饱和层状地基的薄层法基本解及其旁轴边界
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摘要
在Biot饱和多孔介质理论的基础上,应用Fourier级数展开和Hankel变换方法,将Biot波动方程转化为一组二阶常微分方程;运用薄层法在竖向进行离散求解,最终得到层状饱和地基Lamb问题的解答。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通用旁轴近似解答,提高了薄层法的计算效率。最后,通过两个计算实例验证了本文解答。
The dynamic response of saturated layered ground subjected to harmonic force is investigated.By using the Fourier series expansion and Hankel transform,the Biot’s dynamic equations in each layer are reduced into a series of ordinary differential equations,which can be solved by the thin layered method(TLM).And the general solutions for displacement and pore pressure in frequency domain are derived by analytical inversion of the Hankel transform.In order to overcome the problem that TLM only can work in finite depth,a solution of paraxial approxination,which can act as the underlayer boundary of TLM,is obtained by using the Taylor’s series expansion with respect to the stiffness matrix of saturated half-space soil in wave-number domain.Finally,two examples are presented to validate the proposed method.
The dynamic response of saturated layered ground subjected to harmonic force is investigated.By using the Fourier series expansion and Hankel transform,the Biot’s dynamic equations in each layer are reduced into a series of ordinary differential equations,which can be solved by the thin layered method(TLM).And the general solutions for displacement and pore pressure in frequency domain are derived by analytical inversion of the Hankel transform.In order to overcome the problem that TLM only can work in finite depth,a solution of paraxial approxination,which can act as the underlayer boundary of TLM,is obtained by using the Taylor’s series expansion with respect to the stiffness matrix of saturated half-space soil in wave-number domain.Finally,two examples are presented to validate the proposed method.
引文
[1]BOIT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid:I low-frequency range[J].J Acoust Soc Am,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].J Acoust Soc Am,1956,28(2):179–191.
[3]PHILIPPACOPOULOS A J.Lamb’s problem for fluid-saturated porous media[J].Bull Seism Soc Am,1988,78(2):908–923.
[4]SENJUNTICHAI T,RAJAPAKEE RKND.Dynamic Green’s function of homogeneous poroelastic half-space[J].J Eng Mech,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):84–89.(WANG Li-zhong,CHEN Yun-min,WU Shi-ming,et al.A solution to saturated elastic half-space under harmonic force of low frequency[J].Journal of Hydraulic Engineering,1996(2):84–89.(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 Zhi-xin.The axially symmetric mixed boundary-value problem of the vertical vibration of a rigid body on fluid-saturated porous half space[J].Acta Mechanica Sinica,1997,29(6):711–719.(in Chinese))
[7]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 Sheng-li,ZHANG Jian-min,CHEN Long-zhu.Three-dimensional Lamb’s problem of saturated soils subjected to internal excitation[J].Acta Mechanica Solida Sinica,2004,25(2):149–154.(in Chinese))
[8]LU Jian-fei,WANG Jian-hua,SHEN Wei-ping.The Green’s function of the harmonic horizontal force applied at the interior of the saturated half-space soil[J].Acta Seismologica Sinica,2001,14(2):196–202.
[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 Xiang-lian,WANG Jian-hua,LU Jian-fei.Transient response of the surface of saturated soil under the horizontal loads[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.(HUANG Yi,ZHANG Yu-hong.Three-dimensional non-aximmetric Lamb’s problem of saturated soil[J].Science in China(Series E),2000,30(4):375–384.(in Chinese))
[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 Yin-ke,HUANG Yi.Steady dynamic response of elastic saturated porous medium under non-axisymmetrical load[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):56–69.(ZHANG Yin-ke,HUANG Yi.The non-axisymmetrical dynamic response of transversely isotropic saturated poroelastic media[J].Applied Mathematics and Mechanics,2001,22(1):56–69.(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].应用数学与力学,2005,26(11):1278–1286.(WANG Xiao-gang,HUANG Yi.3-D dynamic response of transversely isotropic saturated soils[J].Applied Mathematics 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].应用数学与力学,2006,27(1):75–80.(CAI Yuan-qiang,XU Chang-jie,ZHENG Zao-feng,et,al.Vertical vibration analysis of axisymmetric saturated soil[J].Applied Mathematics and Mechanics,2006,27(1):75–80.(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 Yin-ke,ZAN Hui-ping,HUANG Yi.Transfer matrix method for solving non-axisymmetrical dynamic problem in fluid-saturated poroelastic half-space[J].Journal of Shaanxi Normal University(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 Yu-hong,HUANG Yi,WANG Zhong-jian.Analysis procedures of dynamic response of 3-D non-axisymmetric layered saturated soil[J].Earthquake Engineering and Engineering Vibration,2002,22(4):47–52.(in Chinese))
[17]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 Xiao-gang.The 3-D dynamic response of layered transversely isotropic saturated soils to an abitrary by hamonic source[J].Acta Mechanica Solida Sinica,2006,27(3):223–229.(in Chinese))
[18]KAUSEL E.An explicit solution for the Green functions for dynamic loads in layered media[R].Cambridge:MIT,1981.
