弹性波多重散射的改进算法及在工程中的应用
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摘要
由动力学基本理论,波在一个无限大的均质空间中是以常速度直线传播的,但若是均匀介质中嵌入有障碍物,则波的传播会受到障碍物的影响,产生诸如散射、反射以及衍射等现象。弹性波的多重散射现象在实际工程中应用极为广泛,弹性土体中的振动隔离问题以及隧道、衬砌等构造物周边的动应力集中问题很早就被注意并加以研究。就目前为止,关于非连续屏障的隔振问题,理论解析方法所研究的对象均仅限于单排桩或孔;关于衬砌的动应力集中问题,研究也绝大部分限于双平行衬砌(实际上也是单排的衬砌)。然而,就实际工程而言,为了增强屏障整体刚度以及提升隔振效果,常常使用双排乃至多排桩或孔,同时,地下衬砌的分布也并不一定都是双平行衬砌的形式,因此,从理论上推导双排桩或者多排桩关于振动的隔离,以及地下任意分布衬砌、隧道关于振动的响应显得极为重要。
本文首先基于弹性波基础理论,在Aviles等学者的单排非连续屏障隔振体系计算方法的基础上,运用更为完整的波函数展开式,且结合更为广义上的Graf加法定理,将散射波势函数的表达式变换到同一个圆柱坐标系下,推导了任意分布的多个圆柱体关于弹性波多重散射的散射系数解析解。此种改进算法能够计算任意分布的桩或孔洞关于弹性波的隔离问题,极大地增强了以往算法的适用范围。
其次,将任意分布的刚性圆柱体有规律地排列在两条直线上,则问题演变为双排刚性桩屏障对于弹性波的隔离。基于本文所提出的弹性波多重散射的改进算法,利用桩土间的边界条件,推导了双排刚性桩关于入射弹性波(P波、SV波和SH波)的多重散射系数。引入无量纲位移(屏障后土体的位移与入射平面波引起的位移的比值)和透射系数(屏障宽度范围内无量纲位移的几何平均值)的概念,对双排刚性桩屏障的隔振效果进行了详细的参数分析。
接着,将任意分布的弹性圆柱体有规律地排列在两条平行的直线上,结合弹性桩与土之间的边界条件,研究了双排弹性桩屏障关于入射SV波的隔离问题。同样,引入无量纲位移和透射系数的概念,采用计算机数值方法,对双排弹性桩屏障的隔振效果进行了详细的参数分析。
最后,研究了任意分布的地下深埋衬砌在弹性SH波入射下的响应问题,结合基于本文所提出的弹性波多重散射的改进算法,此处以任意分布的三个衬砌为例,推导了多个衬砌关于弹性SH波多重散射的散射系数解析解。
Waves move at a constant speed rectilinearly in an unbounded, isotropic and homogeneous space. However, propagation of waves would be interrupted if there are some obstacles in the medium. Some secondary waves will be aroused, which is called scattering wave. Multiple scattering of elastic waves has been noticed and researched for a long time, such as vibration isolation by rows of piles and dynamic stress response of linings. Nevertheless, as far as theoretical method is concerned, all of the precedent work was focused on single-row-pile isolation system and double linings. Little work has been carried on multiple-row-pile system or response of multiple linings. This is obviously not the case in the practical engineering. Thus, it is of fatal importance to analyze these problems theoretically.
Firstly, based on the analytical method of Aviles and basic theories of elastic wave, an improved expansion method of wave functions using more complete Fourier-Bessel function series has been adopted. According to Graf addition theorem which is in a more generalized perspective, formulas of scattering wave field can be expressed under one cylinder coordinate system. With the help of boundary conditions, coefficients of multiple scattering of elastic waves by an arbitrary configuration of cylinders have been deduced.
Secondly, these cylinders are assumed to be rigid and rearranged in two parallel lines, which turns the problem into vibration isolation using multiple rows of rigid piles. Based on the improved method raised in this article, multiple scattering coefficients of this two-row rigid piles system under incidence of elastic waves (P, SV and SH waves) can be obtained. Then, normalized displacements (the ratio of the displacement amplitudes of soil in total wave field to the displacement amplitudes only in incident wave field) and transmissibility index (the mean value of normalized displacements across the barrier) were introduced. Detailed parametric study has been done in the following.
Thirdly, these cylinders are assumed to be elastic and rearranged in two parallel lines, which turns the problems into vibration isolation under incidence of plane SV waves using multiple rows of elastic piles. Similarly, normalized displacements and transmissibility index were introduced and parametric study of this isolation system has been done through computational numeric calculation.
