分离式双圆形隧道衬砌对平面SH波的散射
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摘要
采用波函数展开法,将入射、散射和折射SH波的位移势函数展开成Fourier-Bessel函数的级数形式,根据Graf加法定理,得到了任一局部圆柱坐标系下的波场势函数的表达式。根据隧道衬砌与周围岩体都在界面处应力和位移连续及衬砌内边界完全自由的边界条件,得到了分离式双隧道衬砌对入射平面SH波的多重散射问题的理论解。通过数值计算分析得出:入射频率、衬砌间距、衬砌与围岩的模量比及衬砌的厚度等是影响衬砌内侧的无量纲位移和动应力集中因子分布的重要因素,为双衬砌或多衬砌的地震评价提供了理论依据。
Adopting the method of wave function expansion,the potential functions of incident,scattering and infracting SH waves are expanded to the form of series of Fourier-Bessel functions,and the expressions of potential functions in each local cylindrical coordinate are obtained with help of the Graf's addition theorem.The theoretical solutions of multiple scattering of SH waves by two separated tunnel linings are also obtained,according to the conditions that the stresses and displacements are continuous at the boundary of the linings and their adjacent rocks,and the inner sides of the linings are free.The digital calculation result indicats that the incident frequencies,the distances between the linings,the ratio of linings' moduli to adjacent rocks',and the linings' thicknesses are important factors which influence distributions of the normalized displacements and dynamic stress concentration factors on the inner sides of the two linings.The results provide some theoretical basis for earthquake evaluation of two or more linings.
Adopting the method of wave function expansion,the potential functions of incident,scattering and infracting SH waves are expanded to the form of series of Fourier-Bessel functions,and the expressions of potential functions in each local cylindrical coordinate are obtained with help of the Graf's addition theorem.The theoretical solutions of multiple scattering of SH waves by two separated tunnel linings are also obtained,according to the conditions that the stresses and displacements are continuous at the boundary of the linings and their adjacent rocks,and the inner sides of the linings are free.The digital calculation result indicats that the incident frequencies,the distances between the linings,the ratio of linings' moduli to adjacent rocks',and the linings' thicknesses are important factors which influence distributions of the normalized displacements and dynamic stress concentration factors on the inner sides of the two linings.The results provide some theoretical basis for earthquake evaluation of two or more linings.
引文
[1]谢定义.中国土动力学的发展现状与存在的问题[J].西北地震学报,2007,29(1):94-95.
[2]荣棉水,李小军,卢滔.局部地形对入射P波谱特性的影响分析[J].西北地震学报,2007,29(4):297-302.
[3]鲍亦兴,毛昭宙.弹性波的衍射与动应力集中[M].北京:科学出版社,1993:160-178.
[4]Balendra T,Thambiratnam D P,Koh C G,et al.Dynamicresonse of twin circular tunnels due to incident SH waves[J].Earthquake Engineering and Structural Dynamics,1984,12:181-201.
[5]林宏,刘殿魁.半无限空间中圆形空洞周围SH波的散射[J].地震工程与工程振动,2002,22(2):9-16.
[6]孙国富,孙国仓,盖京波.浅埋相邻多个圆形衬砌与SH波的相互作用[J].山东建筑工程学院学报,2006,21(1):19-25.
[7]Lee V W,Karl J.Diffraction of SV waves by underground,circu-lar,cylindrical cavities[J].Soil Dynamics and Earthquake Engi-neering,1992,11:445-456.
[8]梁建文,张浩,Lee V W.平面P波入射下地下洞室群动应力集中问题解析解[J].岩土工程学报,2004,26(6):816-819.
[9]梁建文,张浩,Lee V W.地下双洞室在SV波入射下动力响应问题解析解[J].振动工程学报,2004,17(2):132-140.
[10]杨桂通,张善元.弹性动力学[M].北京:中国铁道出版社,1988.
[11]Avil啨s J,S偄nchez-Sesma F.Piles as barriers for elastic waves[J].J.Geotech.Engng.,1983,119(9):1133-1146.
[12]Abramowitz M,Stegun I A.Handbook of mathematical Functionswith formulas,graphs,and mathematical tables[M].NewYork:Dover Publications,Inc.,1970.
[13]Veksler N,Izbicki J L,Conoir J M.Elastic wave scattering by acylindrical shell[J].Wave Motion,1999,29(3):195-209.
[2]荣棉水,李小军,卢滔.局部地形对入射P波谱特性的影响分析[J].西北地震学报,2007,29(4):297-302.
[3]鲍亦兴,毛昭宙.弹性波的衍射与动应力集中[M].北京:科学出版社,1993:160-178.
[4]Balendra T,Thambiratnam D P,Koh C G,et al.Dynamicresonse of twin circular tunnels due to incident SH waves[J].Earthquake Engineering and Structural Dynamics,1984,12:181-201.
[5]林宏,刘殿魁.半无限空间中圆形空洞周围SH波的散射[J].地震工程与工程振动,2002,22(2):9-16.
[6]孙国富,孙国仓,盖京波.浅埋相邻多个圆形衬砌与SH波的相互作用[J].山东建筑工程学院学报,2006,21(1):19-25.
[7]Lee V W,Karl J.Diffraction of SV waves by underground,circu-lar,cylindrical cavities[J].Soil Dynamics and Earthquake Engi-neering,1992,11:445-456.
[8]梁建文,张浩,Lee V W.平面P波入射下地下洞室群动应力集中问题解析解[J].岩土工程学报,2004,26(6):816-819.
[9]梁建文,张浩,Lee V W.地下双洞室在SV波入射下动力响应问题解析解[J].振动工程学报,2004,17(2):132-140.
[10]杨桂通,张善元.弹性动力学[M].北京:中国铁道出版社,1988.
[11]Avil啨s J,S偄nchez-Sesma F.Piles as barriers for elastic waves[J].J.Geotech.Engng.,1983,119(9):1133-1146.
[12]Abramowitz M,Stegun I A.Handbook of mathematical Functionswith formulas,graphs,and mathematical tables[M].NewYork:Dover Publications,Inc.,1970.
[13]Veksler N,Izbicki J L,Conoir J M.Elastic wave scattering by acylindrical shell[J].Wave Motion,1999,29(3):195-209.
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