二维及轴对称条件下饱和层状地基的Lamb问题解答
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摘要
在Biot饱和两相介质理论的基础上,对二维Biot波动方程进行水平坐标的Fourier变换和相应的变量代换,得到一组以土骨架位移分量与孔压为变量的二阶常微分方程。同时,对轴对称条件下的Biot波动方程进行径向坐标的Hankel变换,得到一组与二维情况相同的常微分方程。为求解该常微分方程组,运用薄层法原理在竖向进行离散求解,最终得到饱和层状地基的动力Lamb问题的基本解答,给出频域内土骨架位移和孔压的显示表达。此外,针对薄层法仅能在有限深度内求解的不足,根据旁轴近似原理,对饱和半空间频率-波数域内的刚度矩阵进行二阶Taylor级数展开,得到适合于薄层法底层边界的旁轴近似解答。最后,通过计算实例验证本文解答。
The dynamic response of saturated layered ground subjected to harmonic force is investigated.By using Fourier transform and variable substitutions,the 2-D Biot's dynamic equations in each layer are reduced into a series of ordinary differential equations.The axisymmetric Biot's dynamic equations are reduced to ordinary differential equations by performing Hankel transform.The equations can be solved by using Thin Layered Method(TLM).The 2-D and axisymmetric solutions for displacement and pore pressure in frequency domain are derived by analytical Fourier transfrom and Hankel transform,respectively.In order to overcome the limitation that TLM only can work in finite depth,a solution of paraxial approximation,which can serve as the under-layer boundary of TLM,is obtained by using Taylor's series expansion with respect to the stiffness matrix of saturated half-space soil in wave-number domain.Several examples are investigated to validate the method.
The dynamic response of saturated layered ground subjected to harmonic force is investigated.By using Fourier transform and variable substitutions,the 2-D Biot's dynamic equations in each layer are reduced into a series of ordinary differential equations.The axisymmetric Biot's dynamic equations are reduced to ordinary differential equations by performing Hankel transform.The equations can be solved by using Thin Layered Method(TLM).The 2-D and axisymmetric solutions for displacement and pore pressure in frequency domain are derived by analytical Fourier transfrom and Hankel transform,respectively.In order to overcome the limitation that TLM only can work in finite depth,a solution of paraxial approximation,which can serve as the under-layer boundary of TLM,is obtained by using Taylor's series expansion with respect to the stiffness matrix of saturated half-space soil in wave-number domain.Several examples are investigated to validate the method.
引文
[1]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous solid I:low-frequency range[J].Journalof the Acoustical Society of America,1956,28(2):168-178
[2]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous solid II:higher frequency range[J].Journal of the Acoustical Society of America,1956,28(2):179-191
[3]Philippacopoulos A J.Lamb’s problem for fluid-saturatedporous media[J].Bulletin of the Seismological Society ofAmerica,1988,78(2):908-932
[4]Senjuntichai T,Rajapakee R K N D.Dynamic Green’sfunction of homogeneous poroelastic half-plane[J].Journalof Engineering Mechanics,1994,120(11):2381-2404
[5]王立忠,陈云敏,吴世明,等.饱和弹性半空间在低频谐和集中力下的积分形式解[J].水利学报,1996,(2):84-88(Wang Lizhong,Chen Yunmin,Wu Shiming,et al.A solution to saturated elastic half-space underharmonic force of low frequency[J].Journal of HydraulicEngineering,1996,(2):84-88(in Chinese))
[6]金波,徐植信.多孔饱和半空间上刚体垂直振动的轴对称混合边值问题[J].力学学报,1997,29(6):711-719(Jin Bo,Xu Zhixin.The axially symmetric mixed boundary-value problem of the vertical vibration of a rigid body onfluid-saturated porous half space[J].Acta MechanicaSinica,1997,29(6):711-719(in Chinese))
[7]Lu Jianfei,Wang Jianhua,Shen Weiping.The green′sfunction of the harmonic horizontal force applied at theinterior of the saturated half-space soil[J].ActaSeismologica Sinica,2001,14(2):196-202
[8]陈胜立,张建民,陈龙珠.饱和土埋置力源的三维动力Lamb问题解答[J].固体力学学报,2004,25(2):149-154(Chen Shengli,Zhang Jianmin,Chen Longzhu.Three-dimensional lamb′s problem of saturated soilssubjected to internal excitation[J].Acta Mechanica SolidaSinica,2004,25(2):149-154(in Chinese))
