非饱和土地基与基础的动力响应研究
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摘要
弹性半空间地基与基础的动力响应问题研究涉及到数学、力学和岩土工程等多个学科,它不仅对弹性动力学的发展有着重要的学术价值,又在建筑工程、水利工程、道路工程和动力机器基础等工程领域有着广泛而重要的应用价值。以往的研究多将土作为单相或两相介质(饱和土)对待,本文以三相多孔介质模型来建立非饱和土的动力控制方程,充分考虑了土骨架与孔隙中水和气之间的黏性和惯性耦合。以此为基础,系统地研究了非饱和土半空间在表面荷载作用下的稳态和瞬态动力响应,并分析了非饱和土地基上刚性和弹性圆板基础的垂直振动问题,主要工作和成果有:
1)在分析和比较已有非饱和多孔介质动力模型的基础上,建立了一组比较实用的非饱和土波动方程。主要改进包括:土水特征曲线采用V-G模型,孔隙流体相对渗透系数根据Mualem理论得到,从而使建立的方程能准确退化到经典的饱和土Biot波动方程;考虑了饱和度对土体动剪切模量的影响,并提出了一个适用于粘性土的动剪切模量与饱和度的关系式。求解波动方程,得到了体波传播的弥散方程,结果表明非饱和土中存在三种压缩波和一种剪切波。通过数值计算,系统地研究了非饱和土中四种体波的传播特性,考察了各主要参数对体波波速和衰减的影响。
2)研究了非饱和土半空间表面在垂直集中力作用下的Lamb问题。利用Hankel变换技术,在圆柱坐标系下求解了非饱和土满足的波动方程,得到了轴对称情况下土体骨架位移、应力分量及孔隙流体压力在变换域内的一般解。考虑表面排水、排气和不排水、不排气两种边界条件,给出了非饱和土半空间表面在垂直集中力作用下动力响应的积分形式解。数值分析表明:在索状饱和状态,地表位移幅值随饱和度的升高而增大;当土体接近完全饱和时,随着饱和度的升高,位移幅值反而出现下降;渗透率、排水(气)条件只有在饱和度非常高时才会对地表位移产生显著的影响。
3)在直角坐标系下研究了非饱和土半空间体的稳态动力响应问题。通过引入位移函数,将波动方程解耦,进而利用双重Fourier变换,化偏微分方程为常微分方程求解,得到了非饱和土半空间表面在任意分布的简谐荷载作用下的稳态动力响应解答。
4)研究了层状非饱和土地基的轴对称和非轴对称稳态动力响应问题。对于轴对称问题,通过引入一组状态向量,在圆柱坐标系下将波动方程转化为一组状态方程,利用Hankel变换对方程进行求解。对于非轴对称问题,通过引入一组状态向量,在直角坐标系下将波动方程转化为一组状态方程,利用双重Fourier变换对方程进行求解。数值算例分析表明:地表动位移主要受临界深度范围内土层的影响,该临界深度大约为10倍荷载作用区域的边长或直径;软硬土层在地层中的相对次序对地表动位移有显著的影响,硬土层靠近地表时位移相对较小。
5)在时间域内研究了非饱和土半空间体在表面垂直荷载作用下的瞬态动力响应问题。利用Laplace-Hankel联合变换,对控制非饱和土振动的偏微分方程组进行解耦,得到了轴对称情况下土体骨架位移的积分形式解。数值算例生动地演示了非饱和土半空间内波的传播过程,结果表明:在索状饱和状态,饱和度增加时,横波波速明显降低,同时地表垂直位移和径向位移幅值明显增加。
6)研究了非饱和土地基上刚性圆板及弹性圆薄板基础的垂直振动问题。首先利用Hankel变换技术得到了变换域内表面位移和应力的关系式,然后按混合边值条件分别建立了非饱和土地基上刚性圆板和弹性圆薄板垂直振动时的对偶积分方程,通过Abel变换,将对偶积分方程化为易于数值求解的第二类Fredholm积分方程,得到了刚性圆板和弹性圆薄板垂直振动的半解析解。结果表明:饱和度对基础板的振动有显著的影响,通常情况下,饱和度越高,地基的刚度越低,阻尼越大;但当土体接近完全饱和时,其变化规律则刚好相反,而且此时饱和度的轻微变化即可对板的振动幅度产生明显的影响。
Studies on dynamic response of subgrade and foundation relate to many subjects, such as mathematics, mechanics and geotechnical engineering etc. It not only has great academic values on development of elastic dynamics, but also is applied widely in fields such as building engineering, hydraulic engineering, road engineering and dynamic machine foundation, etc. In former studies on this subject, soils were taken majorly as one-phase or two-phrase media (saturated soils). In present study, dynamic governing equations for unsaturated soils are established by taking soils as three-phase porous media, and considering the viscous and inertial coupling between pore fluids and soil skeleton. Based on which, we systematically study the steady state and transient dynamic response of unsaturated half space under surface loads, and analyze the vertical vibration of rigid and elastic circular plate on unsaturated soils. The major works and achievements are as follow:
1)After analyzing and comparing the existed dynamic models for unsaturated porous media, a group of practical wave equations for unsaturated soils are established in present. Major improvements include: 1. V-G model for SWCC(soil-water characteristic curve ) is adopted and relative permeability of pore fluids is deduced based on Mualem theory, which make the equations established can be degenerated exactly to the classical Biot’s wave equations for saturated soils. 2. the influence of saturation on dynamic shear modulus is considered and a formula to calculate dynamic shear modulus of unsaturated clay with arbitary saturation is raised. Solving wave equations, dispersive equations of body waves propagation are obtained, which indicate that there are three compressive waves and one shear wave in unsaturated soils. Numerical analysis is performed to study the propogation characteristic of body waves in unsaturated soils. Influences of some parameters on waves velocity and attenuation are examined.