Finally, dynamic response of deeply embedded linings under incidence of plane SH waves has been studied. Herein an arbitrary configuration of three linings was taken as an instance. According to the improved method put forward in this article and boundary conditions of linings which is the key difference with piles, scattering coefficients of multiple linings were deduced.
本文首先基于弹性波基础理论,在Aviles等学者的单排非连续屏障隔振体系计算方法的基础上,运用更为完整的波函数展开式,且结合更为广义上的Graf加法定理,将散射波势函数的表达式变换到同一个圆柱坐标系下,推导了任意分布的多个圆柱体关于弹性波多重散射的散射系数解析解。此种改进算法能够计算任意分布的桩或孔洞关于弹性波的隔离问题,极大地增强了以往算法的适用范围。
其次,将任意分布的刚性圆柱体有规律地排列在两条直线上,则问题演变为双排刚性桩屏障对于弹性波的隔离。基于本文所提出的弹性波多重散射的改进算法,利用桩土间的边界条件,推导了双排刚性桩关于入射弹性波(P波、SV波和SH波)的多重散射系数。引入无量纲位移(屏障后土体的位移与入射平面波引起的位移的比值)和透射系数(屏障宽度范围内无量纲位移的几何平均值)的概念,对双排刚性桩屏障的隔振效果进行了详细的参数分析。
接着,将任意分布的弹性圆柱体有规律地排列在两条平行的直线上,结合弹性桩与土之间的边界条件,研究了双排弹性桩屏障关于入射SV波的隔离问题。同样,引入无量纲位移和透射系数的概念,采用计算机数值方法,对双排弹性桩屏障的隔振效果进行了详细的参数分析。
最后,研究了任意分布的地下深埋衬砌在弹性SH波入射下的响应问题,结合基于本文所提出的弹性波多重散射的改进算法,此处以任意分布的三个衬砌为例,推导了多个衬砌关于弹性SH波多重散射的散射系数解析解。
Waves move at a constant speed rectilinearly in an unbounded, isotropic and homogeneous space. However, propagation of waves would be interrupted if there are some obstacles in the medium. Some secondary waves will be aroused, which is called scattering wave. Multiple scattering of elastic waves has been noticed and researched for a long time, such as vibration isolation by rows of piles and dynamic stress response of linings. Nevertheless, as far as theoretical method is concerned, all of the precedent work was focused on single-row-pile isolation system and double linings. Little work has been carried on multiple-row-pile system or response of multiple linings. This is obviously not the case in the practical engineering. Thus, it is of fatal importance to analyze these problems theoretically.
Firstly, based on the analytical method of Aviles and basic theories of elastic wave, an improved expansion method of wave functions using more complete Fourier-Bessel function series has been adopted. According to Graf addition theorem which is in a more generalized perspective, formulas of scattering wave field can be expressed under one cylinder coordinate system. With the help of boundary conditions, coefficients of multiple scattering of elastic waves by an arbitrary configuration of cylinders have been deduced.
Secondly, these cylinders are assumed to be rigid and rearranged in two parallel lines, which turns the problem into vibration isolation using multiple rows of rigid piles. Based on the improved method raised in this article, multiple scattering coefficients of this two-row rigid piles system under incidence of elastic waves (P, SV and SH waves) can be obtained. Then, normalized displacements (the ratio of the displacement amplitudes of soil in total wave field to the displacement amplitudes only in incident wave field) and transmissibility index (the mean value of normalized displacements across the barrier) were introduced. Detailed parametric study has been done in the following.
Thirdly, these cylinders are assumed to be elastic and rearranged in two parallel lines, which turns the problems into vibration isolation under incidence of plane SV waves using multiple rows of elastic piles. Similarly, normalized displacements and transmissibility index were introduced and parametric study of this isolation system has been done through computational numeric calculation.
Finally, dynamic response of deeply embedded linings under incidence of plane SH waves has been studied. Herein an arbitrary configuration of three linings was taken as an instance. According to the improved method put forward in this article and boundary conditions of linings which is the key difference with piles, scattering coefficients of multiple linings were deduced.
引文
Aboundi J. Elastic waves in half-space with thin barrier. Journal of Engineering Mechanics, ASCE, 1973,99(1):69-83.
Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [M]. New York: Dover Publications, Inc., 1970.
