饱和地基中埋置基础的动力振动特性研究
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摘要
土与基础动力相互作用问题的研究一直受到力学界和土木工程界的广泛关注。它不仅对于弹性土动力学的发展有着重要的作用,而且对机器基础的设计具有重要的指导意义。
本文基于Biot饱和多孔介质理论,首次用解析的方法较系统地研究了饱和地基中埋置刚性圆柱基础的动力振动问题。主要工作有:
对于饱和半空间中埋置刚性圆柱基础的竖向振动,基础中心受到简谐竖向激振力,基础与地基完全粘着接触。假定基底与地基的接触面光滑,接触面为完全透水。地基作用在基础上的总力由基础底面反力和基础侧面反力两部分组成。求解基础底面的反力时,引入Novak简化模型将基底以下的土视为与上覆土层无关的均质饱和半空间。首先运用Hankel积分变换技术求解饱和土动力控制方程,得到其在变换域内的一组通解,然后考虑基底与地基接触面的混合边值条件建立了一组描述刚性基础竖向振动的对偶积分方程,并化该对偶积分方程为第二类Fredholm积分方程,通过解此Fredholm积分方程求得了土体作用在基础底面的反力。求解基础侧面的反力时,假设基础侧壁的土体由若干极薄饱和层所组成,利用极薄饱和层在圆孔周边处的力-位移关系和沿基础侧面积分的方法,求得土体作用在基础侧面的反力。最后结合基础的动力平衡方程,求得了饱和半空间中埋置刚性圆柱基础竖向振动时的动力刚度、振动振幅的表达式。通过与前人研究成果的对比验证了本文结果的正确性。数值算例分析了无量纲激振频率、基础埋深、基础质量比、饱和土粘滞系数、孔隙水对竖向动力响应的影响。
在工程中具有有限深度、下卧有硬层的单层饱和地基很常见。因此文中采用Novak简化计算模型接着研究了下卧基岩单层饱和地基中埋置刚性基础的竖向振动,分析了不同土层厚度下基础的竖向动力刚度、动力放大系数曲线。
对于饱和半空间中埋置刚性圆柱基础的摇摆振动,假定基础与地基完全粘着接触,基底与地基的接触面为完全透水或完全不透水,本文用同样的简化计算模型求解了相应的动力相互作用问题。数值算例分析了无量纲激振频率、基础埋深、基础质量比、饱和土粘滞系数、孔隙水以及接触面透水条件对基础摇摆动力刚度、摇摆角位移幅值的影响。
考虑到地基的成层性,本文还研究了下卧基岩单层饱和地基中埋置刚性圆柱基础的摇摆振动问题,分析了土层厚度、基础埋深、饱和土粘滞系数、孔隙水对埋置基础与单层饱和地基摇摆动力相互作用的影响。
最后,文中采用相同的Novak简化模型,对饱和地基中埋置刚性圆柱基础的扭转振动进行了研究。考虑地基为饱和半空间和下卧基岩单层饱和地基两种情况,求得了刚性埋置基础扭转振动时的动力刚度、扭转角位移幅值。算例分析研究了土层厚度、基础埋深、基础质量比、饱和土粘滞系数、孔隙水对饱和地基中刚性埋置基础扭转动力响应的影响。
The study on dynamic soil-foundation interaction problem has been of considerable interest in the field of geomechanics and civil engineering.It plays an important role in the development of the elastodynamics and has important practical applications in machine foundations design as well.
