SH波入射时等腰三角形与半圆形凸起地形的地震动
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摘要
利用复变函数和移动坐标研究了等腰三角形凸起和半圆形凸起地形对SH波的散射。首先将求解区域分为三个部分,区域Ⅰ为一个带有半圆形弧线的等腰三角形区域,区域Ⅱ为含有两个半圆形凹陷的弹性半空间,区域Ⅲ为包括半圆形凸起在内的圆形区域。在区域Ⅰ中构造满足在腰三角形斜边上应力自由的驻波函数WA,在区域Ⅱ中分别构造两个半圆形凹陷的散射波WB和WC,且两者均预先满足水平界面上应力为零的边界条件,在区域Ⅲ构造一个上半部边界应力为零,而其余部分位移、应力任意的驻波函数。将所研究的问题转换成一个求解满足在"公共边界"上位移应力连续的问题。可利用坐标移动方法,通过傅立叶级数展开,建立该问题代数方程组,并采用截断有限项的方法对其进行求解。最后,给出了一个算例。
In this paper,the scattering of the isosceles triangular hill and the hemi-circular hill to SH wave is studied based on complex function and moving coordinate.The solution is divided into three domains,where domainⅠ involves a isosceles triangle with semi-circular arc,domain II is an elastic half-space with semi-circular depression and domain III is a circle area with semi-circular hill.A standing wave function WA is constructed in the domainⅠ,which satisfies the zero-stress condition at the triangular wedges of the isosceles triangle.In the domain II,two scattering wave functions WB and WC with semi-circular depression which satisfy the condition of stress-free at the horizontal surface are constructed.In the domain III,a standing wave function which satisfies the condition that the stress of the top half of the domain is zero and the displacement and the stress of the others is arbitrary.This is to solve the problem that satisfies the conditions of displacement continuity and stress equilibrium at the interface of the two boundaries.By using moving coordinate method,the result is acquired by constructing a set of linear algebraic equations through Fourier series expansion.Finally a numerical example is given.
In this paper,the scattering of the isosceles triangular hill and the hemi-circular hill to SH wave is studied based on complex function and moving coordinate.The solution is divided into three domains,where domainⅠ involves a isosceles triangle with semi-circular arc,domain II is an elastic half-space with semi-circular depression and domain III is a circle area with semi-circular hill.A standing wave function WA is constructed in the domainⅠ,which satisfies the zero-stress condition at the triangular wedges of the isosceles triangle.In the domain II,two scattering wave functions WB and WC with semi-circular depression which satisfy the condition of stress-free at the horizontal surface are constructed.In the domain III,a standing wave function which satisfies the condition that the stress of the top half of the domain is zero and the displacement and the stress of the others is arbitrary.This is to solve the problem that satisfies the conditions of displacement continuity and stress equilibrium at the interface of the two boundaries.By using moving coordinate method,the result is acquired by constructing a set of linear algebraic equations through Fourier series expansion.Finally a numerical example is given.
引文
[1]Trifunac MD.Scattering of plane SH-waves by a semi-cylindrical canyon[J].Earthquake Engineering and Structural Dynamics,1973,1:267-281.
[2]Liu D K,Han F.Scattering of plane SH-waves by cylindrical canyon of arbitrary shapes in anisotropic media[J].Acta Mechanica Sinica,1990,6:256-266.
[3]Lee V W,Chen S,Hsu I R.Antiplane diffraction from canyon above subsurface unlined tunnel[J].Journal of Engineering Mechanics,1999;125:668-674
[4]袁晓铭,廖振鹏.任意圆弧形凸起地形对平面SH波的散射[J].地震工程与工程振动,1996,16(2):1-13.
[5]Liu D K,Wang G Q.Anti-plane SH-deformation of a semi-cylindrical hill above a subsurface cavity in half-space[J].Acta Mechanica Sinica,2006,38:209-218.
[6]刘宏伟,刘殿魁.界面圆孔对SH波散射的远场解[J].固体力学学报,1999,20(4):349-355.
[7]刘殿魁,王国庆.浅埋圆形孔洞附近的半圆形凸起对SH波的散射[J].力学学报,2006,20(2):209-218.
[8]Nazaret D,Lee VW,Liang J W.Antiplane deformations around arbitrary-shaped canyons on a wedge-shape half space:moment method solutions[J].Earthquake Engineering and Engineering Vibration,2003,2(1):281-287.
[9]Hayir A,Todorovska MI,Trifunac MD.Antiplane response of a dike with flexible soil-structure interface to incident SHwaves[J].Soil Dynamicsand Earthquake Engineering,2001,21(2):603-613.
[10]Qiu F Q,Liu D K.Antiplane response of isosceles triangular hill to incident SHwaves[J].Earthquake Engineering and Engineering Vibration,2005,4(1):1-10.
[2]Liu D K,Han F.Scattering of plane SH-waves by cylindrical canyon of arbitrary shapes in anisotropic media[J].Acta Mechanica Sinica,1990,6:256-266.
[3]Lee V W,Chen S,Hsu I R.Antiplane diffraction from canyon above subsurface unlined tunnel[J].Journal of Engineering Mechanics,1999;125:668-674
[4]袁晓铭,廖振鹏.任意圆弧形凸起地形对平面SH波的散射[J].地震工程与工程振动,1996,16(2):1-13.
[5]Liu D K,Wang G Q.Anti-plane SH-deformation of a semi-cylindrical hill above a subsurface cavity in half-space[J].Acta Mechanica Sinica,2006,38:209-218.
[6]刘宏伟,刘殿魁.界面圆孔对SH波散射的远场解[J].固体力学学报,1999,20(4):349-355.
[7]刘殿魁,王国庆.浅埋圆形孔洞附近的半圆形凸起对SH波的散射[J].力学学报,2006,20(2):209-218.
[8]Nazaret D,Lee VW,Liang J W.Antiplane deformations around arbitrary-shaped canyons on a wedge-shape half space:moment method solutions[J].Earthquake Engineering and Engineering Vibration,2003,2(1):281-287.
[9]Hayir A,Todorovska MI,Trifunac MD.Antiplane response of a dike with flexible soil-structure interface to incident SHwaves[J].Soil Dynamicsand Earthquake Engineering,2001,21(2):603-613.
[10]Qiu F Q,Liu D K.Antiplane response of isosceles triangular hill to incident SHwaves[J].Earthquake Engineering and Engineering Vibration,2005,4(1):1-10.
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