[9]周香莲,王建华,陆建飞.饱和土表面在水平集中荷载作用下的瞬态反应[J].岩土力学,2002,23(3):387-391(Zhou Xianglian,Wang Jianhua,Lu Jianfei.Transientresponse of the surface of saturated soil under the horizontalloads[J].Rock and Soil Mechanics,2002,23(3):387-391(in Chinese))
[10]黄义,张玉红.饱和土三维非轴对称Lamb问题[J].中国科学,E辑,2000,30(4):375-384
[11]张引科,黄义.弹性饱和多孔介质在非轴对称荷载下的稳态动力响应[J].土木工程学报,2002,35(3):41-45(Zhang Yinke,Huang Yi.Steady dynamic response ofelastic saturated porous medium under non-axisymmetricalload[J].China Civil Engineering Journal,2002,35(3):41-45(in Chinese))
[12]张引科,黄义.横观各向同性饱和弹性多孔介质非轴对称动力响应[J].应用数学和力学,2001,22(1):56-70(Zhang Yinke,Huang Yi.The non-axisymmetrical dynamicresponse of transversely isotropic saturated poroelastic media[J].Applied Mathematics and Mechanics,2001,22(1):56-70(in Chinese))
[13]王小岗,黄义.横观各向同性饱和地基的三维动力响应分析[J].应用数学和力学,2005,26(11):1278-1286(Wang Xiaogang,Huang Yi.3-D dynamic response oftransversely isotropic saturated soils[J].AppliedMathematics and Mechanics,2005,26(11):1278-1286(in Chinese))
[14]蔡袁强,徐长节,郑灶峰,等.轴对称饱和地基竖向振动分析[J].应用数学和力学,2006,27(1):75-81(CaiYuanqiang,Xu Changjie,Zheng Zaofeng,et al.Verticalvibration analysis of axisymmetric saturated soil[J].Applied Mathematics and Mechanics,2006,27(1):75-81(in Chinese))
[15]张引科,昝会萍,黄义.求解弹性饱和多孔介质半空间非轴对称动力问题的传递矩阵法[J].陕西师范大学学报:自然科学版,2000,28(2):44-49(Zhang Yinke,Zan Huiping,Huang Yi.Transfer matrix method for solvingnon-axisymmetrical dynamic problem in fluid-saturatedporoelastic half-space[J].Journal of Shanxi NormalUniversity:Natural Science Edition,2000,28(2):44-49(in Chinese))
[16]张玉红,黄义,王忠建.层状饱水软土地基三维非轴对称动力响应分析方法[J].地震工程与工程振动,2002,22(4):47-52(Zhang Yuhong,Huang Yi,WangZhongjian.Analysis procedures of dynamic response of 3-Dnon-axisymmetric layered saturated soil[J].EarthquakeEngineering and Engineering Vibration,2002,22(4):47-52(in Chinese))
[17]王小岗,黄义.横观各向同性饱和层状地基的三维稳态动力响应[J].工程力学,2006,23(5):132-138(Wang Xiaogang,Huang Yi.3-D dynamic response oftransversely isotropic saturated layered soils to an arbitraryharmonious source[J].Engineering Mechanics,2006,23(5):132-138(in Chinese))
[18]黄义,王小岗.横观各向同性饱和层状地基的三维稳态动力响应[J].固体力学学报,2006,27(3):223-229(Huang Yi,Wang Xiaogang.The 3-D dynamic response oflayered transversely isotropic saturated soils to an abitraryby hamonic source[J].Acta Mechanica Solida Sinica,2006,27(3):223-229(in Chinese))
[19]Bath M.Mathematical aspects of seismology[M].NewYork:Elsevier Publication Corporation,1968
[20]Kausel E,Peek R.Dynamic loads in the interior of alayered stratum:an explicit solution[J].Bulletin of theSeismological Society of America,1982,72(5):1459-1481
[21]时刚.饱和地基中Rayleigh波的传播与屏障隔振研究[D].上海:同济大学,2008
[22]Hull S W,Kausel E.Dynamic loads in layered halfspace[C]//Proceedings of 5th Engineering Mechanics DivisionSpecialty Conference.New York:ASCE,1984:201-204
[23]陈龙珠,陈胜立,梁发云.饱和地基竖向振动的衰减特性[J].上海交通大学学报,2002,36(3):376-381(Chen Longzhu,Chen Shengli,Liang Fayun.Vibrationattenuation of saturated strata by vertical surface loads[J].Journal of Shanghai Jiaotong University,2002,36(3):376-381(in Chinese))
[2]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous solid II:higher frequency range[J].Journal of the Acoustical Society of America,1956,28(2):179-191
[3]Philippacopoulos A J.Lamb’s problem for fluid-saturatedporous media[J].Bulletin of the Seismological Society ofAmerica,1988,78(2):908-932
[4]Senjuntichai T,Rajapakee R K N D.Dynamic Green’sfunction of homogeneous poroelastic half-plane[J].Journalof Engineering Mechanics,1994,120(11):2381-2404
[5]王立忠,陈云敏,吴世明,等.饱和弹性半空间在低频谐和集中力下的积分形式解[J].水利学报,1996,(2):84-88(Wang Lizhong,Chen Yunmin,Wu Shiming,et al.A solution to saturated elastic half-space underharmonic force of low frequency[J].Journal of HydraulicEngineering,1996,(2):84-88(in Chinese))
