准饱和土中圆柱形衬砌的瞬态动力响应分析
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摘要
内源瞬态荷载作用下圆柱形孔洞的动力响应解答是土动力学的经典问题之一。已有研究大都假设孔洞周围土体为理想弹性介质或完全饱和多孔介质。然而,实际工程中不存在完全弹性和完全饱和土体。分别视衬砌结构和周围土体为弹性材料和准饱和多孔介质(饱和度Sr≥95%),根据牛顿第二定律、达西定律和Biot波动理论推导出准饱和土体的控制方程。根据边界条件导出衬砌和土体的位移、应力和孔隙压力的Laplace变换空间的解答。利用反Laplace变换数值计算方法,将解答转换为时域解。分析了饱和度对衬砌位移、应力和孔压的影响,结果表明,当95%≤Sr≤99%时,饱和度对径向位移和切向应力的影响较小;99%≤Sr≤100%时,饱和度对径向位移和切向应力的影响较大;但饱和度对孔隙压力的影响远大于对径向位移和切向应力的影响。得出位移、应力和孔压沿径向的衰减规律,当95%≤Sr≤99%时,饱和度对径向位移和切向应力沿径向衰减影响较小,99%≤Sr≤100%时,饱和度对径向位移和切向应力沿径向衰减影响较大,但饱和度对孔压沿径向的衰减影响远大于对径向位移和切向应力沿径向的衰减。
A solution to dynamic response of cylindrical lined cavity under internal transient load is one of the classical soil dynamics problems. Most previous literatures about the problem assumed that soil surrounding with cavity is an elastic medium or full-saturated poroelastic medium. In fact, perfectly elastic and completely saturated soil does not exist in practical engineering. The lining structure and its surrounding soil are treated as an ideal elastic medium and a nearly saturated poroelastic medium, respectively. On the basis of Newton's second law, Darcy's law and Biot's theory, the governing equation of nearly saturated soil is derived. According to boundary conditions, solutions for the radial displacement, hoop stress and pore stress of the lining and saturated soil are obtained in Laplace transform space; and then they are transformed to solutions in time domain using inverse Laplace transforms numerical method. Also, the influence of the saturation degree(Sr) on the dynamic response of the lining and soil is investigated. It is indicated that the saturation degree has a less effect on the radial displacement and hoop stress when 95%≤Sr≤ 99% and has a significant influence on them when 99%≤Sr≤ 100%. But the saturation degree has much more effect on the pore stress than on the former two. The same conclusion can be drawn that the influence of saturation degree on the attenuation of dynamics response along radial direction. The saturation degree has a less effect on the radial displacement and hoop stress when 95%≤Sr≤ 99% and has a significant influence on them when 99%≤Sr≤100%. It also indicates that the saturation degree on the attenuation has more effect on pore stress than that of raidial displacement and hoop stress.
A solution to dynamic response of cylindrical lined cavity under internal transient load is one of the classical soil dynamics problems. Most previous literatures about the problem assumed that soil surrounding with cavity is an elastic medium or full-saturated poroelastic medium. In fact, perfectly elastic and completely saturated soil does not exist in practical engineering. The lining structure and its surrounding soil are treated as an ideal elastic medium and a nearly saturated poroelastic medium, respectively. On the basis of Newton's second law, Darcy's law and Biot's theory, the governing equation of nearly saturated soil is derived. According to boundary conditions, solutions for the radial displacement, hoop stress and pore stress of the lining and saturated soil are obtained in Laplace transform space; and then they are transformed to solutions in time domain using inverse Laplace transforms numerical method. Also, the influence of the saturation degree(Sr) on the dynamic response of the lining and soil is investigated. It is indicated that the saturation degree has a less effect on the radial displacement and hoop stress when 95%≤Sr≤ 99% and has a significant influence on them when 99%≤Sr≤ 100%. But the saturation degree has much more effect on the pore stress than on the former two. The same conclusion can be drawn that the influence of saturation degree on the attenuation of dynamics response along radial direction. The saturation degree has a less effect on the radial displacement and hoop stress when 95%≤Sr≤ 99% and has a significant influence on them when 99%≤Sr≤100%. It also indicates that the saturation degree on the attenuation has more effect on pore stress than that of raidial displacement and hoop stress.