Based on the Biot's poroelastodynamic theory,the dynamic vibrations of a rigid cylindrical embedded foundation in poroelastic soil are studied by analytical method in detail for the first time.The main work is as follows:
For the vertical vibration of a rigid cylindrical foundation embedded in a poroelastic half-space,the foundation is subjected to vertical time-harmonic excitation along the vertical axis and is perfectly bonded to the surrounding soil.It is assumed that the contact surface between the foundation base and the poroelastic soil is smooth and fully permeable.The total dynamic reaction of the soil at the foundation is composed of a reaction acting in the foundation base and a reaction on the foundation vertical sides.To study the reaction acting in the foundation base,the Novak approximate method is introduced based on the assumption that the soil underlying the foundation base is a homogeneous poroelastic half-space which is independent of the overlying soil.First,the dynamic governing equations of poroelastic soil are solved by using the Hankel transform method and the general solutions are formulated in the Hankel transform fields.Then,considering the mixed boundary-value condition at the interface between the foundation base and the poroelastic soil,the vertical vibration of a rigid foundation is formulated in a set of dual integral equations,which are further reduced to Fredholm integral equation of the second kind.Consequently, the soil reaction acting in the foundation base is derived from solving this Fredholm integral equation.To study the reaction acting on the foundation vertical sides,the soil along foundation vertical sides is modeled as an independent stratum composed of a series of infinitesimally thin poroelastic layers.By using the force-displacement relationship of the infinitesimally thin poroelastic layers and the integration along the circumference of the cylinder,the soil reaction acting on the foundation vertical sides can be derived.Finally,the expressions for the dynamic impedance and displacement amplitude are obtained for the vertical vibration of a rigid cylindrical foundation embedded in a poroelastic half-space by combing the dynamic equilibrium equation of the foundation.The accuracy of the present solution is verified by comparisons with the existing solutions obtained by other researchers.Numerical results are presented to demonstrate the influence of nondimensional frequency of excitation, embedment depth,mass ratio of the foundation,internal friction of the poroelastic soil and pore fluid on the vertical dynamic response.
In engineering applications,there are many cases in which the depth of soil layer thickness overlaying rigid base is not large enough.So the vertical vibration of rigid foundation embedded in a single-layered poroelastic soil based on rigid bedrock is then studied by using the same approximate method.The vertical dynamic impedance and dynamic response factor for different values of soil layer thickness are discussed.
For the rocking vibration of a rigid cylindrical foundation embedded in a poroelastic half-space,the dynamic interaction problem is also solved by using the same approximate method.The foundation is assumed to be perfectly bonded to the surrounding soil and the contact surface between the foundation base and the poroelastic soil can be either fully permeable or impermeable.The numerical results are presented to analysis the effect of nondimensional frequency of excitation, embedment depth,mass ratio of the foundation,internal friction of the poroelastic soil, pore fluid and hydraulic boundary condition along the contact surface on the rocking dynamic impendence and angular displacement amplitude of the foundation.
Considering the layered property,of soil,the author further analyse the rocking vibration of a rigid foundation embedded in a single-layered poroelastic soil based on rigid bedrock.The effect of soil layer thickness,embedment depth,internal friction of the poroelastic soil and pore fluid on the rocking dynamic soil-foundation interaction problem is studied.
Finally,the torsional vibration of a rigid cylindrical foundation embedded in poroelastic soil is discussed by the same Novak approximate model.The technique is applied to the computation of torsional dynamic impedance and angular displacement of the rigid foundation embedded in a poroelastic half-space and in a single-layered poroelastic soil based on rigid bedrock.Numerical results are presented to demonstrate the effect of soil layer thickness,embedment depth,mass ratio,internal friction of the poroelastic soil and pore fluid on the torsional dynamic response.