Ahamd S, Al-Hussaini T M. Simplified design for vibration screening by open and in-filled trenches. Journal of Geotechnical Engineering, 1991,117(1):67-88.
Aki K, Richards P G. Quantitative seismology. Vol. I, WH Freeman and Company, San Francisco, California, 1980.
Al-Hussaini T M, Ahamd S. Active isolation of machine foundations by in-filled trench barriers. Journal of Geotechnical Engineering, 1996,122(4):288-294.
Al-Hussaini T M, Ahamd S. Design of wave barriers for prediction of horizontal ground vibration. Journal of Geotechnical Engineering, 1991,117(4):616-636.
Aviles J, Sanchez-Sesma F J. Piles as barriers for elastic waves. Journal of Geotechnical Engineering. ASCE, 1983,109(9):l 133-1146.
Aviles J, Sanchez-Sesma F J. Foundation isolation from vibration using piles as barriers. Journal of Engineering Mechanics, ASCE, 1988, 114(11):1854-1870.
Balendra T, Thambiratnam D P, Koh C G, Lee S L. Dynamic response of twin circular tunnels due to incident SH-waves. Earthquake Engineering and Structural Dynamics, 1984, 12(2):181-201.
Barkan D D. Dynamics of bases and foundation. New York: McGraw-Hill, 1962.
Beskos D E, Leung K L. Vardoulakis I G. Vibration isolation of structures from surface wave in layered soil. Recent Application in Computational Mechanics. New York, 1986.
Boroomand B, Kaynia A M. Vibration isolation by an array of piles. First International Conference on Soil Dynamics and Earthquake Engineering V. Karlsruhe, German}', 1991b: 683-691.
Dasgupta G, Beskos D E, Vardoulakis I G. 3-D Vibration isolation using open trenches. Shaw R P. Innovative Numerical Methods in Engineering. Berlin, 1986.
Beskos D E. Dasgupta B. Vardoulakis I G. Vibration isolation using open or filled trenches, Part I: 2-D homogeneous soil. Computational Mechanics, 1986, 1(1): 43-63.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Screening of plane S waves by an array of rigid piles in poroelastic soil[J]. Journal of Zhejiang University SCIENCE A, 2008, 9(5):589-599.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Amplitude reduction of elastic waves by a row of piles in poroelastic soil[J]. Computers and Geotechnics, 2009,36:463-473.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Vertical amplitude reduction of Rayleigh waves by a row of piles in a poroelastic half-space [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2009,33:1799-1821.
Dasgupta G, Beskos D E, Vardoulakis I G. Vibration isolation using open or filled trenches. Part 2: 3-D homogeneous soil. Computational Mechanics, 1990, 6(2): 129-142.
Emad K, Manolis G D. Shallow trenches and propagation of surface waves. Journal of Engineering Mechanics, ASCE, 1985, 111(2): 279-282.
Flax L, Varadan V K, Varadan V V. Scattering of an obliquely acoustic wave by an infinite cylinder. Journal of the Acoustical Society of America, 1980, 68(6): 1832-1835.
Gao G Y, Li Z Y, Qiu Ch, Yue Z Q. Three-dimensional analysis of rows of piles as passive barriers for ground vibration isolation. Soil Dynamics and Earthquake Engineering. 2006,26(11): 1015-1027.
Haupt W A. Model test on screening of surface waves. Proceedings of the 10th International Conference in Soil Mechanics and Foundation Engineering, Stockholm, 1981. 3:215-222.
Kattis S E, Polyzos D, Beskos D E. Modeling of pile wave barriers by effective trenches and their screening effectiveness. Soil Dynamics and Earthquake Engineering. 1999a, 18(1): 1-10.
Kattis S E, Polyzos D, Beskos D E. Vibration isolation by a row of piles using a 3-D frequency domain BEM. International Journal for Numerical Methods in Engineering. 1999b, 46(5): 713-728.
Lee V W. Trifunac M D. Response of tunnels to incident SH waves. Journal of Engineering Mechanics. ASCE, 1979, 105(4): 643-659.
Lee V W. Karl J. Diffraction of elastic plane P waves by circular, underground unlined tunnels. European Earthquake Engineering, 1993, 6(1): 29-36.
Leung K L, Beskos D E, Vardoulakis I G. Vibration isolation using open or filled trenches, Part 3: 2-D nonhomogeneous soil. Computational Mechanics, 1990, 7: 137-148.