2)Lamb’s problem for unsaturated half-space under vertical surface point load is analyzed. With the help of Hankel transform, solving wave equations in cylindrical coordinates, general solutions of displacement, stress and pore fluids pressure are obtained. Thus the integral solutions are gotten both in the case of drained surface and in the case of undrained surface excited by vertical concentrated harmonic resources. Numerical results indicate that displacement amplitude of ground surface increases with saturation when soils are in the state of funicular saturation. While soils approach completely saturated, with saturation rising, displacement amplitude decreases. And permeability, drainage conditions have significant influence on displacement only when saturation is very high.
3)Analysis about the steady state dynamic response of unsaturated half-space is also performed in Cartesian coordinate. By introducing displacement function, decoupling the wave equations, and then converting partial differential equations into constant differential equations with the help of double Fourier transform, the solutions of steady state dynamic response of unsaturated soils under arbitary distributed loads are presented.
4)The work is extented to axisymmetrical and non-axisymmetrical steady state dynamic response of layered unsaturated soils. In the case of axisymmetrical problem, wave equations are transformed into a group of state equations in cylindrical coordinates by introducing a group of state vectors, then solutions are obtained with the help of Hankel transform. In the case of non-axisymmetrical problem, wave equations are transformed into a group of state equations in the Cartesian coordinates by introducing a group of state vectors, then solutions are obtained by means of double Fourier transform. Numerical results indicate that displacement of ground surface is influenced mainly by stratum in critical depth. The critical depth is about 10 times of side length or diameter of the area which loads exerted on. The relative position of soft and hard strata has significant influence on displacement, which is relatively small when hard stratum lies close to ground surface.
5)Transient dynamic response of unsaturated half-space under vertical surface loads is presented in time domain. Using Laplace-Hankel transfer to decouple the partial differential equations of unsaturated soils, the integral solutions of displacement for axisymmetrical problem are obtained. Numerical examples show vividly the propagation process of waves in unsaturated half-space. The results indicate that, with saturation rising, the speed of shear wave decreases obviously, and the vertical and radial displacement amplitude increases significantly at the same time when soils are in the state of funicular saturation.
6)Vertical vibration of rigid circular plate and elastic circular thin plate on unsaturated subgrade are invesgated. By using Hankel transform technology, the relations between surface displacement and stress are obtained firstly. Considering mixed boundary value conditions, the dual integral equations for vertical vibration of rigid circular plate and elastic thin circular thin plate on unsaturated subgrade are established. Then by means of Abel transform, the dual integral equations are transformed into Fredholm integral equation of the second kind, which can be solved easily by numerical methods. At last, the semi-analytical solutions for vertical vibration of rigid circular plate and elastic thin circular thin plates are obtained. Numerical results indicate that saturation has significant influence on vibration of plate. In general cases, with saturation increasing, the stiffness of subgrade decreases, and the damping increases. But when soils approaches completely saturated, the change is just on the contrary, furthermore, a slight change of saturation will make significant influence on vibration amplitude of plate.