本文基于Biot饱和多孔介质理论,首次用解析的方法较系统地研究了饱和地基中埋置刚性圆柱基础的动力振动问题。主要工作有:
对于饱和半空间中埋置刚性圆柱基础的竖向振动,基础中心受到简谐竖向激振力,基础与地基完全粘着接触。假定基底与地基的接触面光滑,接触面为完全透水。地基作用在基础上的总力由基础底面反力和基础侧面反力两部分组成。求解基础底面的反力时,引入Novak简化模型将基底以下的土视为与上覆土层无关的均质饱和半空间。首先运用Hankel积分变换技术求解饱和土动力控制方程,得到其在变换域内的一组通解,然后考虑基底与地基接触面的混合边值条件建立了一组描述刚性基础竖向振动的对偶积分方程,并化该对偶积分方程为第二类Fredholm积分方程,通过解此Fredholm积分方程求得了土体作用在基础底面的反力。求解基础侧面的反力时,假设基础侧壁的土体由若干极薄饱和层所组成,利用极薄饱和层在圆孔周边处的力-位移关系和沿基础侧面积分的方法,求得土体作用在基础侧面的反力。最后结合基础的动力平衡方程,求得了饱和半空间中埋置刚性圆柱基础竖向振动时的动力刚度、振动振幅的表达式。通过与前人研究成果的对比验证了本文结果的正确性。数值算例分析了无量纲激振频率、基础埋深、基础质量比、饱和土粘滞系数、孔隙水对竖向动力响应的影响。
在工程中具有有限深度、下卧有硬层的单层饱和地基很常见。因此文中采用Novak简化计算模型接着研究了下卧基岩单层饱和地基中埋置刚性基础的竖向振动,分析了不同土层厚度下基础的竖向动力刚度、动力放大系数曲线。
对于饱和半空间中埋置刚性圆柱基础的摇摆振动,假定基础与地基完全粘着接触,基底与地基的接触面为完全透水或完全不透水,本文用同样的简化计算模型求解了相应的动力相互作用问题。数值算例分析了无量纲激振频率、基础埋深、基础质量比、饱和土粘滞系数、孔隙水以及接触面透水条件对基础摇摆动力刚度、摇摆角位移幅值的影响。
考虑到地基的成层性,本文还研究了下卧基岩单层饱和地基中埋置刚性圆柱基础的摇摆振动问题,分析了土层厚度、基础埋深、饱和土粘滞系数、孔隙水对埋置基础与单层饱和地基摇摆动力相互作用的影响。
最后,文中采用相同的Novak简化模型,对饱和地基中埋置刚性圆柱基础的扭转振动进行了研究。考虑地基为饱和半空间和下卧基岩单层饱和地基两种情况,求得了刚性埋置基础扭转振动时的动力刚度、扭转角位移幅值。算例分析研究了土层厚度、基础埋深、基础质量比、饱和土粘滞系数、孔隙水对饱和地基中刚性埋置基础扭转动力响应的影响。
The study on dynamic soil-foundation interaction problem has been of considerable interest in the field of geomechanics and civil engineering.It plays an important role in the development of the elastodynamics and has important practical applications in machine foundations design as well.
Based on the Biot's poroelastodynamic theory,the dynamic vibrations of a rigid cylindrical embedded foundation in poroelastic soil are studied by analytical method in detail for the first time.The main work is as follows:
For the vertical vibration of a rigid cylindrical foundation embedded in a poroelastic half-space,the foundation is subjected to vertical time-harmonic excitation along the vertical axis and is perfectly bonded to the surrounding soil.It is assumed that the contact surface between the foundation base and the poroelastic soil is smooth and fully permeable.The total dynamic reaction of the soil at the foundation is composed of a reaction acting in the foundation base and a reaction on the foundation vertical sides.To study the reaction acting in the foundation base,the Novak approximate method is introduced based on the assumption that the soil underlying the foundation base is a homogeneous poroelastic half-space which is independent of the overlying soil.First,the dynamic governing equations of poroelastic soil are solved by using the Hankel transform method and the general solutions are formulated in the Hankel transform fields.Then,considering the mixed boundary-value condition at the interface between the foundation base and the poroelastic soil,the vertical vibration of a rigid foundation is formulated in a set of dual integral equations,which are further reduced to Fredholm integral equation of the second kind.Consequently, the soil reaction acting in the foundation base is derived from solving this Fredholm integral equation.To study the reaction acting on the foundation vertical sides,the soil along foundation vertical sides is modeled as an independent stratum composed of a series of infinitesimally thin poroelastic layers.By using the force-displacement relationship of the infinitesimally thin poroelastic layers and the integration along the circumference of the cylinder,the soil reaction acting on the foundation vertical sides can be derived.Finally,the expressions for the dynamic impedance and displacement amplitude are obtained for the vertical vibration of a rigid cylindrical foundation embedded in a poroelastic half-space by combing the dynamic equilibrium equation of the foundation.The accuracy of the present solution is verified by comparisons with the existing solutions obtained by other researchers.Numerical results are presented to demonstrate the influence of nondimensional frequency of excitation, embedment depth,mass ratio of the foundation,internal friction of the poroelastic soil and pore fluid on the vertical dynamic response.