Leung K L, Vardoulakis I G, Beskos D E, Tassoulas J L. Vibration isolation by trenches in continuously nonhomogeneous soil by the BEM. Soil Dynamics and Earthquake Engineering, 1991,10(3): 172-179.
Liao S, Sangrey D A. Use of piles as isolation barriers. Journal of Geotechnical Engineering Division, ASCE, 1978, 104(9): 1139-1152.
Pao Y H, Mow C C. Diffraction of elastic waves and dynamic stress concentrations. New York: Crane and Russak, 1973.
Richart F E, Hall J R, Woods R D. Vibration of soils and foundations. Englewood cliffs, NJ: Prentice-Hall, 1970.
Sancar S, Pao Y H. Spectral analysis of elastic pulses backscattered from two cylindrical cavities in a solid. Part I. Journal of the Acoustical Society of America, 1981, 69(6): 1591-1596.
Segol G, Lee C Y, Abel J F. Amplitude reduction of surface waves by trenches. Journal of Engineering Mechanics, ASCE. 1978, 104(3): 621-641.
Sezawa K. Scattering of elastic waves and some allied problems. Bulletin of the Earthquake Research Institute, Tokyo University, Japan, 1927. 3: 19-27.
Thai P H, Feng Z Y. Jen T L. Three-dimensional analysis of the screening effectiveness of hollow pile barriers for foundation-induced vertical vibration. Computers and Geotechnics, 2008, 35(3): 489-499.
Twersky V. Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders. Journal of Applied Physics, 1952, 23(4): 407-414.
Waterman P C. Matrix theory of elastic wave scattering. Journal of the Acoustical Society of America. 1976. 60(3): 567-578.
Watson. G.N.(1966). A Treatise on the Theory of Bessel Functions. Cambridge University Press, London. N.W.. England.
Woods R D. Screening of surface waves in soils. Journal of the Soil Mechanics and Foundations Division,ASCE,1968,94(4):951-979.
蔡袁强,丁光亚,徐长节.饱和土中排桩对入射S波隔离的三维分析[J].自然灾害学报,2008,17(2):1-7.
蔡袁强,丁光亚,徐长节.饱和土中单排弹性桩对平面S波的屏蔽[J].岩土工程学报,2008,30(7):964-969.
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Ahamd S, Al-Hussaini T M. Simplified design for vibration screening by open and in-filled trenches. Journal of Geotechnical Engineering, 1991,117(1):67-88.
Aki K, Richards P G. Quantitative seismology. Vol. I, WH Freeman and Company, San Francisco, California, 1980.
Al-Hussaini T M, Ahamd S. Active isolation of machine foundations by in-filled trench barriers. Journal of Geotechnical Engineering, 1996,122(4):288-294.
Al-Hussaini T M, Ahamd S. Design of wave barriers for prediction of horizontal ground vibration. Journal of Geotechnical Engineering, 1991,117(4):616-636.
Aviles J, Sanchez-Sesma F J. Piles as barriers for elastic waves. Journal of Geotechnical Engineering. ASCE, 1983,109(9):l 133-1146.
Aviles J, Sanchez-Sesma F J. Foundation isolation from vibration using piles as barriers. Journal of Engineering Mechanics, ASCE, 1988, 114(11):1854-1870.
Balendra T, Thambiratnam D P, Koh C G, Lee S L. Dynamic response of twin circular tunnels due to incident SH-waves. Earthquake Engineering and Structural Dynamics, 1984, 12(2):181-201.
Barkan D D. Dynamics of bases and foundation. New York: McGraw-Hill, 1962.
Beskos D E, Leung K L. Vardoulakis I G. Vibration isolation of structures from surface wave in layered soil. Recent Application in Computational Mechanics. New York, 1986.
Boroomand B, Kaynia A M. Vibration isolation by an array of piles. First International Conference on Soil Dynamics and Earthquake Engineering V. Karlsruhe, German}', 1991b: 683-691.
Dasgupta G, Beskos D E, Vardoulakis I G. 3-D Vibration isolation using open trenches. Shaw R P. Innovative Numerical Methods in Engineering. Berlin, 1986.
Beskos D E. Dasgupta B. Vardoulakis I G. Vibration isolation using open or filled trenches, Part I: 2-D homogeneous soil. Computational Mechanics, 1986, 1(1): 43-63.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Screening of plane S waves by an array of rigid piles in poroelastic soil[J]. Journal of Zhejiang University SCIENCE A, 2008, 9(5):589-599.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Amplitude reduction of elastic waves by a row of piles in poroelastic soil[J]. Computers and Geotechnics, 2009,36:463-473.
