SH波斜入射时有阻尼成层介质自由场的一维化时域算法
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摘要
在进行大型大跨结构抗震设计时,需要考虑地震波斜入射的影响。在刘晶波等提出的一维化时域算法基础上,采用有限差分方法,推导建立了SH波斜入射情形下、考虑阻尼影响的水平成层弹性介质出平面自由场求解的显式数值逐步法公式,并采用Fortan程序语言编制了相应的数值计算程序。综合采用有限元法和数值逐步法,进行SH波斜入射下单层土和双层土的算例分析。结果表明:土层介质阻尼仅影响到SH波斜入射下自由场的位移幅值,而对位移时程的波形没有影响;SH波斜入射下有阻尼与无阻尼情形相比位移幅值有明显衰减,对单层土,30°斜入射下衰减幅度可达40.5%;对双层土,最大衰减幅度可达39.7%;有阻尼土层位移幅值的衰减幅度从底部边界向自由表面逐渐增大。
The influence of obliquely incident seismic wave on the seismic responses of large and long-span structures need to be considered in seismic design.Based on a 1D time-domain method for the out-of-plane wave motion of the free field in a layered half space proposed by Liu and Wang,formulas of an explicit time-stepping method to solve the out-of-plane wave motion of the free field in layered media with damping under obliquely incident SH wave were established by using the finite difference method,and its numerical program was developed with Fortran programming language.A cases study of out-of-plane wave motions of the free field in one-layer soil and two-layer soil with damping under obliquely incident SH wave were carried out respectively.The results show that:soil damping only has effect on the peak displacements of free field motions,and has no effect on the waveforms of the displacement time histories;the displacement amplitudes in layered soil with damping reduce significantly compared with that of without damping under obliquely incident SH wave,and the displacement amplitudes can be reduced by 40.5% and 39.7% for one-layer soil and two-layer soil,respectively.The case study also indicates that the attenuation of displacement amplitudes in layered soil with damping gradually increased from the bottom boundary to the free surface.
The influence of obliquely incident seismic wave on the seismic responses of large and long-span structures need to be considered in seismic design.Based on a 1D time-domain method for the out-of-plane wave motion of the free field in a layered half space proposed by Liu and Wang,formulas of an explicit time-stepping method to solve the out-of-plane wave motion of the free field in layered media with damping under obliquely incident SH wave were established by using the finite difference method,and its numerical program was developed with Fortran programming language.A cases study of out-of-plane wave motions of the free field in one-layer soil and two-layer soil with damping under obliquely incident SH wave were carried out respectively.The results show that:soil damping only has effect on the peak displacements of free field motions,and has no effect on the waveforms of the displacement time histories;the displacement amplitudes in layered soil with damping reduce significantly compared with that of without damping under obliquely incident SH wave,and the displacement amplitudes can be reduced by 40.5% and 39.7% for one-layer soil and two-layer soil,respectively.The case study also indicates that the attenuation of displacement amplitudes in layered soil with damping gradually increased from the bottom boundary to the free surface.
引文
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[4]杜修力,陈维,李亮,等.斜入射条件下地下结构时域地震反应分析初探[J].震灾防御技术,2007,2(3):290-296.DU Xiuli,CHEN Wei,LI Liang,et al.Preliminary study of time-domain seismic response for underground structures to obliquely incident seismic waves[J].Technology for Earthquake Disaster Prevention,2007,2(3):290-296.
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[6]廖振鹏.工程波动理论导论[M].2版.北京:科学出版社,2002.
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[10]廖振鹏,黄孔亮,杨柏坡,等.暂态波透射边界[J].中国科学(A辑),1984(6):556-564.LIAO Zhenpeng,HUANG Kongliang,YANG Baipo,et al.A transmitting boundary for transient wave[J].Science in China(Series A),1984(6):556-564.
[11]刘晶波,王艳.成层半空间出平面自由波场的一维化时域算法[J].力学学报,2006,38(2):219-225.LIU Jingbo,WANG Yan.A 1D time-domain method for 2D wave motion in elastic layered half-space by antiplane wave oblique incidence[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):219-225.
[12]刘晶波,王艳.成层介质中平面内自由波场的一维化时域算法[J].工程力学,2007,24(7):16-22.LIU Jingbo,WANG Yan.A 1D time-domain method for inplane wave motion of free field in layered media[J].Engineering Mechanics,2007,24(7):16-22.