In engineering applications,there are many cases in which the depth of soil layer thickness overlaying rigid base is not large enough.So the vertical vibration of rigid foundation embedded in a single-layered poroelastic soil based on rigid bedrock is then studied by using the same approximate method.The vertical dynamic impedance and dynamic response factor for different values of soil layer thickness are discussed.
For the rocking vibration of a rigid cylindrical foundation embedded in a poroelastic half-space,the dynamic interaction problem is also solved by using the same approximate method.The foundation is assumed to be perfectly bonded to the surrounding soil and the contact surface between the foundation base and the poroelastic soil can be either fully permeable or impermeable.The numerical results are presented to analysis the effect of nondimensional frequency of excitation, embedment depth,mass ratio of the foundation,internal friction of the poroelastic soil, pore fluid and hydraulic boundary condition along the contact surface on the rocking dynamic impendence and angular displacement amplitude of the foundation.
Considering the layered property,of soil,the author further analyse the rocking vibration of a rigid foundation embedded in a single-layered poroelastic soil based on rigid bedrock.The effect of soil layer thickness,embedment depth,internal friction of the poroelastic soil and pore fluid on the rocking dynamic soil-foundation interaction problem is studied.
Finally,the torsional vibration of a rigid cylindrical foundation embedded in poroelastic soil is discussed by the same Novak approximate model.The technique is applied to the computation of torsional dynamic impedance and angular displacement of the rigid foundation embedded in a poroelastic half-space and in a single-layered poroelastic soil based on rigid bedrock.Numerical results are presented to demonstrate the effect of soil layer thickness,embedment depth,mass ratio,internal friction of the poroelastic soil and pore fluid on the torsional dynamic response.
引文
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Reissner E. Stationare, axialsymmetrische, durch eine schuttelide masse erregte Schwingungen einrs homogenen elastischen halfbraumes. Archive of Applied Mechanics, 1936,7: 381-396.
Robertson IA. Forced vertical vibration of a rigid circular disk on a semi-infinite elastic solid. Mathematical Proceedings of the Cambridge Philosophical Society, 1966,62A: 547-553.
Robertson IA. On a proposed determination of the shear modulus of an isotropic half-space by the forced torsional oscillations of a circular disc. Applied Scientific Research, 1966, 17: 305-312.
Sankaran KS, Krishnaswamy NR, Nair PGB. Torsion vibration tests on embedded footings.Journal of the Geotechnical Engineering, 1980,106(GT3): 325-331.
Selvadurai APS. Rotary oscillations of a rigid disc inclusion embedded in an isotropic elastic infinite space. International Journal of Solids and Structures, 1981,17:493-498.
Senjuntichai T, Mani S, Rajapakse RKND. Vertical vibration of an embedded rigid foundation in a poroelastic soil. Soil Dynamics and Earthquake Engineering, 2006,26(6-7): 626-636.
Senjuntichai T, Rajapakse RKND. Dynamics of a rigid strip bonded to a multilayered poroelastic half-space. Mechanics of poroelastic media, 1996, A.P.S. Selvadurai, editor, Kluwer Academic Publisher, Dordrech1, The Netherlands, Boston and Londan: 353-369.
Senjuntichai T, Sapsathiarn Y. Forced vertical vibration of circular plate in multilayered poroelastic medium. Journal of the Engineering Mechanics Division, ASCE, 2003, 129(11):1330-1341.
Sneddon I. The use of integral transforms. New York: Mcgraw-Hill, 1970.
Spyrakos CC, Xu CJ. Dynamic analysis of flexible massive strip-foundations embedded in layered soils by hybrid BEM-FEM.Computers & Structures,2004,82(29-30):2541-2550.
Takemiya H,Guan F.Transient Lamb's solution strip impulses.Journal of Engineering Mechanics,1993,119:2385-2403.