1)在分析和比较已有非饱和多孔介质动力模型的基础上,建立了一组比较实用的非饱和土波动方程。主要改进包括:土水特征曲线采用V-G模型,孔隙流体相对渗透系数根据Mualem理论得到,从而使建立的方程能准确退化到经典的饱和土Biot波动方程;考虑了饱和度对土体动剪切模量的影响,并提出了一个适用于粘性土的动剪切模量与饱和度的关系式。求解波动方程,得到了体波传播的弥散方程,结果表明非饱和土中存在三种压缩波和一种剪切波。通过数值计算,系统地研究了非饱和土中四种体波的传播特性,考察了各主要参数对体波波速和衰减的影响。
2)研究了非饱和土半空间表面在垂直集中力作用下的Lamb问题。利用Hankel变换技术,在圆柱坐标系下求解了非饱和土满足的波动方程,得到了轴对称情况下土体骨架位移、应力分量及孔隙流体压力在变换域内的一般解。考虑表面排水、排气和不排水、不排气两种边界条件,给出了非饱和土半空间表面在垂直集中力作用下动力响应的积分形式解。数值分析表明:在索状饱和状态,地表位移幅值随饱和度的升高而增大;当土体接近完全饱和时,随着饱和度的升高,位移幅值反而出现下降;渗透率、排水(气)条件只有在饱和度非常高时才会对地表位移产生显著的影响。
3)在直角坐标系下研究了非饱和土半空间体的稳态动力响应问题。通过引入位移函数,将波动方程解耦,进而利用双重Fourier变换,化偏微分方程为常微分方程求解,得到了非饱和土半空间表面在任意分布的简谐荷载作用下的稳态动力响应解答。
4)研究了层状非饱和土地基的轴对称和非轴对称稳态动力响应问题。对于轴对称问题,通过引入一组状态向量,在圆柱坐标系下将波动方程转化为一组状态方程,利用Hankel变换对方程进行求解。对于非轴对称问题,通过引入一组状态向量,在直角坐标系下将波动方程转化为一组状态方程,利用双重Fourier变换对方程进行求解。数值算例分析表明:地表动位移主要受临界深度范围内土层的影响,该临界深度大约为10倍荷载作用区域的边长或直径;软硬土层在地层中的相对次序对地表动位移有显著的影响,硬土层靠近地表时位移相对较小。
5)在时间域内研究了非饱和土半空间体在表面垂直荷载作用下的瞬态动力响应问题。利用Laplace-Hankel联合变换,对控制非饱和土振动的偏微分方程组进行解耦,得到了轴对称情况下土体骨架位移的积分形式解。数值算例生动地演示了非饱和土半空间内波的传播过程,结果表明:在索状饱和状态,饱和度增加时,横波波速明显降低,同时地表垂直位移和径向位移幅值明显增加。
6)研究了非饱和土地基上刚性圆板及弹性圆薄板基础的垂直振动问题。首先利用Hankel变换技术得到了变换域内表面位移和应力的关系式,然后按混合边值条件分别建立了非饱和土地基上刚性圆板和弹性圆薄板垂直振动时的对偶积分方程,通过Abel变换,将对偶积分方程化为易于数值求解的第二类Fredholm积分方程,得到了刚性圆板和弹性圆薄板垂直振动的半解析解。结果表明:饱和度对基础板的振动有显著的影响,通常情况下,饱和度越高,地基的刚度越低,阻尼越大;但当土体接近完全饱和时,其变化规律则刚好相反,而且此时饱和度的轻微变化即可对板的振动幅度产生明显的影响。
Studies on dynamic response of subgrade and foundation relate to many subjects, such as mathematics, mechanics and geotechnical engineering etc. It not only has great academic values on development of elastic dynamics, but also is applied widely in fields such as building engineering, hydraulic engineering, road engineering and dynamic machine foundation, etc. In former studies on this subject, soils were taken majorly as one-phase or two-phrase media (saturated soils). In present study, dynamic governing equations for unsaturated soils are established by taking soils as three-phase porous media, and considering the viscous and inertial coupling between pore fluids and soil skeleton. Based on which, we systematically study the steady state and transient dynamic response of unsaturated half space under surface loads, and analyze the vertical vibration of rigid and elastic circular plate on unsaturated soils. The major works and achievements are as follow:
1)After analyzing and comparing the existed dynamic models for unsaturated porous media, a group of practical wave equations for unsaturated soils are established in present. Major improvements include: 1. V-G model for SWCC(soil-water characteristic curve ) is adopted and relative permeability of pore fluids is deduced based on Mualem theory, which make the equations established can be degenerated exactly to the classical Biot’s wave equations for saturated soils. 2. the influence of saturation on dynamic shear modulus is considered and a formula to calculate dynamic shear modulus of unsaturated clay with arbitary saturation is raised. Solving wave equations, dispersive equations of body waves propagation are obtained, which indicate that there are three compressive waves and one shear wave in unsaturated soils. Numerical analysis is performed to study the propogation characteristic of body waves in unsaturated soils. Influences of some parameters on waves velocity and attenuation are examined.