[19]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.(SHI Gang.The propagation of Rayleigh wave and vibration isolation by wave barriers in saturated soil[D].Shanghai:Tongji University,2008.(in Chinese))
[20]KAUSEL E,PEEK R.Dynamic loads in the interior of a layered stratum:an explicit solution[J].Bulletin of the Seismological Society of America,1982,72(5):1459–1480
[21]SEAL S H.Dynamic loads in layered half-spaces[D].Cambridge:Department of Civil Engineering,MIT,1985.
[22]王贻荪.地面波动分析若干问题[J].建筑结构学报,1982,4(2):56–67.(WANG Yi-sun.Some problems in the ground vibration analysis[J].Journal of Building Structures.1982,4(2):56–67.(in Chinese))
[2]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid:II higher frequency range[J].J Acoust Soc Am,1956,28(2):179–191.
[3]PHILIPPACOPOULOS A J.Lamb’s problem for fluid-saturated porous media[J].Bull Seism Soc Am,1988,78(2):908–923.
[4]SENJUNTICHAI T,RAJAPAKEE RKND.Dynamic Green’s function of homogeneous poroelastic half-space[J].J Eng Mech,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):84–89.(WANG Li-zhong,CHEN Yun-min,WU Shi-ming,et al.A solution to saturated elastic half-space under harmonic force of low frequency[J].Journal of Hydraulic Engineering,1996(2):84–89.(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 Zhi-xin.The axially symmetric mixed boundary-value problem of the vertical vibration of a rigid body on fluid-saturated porous half space[J].Acta Mechanica Sinica,1997,29(6):711–719.(in Chinese))
[7]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 Sheng-li,ZHANG Jian-min,CHEN Long-zhu.Three-dimensional Lamb’s problem of saturated soils subjected to internal excitation[J].Acta Mechanica Solida Sinica,2004,25(2):149–154.(in Chinese))
[8]LU Jian-fei,WANG Jian-hua,SHEN Wei-ping.The Green’s function of the harmonic horizontal force applied at the interior of the saturated half-space soil[J].Acta Seismologica Sinica,2001,14(2):196–202.
[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 Xiang-lian,WANG Jian-hua,LU Jian-fei.Transient response of the surface of saturated soil under the horizontal loads[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.(HUANG Yi,ZHANG Yu-hong.Three-dimensional non-aximmetric Lamb’s problem of saturated soil[J].Science in China(Series E),2000,30(4):375–384.(in Chinese))
[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 Yin-ke,HUANG Yi.Steady dynamic response of elastic saturated porous medium under non-axisymmetrical load[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):56–69.(ZHANG Yin-ke,HUANG Yi.The non-axisymmetrical dynamic response of transversely isotropic saturated poroelastic media[J].Applied Mathematics and Mechanics,2001,22(1):56–69.(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].应用数学与力学,2005,26(11):1278–1286.(WANG Xiao-gang,HUANG Yi.3-D dynamic response of transversely isotropic saturated soils[J].Applied Mathematics 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].应用数学与力学,2006,27(1):75–80.(CAI Yuan-qiang,XU Chang-jie,ZHENG Zao-feng,et,al.Vertical vibration analysis of axisymmetric saturated soil[J].Applied Mathematics and Mechanics,2006,27(1):75–80.(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 Yin-ke,ZAN Hui-ping,HUANG Yi.Transfer matrix method for solving non-axisymmetrical dynamic problem in fluid-saturated poroelastic half-space[J].Journal of Shaanxi Normal University(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 Yu-hong,HUANG Yi,WANG Zhong-jian.Analysis procedures of dynamic response of 3-D non-axisymmetric layered saturated soil[J].Earthquake Engineering and Engineering Vibration,2002,22(4):47–52.(in Chinese))
[17]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 Xiao-gang.The 3-D dynamic response of layered transversely isotropic saturated soils to an abitrary by hamonic source[J].Acta Mechanica Solida Sinica,2006,27(3):223–229.(in Chinese))
[18]KAUSEL E.An explicit solution for the Green functions for dynamic loads in layered media[R].Cambridge:MIT,1981.
[19]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.(SHI Gang.The propagation of Rayleigh wave and vibration isolation by wave barriers in saturated soil[D].Shanghai:Tongji University,2008.(in Chinese))
[20]KAUSEL E,PEEK R.Dynamic loads in the interior of a layered stratum:an explicit solution[J].Bulletin of the Seismological Society of America,1982,72(5):1459–1480
[21]SEAL S H.Dynamic loads in layered half-spaces[D].Cambridge:Department of Civil Engineering,MIT,1985.
[22]王贻荪.地面波动分析若干问题[J].建筑结构学报,1982,4(2):56–67.(WANG Yi-sun.Some problems in the ground vibration analysis[J].Journal of Building Structures.1982,4(2):56–67.(in Chinese))
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