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Zeng X,Cakmak AS.Dynamic structure-soil interaction under the action of vertical ground motion,Part Ⅱ:the plane problem.Soil Dynamics and Earthquake Engineering,1984b,3(4):211-219.
Zeng X,Rajapakse RKND.Vertical vibrations of a rigid disk embedded in a poroelastic medium.International Journal for Numerical and Analytical Methods in Geomechanics,1999,23(15):2075-2095.
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陈胜立.饱和地基上基础与板的竖向振动特性研究[博士学位论文].杭州,浙江大学,2000.
陈胜立,张建民,陈龙珠.饱和地基中埋置点源荷载的动力Green函数.岩土工程学报,2001.23(4):423-426.
陈胜立,张建民,陈龙珠.下卧刚性基岩的饱和地基上基础的动力分析.固体力学学报,2002. 23(3):325-329.
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黄秋菊.瑞利波在饱和土中的传播[硕士学位论文].杭州,浙江大学,1997.
金波.层状地基及其动力基础的计算[博士学位论文].杭州,浙江大学,1994.
罗松南,郭平.粘弹性层状半空间的动力响应及其在动力基础中的应用.湖南大学学报,1993,20(1):57-64.
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王建华,陆建飞,沈为平.半空间饱和土在内部简谐垂直力作用下的Green函数.水利学报,2001.3:54-57.
王国才.饱和地基上基础的扭转特性研究[博士学位论文].杭州,浙江大学,2002.
吴大志,张政伟。徐长节,刘飞禹.下卧基岩双层地基上刚性圆板的扭转振动.岩石力学与工程学报,2005,24(18):3316-3321.
杨峻,吴世明.非均质流固耦合介质轴对称动力问题的时域解.力学学报,1996,28(3):308-318.
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Rajapakse RKND, Shan AH, Datta SK. Torsional vibrations of elastic foundations embedded in an elastic half-space. Earthquake Engineering and Structural Dynamics, 1987,15: 279-297.
Reissner E. Stationare, axialsymmetrische, durch eine schuttelide masse erregte Schwingungen einrs homogenen elastischen halfbraumes. Archive of Applied Mechanics, 1936,7: 381-396.
Robertson IA. Forced vertical vibration of a rigid circular disk on a semi-infinite elastic solid. Mathematical Proceedings of the Cambridge Philosophical Society, 1966,62A: 547-553.
Robertson IA. On a proposed determination of the shear modulus of an isotropic half-space by the forced torsional oscillations of a circular disc. Applied Scientific Research, 1966, 17: 305-312.
Sankaran KS, Krishnaswamy NR, Nair PGB. Torsion vibration tests on embedded footings.Journal of the Geotechnical Engineering, 1980,106(GT3): 325-331.
Selvadurai APS. Rotary oscillations of a rigid disc inclusion embedded in an isotropic elastic infinite space. International Journal of Solids and Structures, 1981,17:493-498.
Senjuntichai T, Mani S, Rajapakse RKND. Vertical vibration of an embedded rigid foundation in a poroelastic soil. Soil Dynamics and Earthquake Engineering, 2006,26(6-7): 626-636.
Senjuntichai T, Rajapakse RKND. Dynamics of a rigid strip bonded to a multilayered poroelastic half-space. Mechanics of poroelastic media, 1996, A.P.S. Selvadurai, editor, Kluwer Academic Publisher, Dordrech1, The Netherlands, Boston and Londan: 353-369.
Senjuntichai T, Sapsathiarn Y. Forced vertical vibration of circular plate in multilayered poroelastic medium. Journal of the Engineering Mechanics Division, ASCE, 2003, 129(11):1330-1341.
Sneddon I. The use of integral transforms. New York: Mcgraw-Hill, 1970.
Spyrakos CC, Xu CJ. Dynamic analysis of flexible massive strip-foundations embedded in layered soils by hybrid BEM-FEM.Computers & Structures,2004,82(29-30):2541-2550.