2)Lamb’s problem for unsaturated half-space under vertical surface point load is analyzed. With the help of Hankel transform, solving wave equations in cylindrical coordinates, general solutions of displacement, stress and pore fluids pressure are obtained. Thus the integral solutions are gotten both in the case of drained surface and in the case of undrained surface excited by vertical concentrated harmonic resources. Numerical results indicate that displacement amplitude of ground surface increases with saturation when soils are in the state of funicular saturation. While soils approach completely saturated, with saturation rising, displacement amplitude decreases. And permeability, drainage conditions have significant influence on displacement only when saturation is very high.
3)Analysis about the steady state dynamic response of unsaturated half-space is also performed in Cartesian coordinate. By introducing displacement function, decoupling the wave equations, and then converting partial differential equations into constant differential equations with the help of double Fourier transform, the solutions of steady state dynamic response of unsaturated soils under arbitary distributed loads are presented.
4)The work is extented to axisymmetrical and non-axisymmetrical steady state dynamic response of layered unsaturated soils. In the case of axisymmetrical problem, wave equations are transformed into a group of state equations in cylindrical coordinates by introducing a group of state vectors, then solutions are obtained with the help of Hankel transform. In the case of non-axisymmetrical problem, wave equations are transformed into a group of state equations in the Cartesian coordinates by introducing a group of state vectors, then solutions are obtained by means of double Fourier transform. Numerical results indicate that displacement of ground surface is influenced mainly by stratum in critical depth. The critical depth is about 10 times of side length or diameter of the area which loads exerted on. The relative position of soft and hard strata has significant influence on displacement, which is relatively small when hard stratum lies close to ground surface.
5)Transient dynamic response of unsaturated half-space under vertical surface loads is presented in time domain. Using Laplace-Hankel transfer to decouple the partial differential equations of unsaturated soils, the integral solutions of displacement for axisymmetrical problem are obtained. Numerical examples show vividly the propagation process of waves in unsaturated half-space. The results indicate that, with saturation rising, the speed of shear wave decreases obviously, and the vertical and radial displacement amplitude increases significantly at the same time when soils are in the state of funicular saturation.
6)Vertical vibration of rigid circular plate and elastic circular thin plate on unsaturated subgrade are invesgated. By using Hankel transform technology, the relations between surface displacement and stress are obtained firstly. Considering mixed boundary value conditions, the dual integral equations for vertical vibration of rigid circular plate and elastic thin circular thin plate on unsaturated subgrade are established. Then by means of Abel transform, the dual integral equations are transformed into Fredholm integral equation of the second kind, which can be solved easily by numerical methods. At last, the semi-analytical solutions for vertical vibration of rigid circular plate and elastic thin circular thin plates are obtained. Numerical results indicate that saturation has significant influence on vibration of plate. In general cases, with saturation increasing, the stiffness of subgrade decreases, and the damping increases. But when soils approaches completely saturated, the change is just on the contrary, furthermore, a slight change of saturation will make significant influence on vibration amplitude of plate.
引文
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