引文
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[2]GAO M,WANG Y,GAO G Y,et al.An analytical solution for the transient response of a cylindrical lined cavity in a poroelastic medium[J].Soil Dynamics and Earthquake Engineering,2013,46(3):30-40.
[3]高盟,高广运,李大勇.考虑耦合质量影响的均布突加荷载作用下衬砌结构的瞬态响应[J].岩土工程学报,2011,33(6):862-868.GAO Meng,GAO Guang-yun,LI Da-yong.Transient response of lining structure subjected to sudden internal uniform loading considering effects of coupling mass[J].Chinese Journal of Geotechnical Engineering,2011,33(6):862-868.
[4]高盟,高广运.饱和地基任意形状衬砌内部均布突加荷载作用下的动力响应[J].固体力学学报,2011,32(6):655-664.GAO Meng,GAO Guang-yun.Dynamic response of lining in saturated foundation subjected to sudden internal uniform loading[J].Chinese Journal of Solid Mechanics,2011,32(6):655-664.
[5]蔡袁强,陈成振,孙宏磊.爆炸荷载作用下饱和土中隧道的瞬态动力响应[J].岩土工程学报,2011,33(3):361-367.CAI Yuan-qiang,CHEN Cheng-zhen,SUN Hong-lei.Transient dynamic response of tunnels subjected to blast loads in saturated soil[J].Chinese Journal of Geotechnical Engineering,2011,33(3):361-367.
[6]闻敏杰,杨骁,高华喜.具有球形空腔饱和土分数导数黏弹性土的动力特性[J].岩土力学,2013,34(4):1001-1008.WEN Min-jie,YANG Xiao,Gao Hua-xi.Dynamic characteristics of saturated fractional derivative viscoelastic soil with a spherical cavity[J].Rock and Soil Mechanics,2013,34(4):1001-1008.
[7]RAATS P A C,KLUTE A.Transport in soils:the balance of mass[J].Journal of the Acoustical Society of America,1968,32(2):161-166.
[8]RAATS P A C,KLUTE A.Transport in soils:the balance of momentum[J].Journal of the Acoustical Society of America,1968,32(4):452-456.
[9]VERRUIJT A.In:De Wiest R.J.M.editor.Flow through porous media.New York and London:Academic Press;1969.
[10]VERRUIJT A.Some remarks on the principle of effective stress[C]//IUTAM Conference on Deformation and Failure of Granular Materials.Delft,Netherlands:[s.n.],1982:167-181.
[11]YAMAMOTO T,KONING H L,SELLMEIJER H,et al.On the response of a poro-elastic bed to water waves[J].Journal of Fluid Mechanics,1978,87(1):193-206.
[12]MADSEN O S.Wave-induced pore pressures and effective stresses in porous bed[J].Geotechnique,1978,28(4):377-393.
[13]MEI C C,FODA M A.Wave-induced responses in a fluid filled poro-elastic solid with a free surface[J].Geophysical Journal of the Royal Astronomical Society,1981,66(3):579-596.
[14]VARDOULAKIS I,BESKOS D E.On the dynamic behavior of nearly saturated granular media[C]//Nemat Nasser editor,Geomechanics 1983;AMD 57,ASME,New York,65-73.
[15]VARDOULAKIS I,BESKOS D E.Dynamic behavior of nearly saturated porous media[J].Mechanics of Materials,1986,5:87-108
[16]BARDET J P,SAYED H.Velocity and attenuation of compressional waves in nearly saturated soils[J].Soil Dynamics and Earthquake Engineering,1993,12(7):391-401.