Takemiya H,Guan F.Transient Lamb's solution strip impulses.Journal of Engineering Mechanics,1993,119:2385-2403.
Thomas DP.A note on the torsional oscillations of an elastic half space.International Journal of Engineering Science,1968,6:565-570.
Thomson WT.Transmission of elastic waves through a stratified medium.Journal of Applied Mechanics,1950,21:89-93.
Titchmarsh EC.Introduction to the theory of Fourier integrals.Oxford,1937.
Tranter CJ.On some dual integral equations occurring in potential problems with axial symmetry.Quartly Journal of Mechanics and Applied Mathematics,1950,3:411-419.
Tranter CJ.A further note on dual integral equations and application to the diffraction of electromagnetic waves.Quartly Journal of Mechanics and Applied Mathematics,1954,Ⅶ:315-325.
Tsai YM.Torsional vibration of a circular disk on an infinite transversely isotropic.International Journal of Solids and Structures,1989,25(9):1069-1076.
Veletsos AS,Verbic B.Vibration of viscoelastic foundations.Earthquake Engineering and Structural Dynamics,1973,2(1):87-102.
Warburton GB.Forced vibration of a body on an elastic stratum.Journal of Applied Mechanics,1957,24:55-58.
Watson GN.A treatise of the theory of Bessel functions.Cambridge University Press,London,1944.
Wong HL,Luco JE.Dynamic response of rigid foundations of arbitrary shape.Earthquake Engineering & Structural Dynamics,1976,4:579-587.
Zeng X,Cakmak AS.Dynamic structure-soil interaction under the action of vertical ground motion,Part Ⅰ:the axisymmetric problem.Soil Dynamics and Earthquake Engineering,1984a,3(4):200-210.
Zeng X,Cakmak AS.Dynamic structure-soil interaction under the action of vertical ground motion,Part Ⅱ:the plane problem.Soil Dynamics and Earthquake Engineering,1984b,3(4):211-219.
Zeng X,Rajapakse RKND.Vertical vibrations of a rigid disk embedded in a poroelastic medium.International Journal for Numerical and Analytical Methods in Geomechanics,1999,23(15):2075-2095.
陈龙珠.饱和土中弹性波的传播速度及其应用[博士学位论文].杭州,浙江大学.1987.
陈胜立.饱和地基上基础与板的竖向振动特性研究[博士学位论文].杭州,浙江大学,2000.
陈胜立,张建民,陈龙珠.饱和地基中埋置点源荷载的动力Green函数.岩土工程学报,2001.23(4):423-426.
陈胜立,张建民,陈龙珠.下卧刚性基岩的饱和地基上基础的动力分析.固体力学学报,2002. 23(3):325-329.
付兵.饱和地基上基础的摇摆振动特性研究[博士学位论文].杭州,浙江大学,2005.
黄秋菊.瑞利波在饱和土中的传播[硕士学位论文].杭州,浙江大学,1997.
金波.层状地基及其动力基础的计算[博士学位论文].杭州,浙江大学,1994.
罗松南,郭平.粘弹性层状半空间的动力响应及其在动力基础中的应用.湖南大学学报,1993,20(1):57-64.
门福禄.波在饱水孔隙弹性介质中的传播.地球物理学报,1965,14(2):107-144.
门福禄.波在饱含流体的孔隙介质中的传播问题.地球物理学报,1981,24(1):64-67.
王立忠,陈云敏,吴世明 丁皓江.饱和弹性半空间在低频谐和集中力下的积分形式解.水利学报,1996,2:84-88.
王建华,陆建飞,沈为平.半空间饱和土在内部简谐垂直力作用下的Green函数.水利学报,2001.3:54-57.
王国才.饱和地基上基础的扭转特性研究[博士学位论文].杭州,浙江大学,2002.
吴大志,张政伟。徐长节,刘飞禹.下卧基岩双层地基上刚性圆板的扭转振动.岩石力学与工程学报,2005,24(18):3316-3321.
杨峻,吴世明.非均质流固耦合介质轴对称动力问题的时域解.力学学报,1996,28(3):308-318.