decrease of the soil permeability. There is a significant difference in the isolation effectiveness of a row of piles between the poroelastic soil and the elastic soil.
Thirdly, the effect of a row of elastic piles in reducing the vertical displacement (in the direction of pile axis) amplitudes of poroelastic soil is investigated. The wave field near certain pile is constructed as the sum of incident waves and scattering waves from all piles with the translational addition theorems for cylindrical wave functions. By imposing continuity conditions at the soil-pile interfaces, the unknown coefficients for potential functions of scattered waves are determined. Then, the soil displacements behind the pile barrier can be calculated. It is found that the modulus ratio between the pile and soil has significant influence on the reduction of vertical displacements, while the soil permeability has little influence on it.
Finally, on the basis of Biot's theory without considering the inertial coupling effects between the solid-fluid phases of the medium, the piles are modeled as Euler-Bernoulli beams and the diffracted field by each pile is constructed only with Rayleigh waves. Six infinite linear systems of algebraic equations are obtained according to Graff's addition theorem and the application of suitable boundary conditions. The systems are subsequently solved in the complex least squares sense. Numerical calculation results show that the influence of soil permeability on the isolation of Rayleigh waves in poroelastic soil is different from that of plane S waves, and the pore water pressure behind the pile barrier can be reduced remarkably.
本文首先在Biot波动理论的基础上,考虑饱和土体中快纵波、慢纵波和横波的作用,利用圆柱波势函数推导出单个圆柱体对倾斜入射弹性波的散射方程,并给出了背向散射谱和总散射截面积的计算公式。数值计算结果表明,饱和土渗透性对背向散射谱影响较大;随着倾斜入射角的增大,总散射截面积在减小。
其次,分别采用刚性和弹性排桩作为屏障,对全空间饱和土中排桩对平面纵波和横波的隔离问题进行了研究。使用波函数展开法,利用一组圆柱坐标系统和Graff加法转换定理,将散射波势函数的表达式变换到同一个坐标系。通过施加问题的连续条件和平衡条件,利用三角函数的正交性以及不同阶三角函数线性无关的性质,得到问题的解。结果表明随着饱和土体渗透性降低,排桩对横波的隔离效果越好;排桩隔离效果在饱和土与相应的弹性土中差异较大。
然后对排桩后饱和土体竖向(与桩轴线方向一致)位移衰减进行了研究,使用坐标转换和加法转换定理,通过施加桩土界面处的应力和位移连续的边界条件,求得弹性波倾斜入射时散射波势函数的待定系数,进而得到排桩后相应位置处的土体位移。计算结果表明桩土弹性模量比变化对竖向位移衰减有较大影响,但饱和土渗透性对其影响较小。
最后采用不考虑固-液惯性耦合效应的饱和土波动方程,将桩视为欧拉-伯努力梁,根据求出的饱和半空间中瑞利波的膨胀和旋转势函数表达式,将散射瑞利波势函数表达式用一组圆柱坐标系统进行转换,通过施加桩顶和桩底的边界条件以及桩的动力平衡条件,使用复数域内的最小二乘法得到问题的近似解。数值分析结果表明在饱和土中瑞利波作用下,土体渗透性对排桩隔离效果的影响与横波作用时情况不同,排桩的存在可以显著减小入射瑞利波引起的孔隙水压力。
Vibration isolation of structures from ground transmitted waves generated by machines, traffic, blasting or pile driving has received considerable attention in single phase soil, while little work has been done in poroelastic soil. With the rapid development of modern industry and urban expansion, the vibration in poroelastic soil has increasingly important influences on the surroundings. It is important to investigate how to impede or reduce the propagation of elastic waves in poroelastic soil, which not only provides a useful way for revealing the action mechanism of barrier and waves in poroelastic soil, but also has important application value in engineering.
Firstly, based on Biot's poroelastic theory, the scattering of an elastic wave obliquely incident on a cylinder is studied and the scattering equation is given using cylindrical wave functions. The expressions for the backscattering spectrum and total scattering cross-section are subsequently calculated. The numerical results show that the influence of soil permeability on the backscattering spectrum is remarkable, and the total scattering cross-section decreases with an increase of obliquely incident angle.
Secondly, the isolation of plane P and S waves by using a row of rigid or elastic piles as a barrier is investigated in a homogeneous unbounded space. Utilizing wave function expansion method, all of the potential functions of scattered waves are transformed into one coordinate system with the aid of Graff's addition theorem. Taking into account the orthogonality and linear independence of trigonometric functions, closed-form expressions for scattering coefficients are developed by imposing continuity conditions of pile-soil interfaces and equilibrium conditions of piles. The results indicate that the isolation effectiveness for plane S waves will become better with the decrease of the soil permeability. There is a significant difference in the isolation effectiveness of a row of piles between the poroelastic soil and the elastic soil.
Thirdly, the effect of a row of elastic piles in reducing the vertical displacement (in the direction of pile axis) amplitudes of poroelastic soil is investigated. The wave field near certain pile is constructed as the sum of incident waves and scattering waves from all piles with the translational addition theorems for cylindrical wave functions. By imposing continuity conditions at the soil-pile interfaces, the unknown coefficients for potential functions of scattered waves are determined. Then, the soil displacements behind the pile barrier can be calculated. It is found that the modulus ratio between the pile and soil has significant influence on the reduction of vertical displacements, while the soil permeability has little influence on it.
Finally, on the basis of Biot's theory without considering the inertial coupling effects between the solid-fluid phases of the medium, the piles are modeled as Euler-Bernoulli beams and the diffracted field by each pile is constructed only with Rayleigh waves. Six infinite linear systems of algebraic equations are obtained according to Graff's addition theorem and the application of suitable boundary conditions. The systems are subsequently solved in the complex least squares sense. Numerical calculation results show that the influence of soil permeability on the isolation of Rayleigh waves in poroelastic soil is different from that of plane S waves, and the pore water pressure behind the pile barrier can be reduced remarkably.
引文
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