[6]金波,徐植信.多孔饱和半空间上刚体垂直振动的轴对称混合边值问题[J].力学学报,1997,29(6):711-719(Jin Bo,Xu Zhixin.The axially symmetric mixed boundary-value problem of the vertical vibration of a rigid body onfluid-saturated porous half space[J].Acta MechanicaSinica,1997,29(6):711-719(in Chinese))
[7]Lu Jianfei,Wang Jianhua,Shen Weiping.The green′sfunction of the harmonic horizontal force applied at theinterior of the saturated half-space soil[J].ActaSeismologica Sinica,2001,14(2):196-202
[8]陈胜立,张建民,陈龙珠.饱和土埋置力源的三维动力Lamb问题解答[J].固体力学学报,2004,25(2):149-154(Chen Shengli,Zhang Jianmin,Chen Longzhu.Three-dimensional lamb′s problem of saturated soilssubjected to internal excitation[J].Acta Mechanica SolidaSinica,2004,25(2):149-154(in Chinese))
[9]周香莲,王建华,陆建飞.饱和土表面在水平集中荷载作用下的瞬态反应[J].岩土力学,2002,23(3):387-391(Zhou Xianglian,Wang Jianhua,Lu Jianfei.Transientresponse of the surface of saturated soil under the horizontalloads[J].Rock and Soil Mechanics,2002,23(3):387-391(in Chinese))
[10]黄义,张玉红.饱和土三维非轴对称Lamb问题[J].中国科学,E辑,2000,30(4):375-384
[11]张引科,黄义.弹性饱和多孔介质在非轴对称荷载下的稳态动力响应[J].土木工程学报,2002,35(3):41-45(Zhang Yinke,Huang Yi.Steady dynamic response ofelastic saturated porous medium under non-axisymmetricalload[J].China Civil Engineering Journal,2002,35(3):41-45(in Chinese))
[12]张引科,黄义.横观各向同性饱和弹性多孔介质非轴对称动力响应[J].应用数学和力学,2001,22(1):56-70(Zhang Yinke,Huang Yi.The non-axisymmetrical dynamicresponse of transversely isotropic saturated poroelastic media[J].Applied Mathematics and Mechanics,2001,22(1):56-70(in Chinese))
[13]王小岗,黄义.横观各向同性饱和地基的三维动力响应分析[J].应用数学和力学,2005,26(11):1278-1286(Wang Xiaogang,Huang Yi.3-D dynamic response oftransversely isotropic saturated soils[J].AppliedMathematics and Mechanics,2005,26(11):1278-1286(in Chinese))
[14]蔡袁强,徐长节,郑灶峰,等.轴对称饱和地基竖向振动分析[J].应用数学和力学,2006,27(1):75-81(CaiYuanqiang,Xu Changjie,Zheng Zaofeng,et al.Verticalvibration analysis of axisymmetric saturated soil[J].Applied Mathematics and Mechanics,2006,27(1):75-81(in Chinese))
[15]张引科,昝会萍,黄义.求解弹性饱和多孔介质半空间非轴对称动力问题的传递矩阵法[J].陕西师范大学学报:自然科学版,2000,28(2):44-49(Zhang Yinke,Zan Huiping,Huang Yi.Transfer matrix method for solvingnon-axisymmetrical dynamic problem in fluid-saturatedporoelastic half-space[J].Journal of Shanxi NormalUniversity:Natural Science Edition,2000,28(2):44-49(in Chinese))
[16]张玉红,黄义,王忠建.层状饱水软土地基三维非轴对称动力响应分析方法[J].地震工程与工程振动,2002,22(4):47-52(Zhang Yuhong,Huang Yi,WangZhongjian.Analysis procedures of dynamic response of 3-Dnon-axisymmetric layered saturated soil[J].EarthquakeEngineering and Engineering Vibration,2002,22(4):47-52(in Chinese))
[17]王小岗,黄义.横观各向同性饱和层状地基的三维稳态动力响应[J].工程力学,2006,23(5):132-138(Wang Xiaogang,Huang Yi.3-D dynamic response oftransversely isotropic saturated layered soils to an arbitraryharmonious source[J].Engineering Mechanics,2006,23(5):132-138(in Chinese))
[18]黄义,王小岗.横观各向同性饱和层状地基的三维稳态动力响应[J].固体力学学报,2006,27(3):223-229(Huang Yi,Wang Xiaogang.The 3-D dynamic response oflayered transversely isotropic saturated soils to an abitraryby hamonic source[J].Acta Mechanica Solida Sinica,2006,27(3):223-229(in Chinese))
[19]Bath M.Mathematical aspects of seismology[M].NewYork:Elsevier Publication Corporation,1968
[20]Kausel E,Peek R.Dynamic loads in the interior of alayered stratum:an explicit solution[J].Bulletin of theSeismological Society of America,1982,72(5):1459-1481
[21]时刚.饱和地基中Rayleigh波的传播与屏障隔振研究[D].上海:同济大学,2008
[22]Hull S W,Kausel E.Dynamic loads in layered halfspace[C]//Proceedings of 5th Engineering Mechanics DivisionSpecialty Conference.New York:ASCE,1984:201-204
[23]陈龙珠,陈胜立,梁发云.饱和地基竖向振动的衰减特性[J].上海交通大学学报,2002,36(3):376-381(Chen Longzhu,Chen Shengli,Liang Fayun.Vibrationattenuation of saturated strata by vertical surface loads[J].Journal of Shanghai Jiaotong University,2002,36(3):376-381(in Chinese))
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