[17]李保忠,蔡袁强.饱和度对横观各向同性准饱和多孔介质中弹性波传播的影响[J].水利学报,2003,34(9):94-101.LI Bao-zhong,CAI Yuan-qiang.Influence of water saturation on elastic wave propagation in transversely isotropic quasi-saturated porous media[J].Journal of Hydraulic Engineering,2003,34(9):94-101.
[18]马晓华,徐长节,蔡袁强.黏弹性土层中竖向振动放大系数的研究[J].岩石力学与工程学报,2005,24(24):4534-4539.MA Xiao-hua,XU Chang-jie,CAI Yuan-qiang.Analytical study on vertical vibration amplification of a viscoelastic soil layer[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(24):4534-4539.
[19]徐长节,马晓华,蔡袁强.弹性准饱和土中球空腔的动力响应[J].岩土工程学报,2005,27(8):934-938.XU Chang-jie,MA Xiao-hua,CAI Yuan-qiang.Dynamic response of spherical cavity in nearly saturated elastic soils[J].Chinese Journal of Geotechnical Engineering,2005,27(8):934-938.
[20]徐平,夏唐代.饱和度对准饱和土体中瑞利波传播特性的影响[J].振动与冲击,2008,27(4):10-13.XU Ping,XIA Tang-dai.Influence of saturation degree on propagation characteristics of Rayleigh waves in nearly saturated soils[J].Journal of Vibration and Shock,2008,27(4):10-13.
[21]高华喜,闻敏杰.黏弹性准饱和土中球空腔动力特性.力学学报,2012,44(4):753-761.GAO Hua-xi,WEN Min-jie.Dynamic characteristics of a spherical cavity in nearly saturated viscoelastic soil[J].Acta Mechanica Sinica,2012,44(4):753-761.
[22]BIOT M A.Theory of propagation of elastic waves in fluid-saturated porous solid,Ⅰ:low frequency range[J].Journal of the Acoustical Society of America,1956,28(2):168-179.
[23]DURBIN F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate’s method[J].The Computer Journal,1974,17(4):371-376.
[2]GAO M,WANG Y,GAO G Y,et al.An analytical solution for the transient response of a cylindrical lined cavity in a poroelastic medium[J].Soil Dynamics and Earthquake Engineering,2013,46(3):30-40.
[3]高盟,高广运,李大勇.考虑耦合质量影响的均布突加荷载作用下衬砌结构的瞬态响应[J].岩土工程学报,2011,33(6):862-868.GAO Meng,GAO Guang-yun,LI Da-yong.Transient response of lining structure subjected to sudden internal uniform loading considering effects of coupling mass[J].Chinese Journal of Geotechnical Engineering,2011,33(6):862-868.
[4]高盟,高广运.饱和地基任意形状衬砌内部均布突加荷载作用下的动力响应[J].固体力学学报,2011,32(6):655-664.GAO Meng,GAO Guang-yun.Dynamic response of lining in saturated foundation subjected to sudden internal uniform loading[J].Chinese Journal of Solid Mechanics,2011,32(6):655-664.
[5]蔡袁强,陈成振,孙宏磊.爆炸荷载作用下饱和土中隧道的瞬态动力响应[J].岩土工程学报,2011,33(3):361-367.CAI Yuan-qiang,CHEN Cheng-zhen,SUN Hong-lei.Transient dynamic response of tunnels subjected to blast loads in saturated soil[J].Chinese Journal of Geotechnical Engineering,2011,33(3):361-367.
[6]闻敏杰,杨骁,高华喜.具有球形空腔饱和土分数导数黏弹性土的动力特性[J].岩土力学,2013,34(4):1001-1008.WEN Min-jie,YANG Xiao,Gao Hua-xi.Dynamic characteristics of saturated fractional derivative viscoelastic soil with a spherical cavity[J].Rock and Soil Mechanics,2013,34(4):1001-1008.