Cai Yuanqiang, Ding Guangya, Xu Changjie. Vertical amplitude reduction of Rayleigh waves by a row of piles in a poroelastic half-space [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2009,33:1799-1821.
Dasgupta G, Beskos D E, Vardoulakis I G. Vibration isolation using open or filled trenches. Part 2: 3-D homogeneous soil. Computational Mechanics, 1990, 6(2): 129-142.
Emad K, Manolis G D. Shallow trenches and propagation of surface waves. Journal of Engineering Mechanics, ASCE, 1985, 111(2): 279-282.
Flax L, Varadan V K, Varadan V V. Scattering of an obliquely acoustic wave by an infinite cylinder. Journal of the Acoustical Society of America, 1980, 68(6): 1832-1835.
Gao G Y, Li Z Y, Qiu Ch, Yue Z Q. Three-dimensional analysis of rows of piles as passive barriers for ground vibration isolation. Soil Dynamics and Earthquake Engineering. 2006,26(11): 1015-1027.
Haupt W A. Model test on screening of surface waves. Proceedings of the 10th International Conference in Soil Mechanics and Foundation Engineering, Stockholm, 1981. 3:215-222.
Kattis S E, Polyzos D, Beskos D E. Modeling of pile wave barriers by effective trenches and their screening effectiveness. Soil Dynamics and Earthquake Engineering. 1999a, 18(1): 1-10.
Kattis S E, Polyzos D, Beskos D E. Vibration isolation by a row of piles using a 3-D frequency domain BEM. International Journal for Numerical Methods in Engineering. 1999b, 46(5): 713-728.
Lee V W. Trifunac M D. Response of tunnels to incident SH waves. Journal of Engineering Mechanics. ASCE, 1979, 105(4): 643-659.
Lee V W. Karl J. Diffraction of elastic plane P waves by circular, underground unlined tunnels. European Earthquake Engineering, 1993, 6(1): 29-36.
Leung K L, Beskos D E, Vardoulakis I G. Vibration isolation using open or filled trenches, Part 3: 2-D nonhomogeneous soil. Computational Mechanics, 1990, 7: 137-148.
Leung K L, Vardoulakis I G, Beskos D E, Tassoulas J L. Vibration isolation by trenches in continuously nonhomogeneous soil by the BEM. Soil Dynamics and Earthquake Engineering, 1991,10(3): 172-179.
Liao S, Sangrey D A. Use of piles as isolation barriers. Journal of Geotechnical Engineering Division, ASCE, 1978, 104(9): 1139-1152.
Pao Y H, Mow C C. Diffraction of elastic waves and dynamic stress concentrations. New York: Crane and Russak, 1973.
Richart F E, Hall J R, Woods R D. Vibration of soils and foundations. Englewood cliffs, NJ: Prentice-Hall, 1970.
Sancar S, Pao Y H. Spectral analysis of elastic pulses backscattered from two cylindrical cavities in a solid. Part I. Journal of the Acoustical Society of America, 1981, 69(6): 1591-1596.
Segol G, Lee C Y, Abel J F. Amplitude reduction of surface waves by trenches. Journal of Engineering Mechanics, ASCE. 1978, 104(3): 621-641.
Sezawa K. Scattering of elastic waves and some allied problems. Bulletin of the Earthquake Research Institute, Tokyo University, Japan, 1927. 3: 19-27.
Thai P H, Feng Z Y. Jen T L. Three-dimensional analysis of the screening effectiveness of hollow pile barriers for foundation-induced vertical vibration. Computers and Geotechnics, 2008, 35(3): 489-499.
Twersky V. Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders. Journal of Applied Physics, 1952, 23(4): 407-414.
Waterman P C. Matrix theory of elastic wave scattering. Journal of the Acoustical Society of America. 1976. 60(3): 567-578.
Watson. G.N.(1966). A Treatise on the Theory of Bessel Functions. Cambridge University Press, London. N.W.. England.
Woods R D. Screening of surface waves in soils. Journal of the Soil Mechanics and Foundations Division,ASCE,1968,94(4):951-979.
蔡袁强,丁光亚,徐长节.饱和土中排桩对入射S波隔离的三维分析[J].自然灾害学报,2008,17(2):1-7.
蔡袁强,丁光亚,徐长节.饱和土中单排弹性桩对平面S波的屏蔽[J].岩土工程学报,2008,30(7):964-969.
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