[7]RAATS P A C,KLUTE A.Transport in soils:the balance of mass[J].Journal of the Acoustical Society of America,1968,32(2):161-166.
[8]RAATS P A C,KLUTE A.Transport in soils:the balance of momentum[J].Journal of the Acoustical Society of America,1968,32(4):452-456.
[9]VERRUIJT A.In:De Wiest R.J.M.editor.Flow through porous media.New York and London:Academic Press;1969.
[10]VERRUIJT A.Some remarks on the principle of effective stress[C]//IUTAM Conference on Deformation and Failure of Granular Materials.Delft,Netherlands:[s.n.],1982:167-181.
[11]YAMAMOTO T,KONING H L,SELLMEIJER H,et al.On the response of a poro-elastic bed to water waves[J].Journal of Fluid Mechanics,1978,87(1):193-206.
[12]MADSEN O S.Wave-induced pore pressures and effective stresses in porous bed[J].Geotechnique,1978,28(4):377-393.
[13]MEI C C,FODA M A.Wave-induced responses in a fluid filled poro-elastic solid with a free surface[J].Geophysical Journal of the Royal Astronomical Society,1981,66(3):579-596.
[14]VARDOULAKIS I,BESKOS D E.On the dynamic behavior of nearly saturated granular media[C]//Nemat Nasser editor,Geomechanics 1983;AMD 57,ASME,New York,65-73.
[15]VARDOULAKIS I,BESKOS D E.Dynamic behavior of nearly saturated porous media[J].Mechanics of Materials,1986,5:87-108
[16]BARDET J P,SAYED H.Velocity and attenuation of compressional waves in nearly saturated soils[J].Soil Dynamics and Earthquake Engineering,1993,12(7):391-401.
[17]李保忠,蔡袁强.饱和度对横观各向同性准饱和多孔介质中弹性波传播的影响[J].水利学报,2003,34(9):94-101.LI Bao-zhong,CAI Yuan-qiang.Influence of water saturation on elastic wave propagation in transversely isotropic quasi-saturated porous media[J].Journal of Hydraulic Engineering,2003,34(9):94-101.
[18]马晓华,徐长节,蔡袁强.黏弹性土层中竖向振动放大系数的研究[J].岩石力学与工程学报,2005,24(24):4534-4539.MA Xiao-hua,XU Chang-jie,CAI Yuan-qiang.Analytical study on vertical vibration amplification of a viscoelastic soil layer[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(24):4534-4539.
[19]徐长节,马晓华,蔡袁强.弹性准饱和土中球空腔的动力响应[J].岩土工程学报,2005,27(8):934-938.XU Chang-jie,MA Xiao-hua,CAI Yuan-qiang.Dynamic response of spherical cavity in nearly saturated elastic soils[J].Chinese Journal of Geotechnical Engineering,2005,27(8):934-938.
[20]徐平,夏唐代.饱和度对准饱和土体中瑞利波传播特性的影响[J].振动与冲击,2008,27(4):10-13.XU Ping,XIA Tang-dai.Influence of saturation degree on propagation characteristics of Rayleigh waves in nearly saturated soils[J].Journal of Vibration and Shock,2008,27(4):10-13.
[21]高华喜,闻敏杰.黏弹性准饱和土中球空腔动力特性.力学学报,2012,44(4):753-761.GAO Hua-xi,WEN Min-jie.Dynamic characteristics of a spherical cavity in nearly saturated viscoelastic soil[J].Acta Mechanica Sinica,2012,44(4):753-761.
[22]BIOT M A.Theory of propagation of elastic waves in fluid-saturated porous solid,Ⅰ:low frequency range[J].Journal of the Acoustical Society of America,1956,28(2):168-179.
[23]DURBIN F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate’s method[J].The Computer Journal,1974,17(4):371-376.
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