[13]LIU Jingbo,WANG Yan.A 1D time-domain method for inplane wave motions in a layered half-space[J].Acta Mechanica Sinica,2007,23(6):673-680.
[14]赵密,杜修力,刘晶波,等.P-SV波斜入射时成层半空间自由场的时域算法[J].地震工程学报,2013,35(1):84-90.ZHAO Mi,DU Xiuli,LIU Jingbo,et al.Time-domain method for free field in layered half space under P-SV waves of oblique incidence[J].China Earthquake Engineering Journal,2013,35(1):84-90.
[15]范留明,赵钦.成层半空间出平面自由波场的界面子波算法[J].地震工程学报,2014,36(4):765-772.FAN Liuming,ZHAO Qin.Interfacial wavelet superposition method for out-of-plane wave motion of a free field in a layered half space[J].China Earthquake Engineering Journal,2014,36(4):765-772.
[16]王笃国,赵成刚.地震波斜入射时二维成层介质自由场求解的等效线性化方法[J].岩土工程学报,2016,38(3):554-561.WANG Duguo,ZHAO Chenggang.Two-dimensional equivalent linear seismic analysis of free field in layered halfspace due to oblique incidence[J].Chinese Journal of Geotechnical Engineering,2016,38(3):554-561.
[17]LYSMER J,KULEMEYER R L.Finite dynamic model for infinite media[J].Journal of Engineering Mechanics,1969,95(4):859-877.
[18]WOLF P J.A comparison of time-domain transmitting boundaries[J].Earthquake Engineering and Structure Dynamics,1986,14(4):655-673.
[19]DEEKS A J,RANDOLPH M F.Axisymmetric time-domain transmitting boundaries[J].Journal of Engineering Mechanics,1994,120(1):25-42.
[20]王振宇,刘晶波.成层地基非线性波动问题人工边界与波动输入研究[J].岩石力学与工程学报,2004,23(7):1169-1173.WANG Zhenyu,LIU Jingbo.Study on wave motion input and artificial boundary for problem of nonlinear wave motion in layered soil[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(7):1169-1173.
[21]刘晶波,王振宇,杜修力,等.波动问题中的三维时域黏弹性人工边界[J].工程力学,2005,22(6):46-51.LIU Jingbo,WANG Zhenyu,DU Xiuli,et al.Threedimensional visco-elastic artificial boundaries in time domain for wave motion problems[J].Engineering Mechanics,2005,22(6):46-51.
[22]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界单元[J].岩土工程学报,2006,28(9):1070-1075.LIU Jingbo,GU Yin,DU Yixin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070-1075.
[23]杜修力,赵密,王进廷.近场波动模拟的人工应力边界条件[J].力学学报,2006,38(1):49-56.DU Xiuli,ZHAO Mi,WANG Jinting.A stress artificial boundary in FEA for near-field wave problem[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(1):49-56.
[24]卢华喜,梁平英,尚守平.地基非线性波动问题中黏-弹性人工边界研究[J].岩土力学,2008,29(7):1911-1916.LU Huaxi,LIANG Pingying,SHANG Shouping.Research on viscoelastic artificial boundary for problem of nonlinear wave motion in soil[J].Rock and Soil Mechanics,2008,29(7):1911-1916.
[25]张波,李术才,杨学英,等.三维黏弹性介质人工边界研究[J].岩土力学,2009,30(11):3469-3475.ZHANG Bo,LI Shucai,YANG Xueying,et al.Study of three dimensional viscoelastic medium artificial boundary[J].Rock and Soil Mechanics,2009,30(11):3469-3475.
[26]赵密,杜修力,刘晶波.一种高阶精度人工边界条件:出平面外域波动问题[J].工程力学,2012,29(4):7-14.ZHAO Mi,DU Xiuli,LIU Jingbo.A high-order accurate artificial boundary condition:out-of-plane exterior waver problem[J].Engineering Mechanics,2012,29(4):7-14.
[27]CHOPRA A K.Dynamics of structures:theory and applications to earthquake engineering[M].3rd ed.New Jersey:Pearson Education Inc.,2007.
[2]李山有,廖振鹏,周正华.大型结构地震反应数值模拟中的波动输入[J].地震工程与工程振动,2001,6(2):1-5.LI Shanyou,LIAO Zhenpeng,ZHOU Zhenghua.Wave motion input in numerical simulation of seismic response for1arge-scale structure[J].Earthquake Engineering and Engineering Vibration,2001,6(2):1-5.
[3]潘旦光,楼梦麟,范立础.多点输入下大跨度结构地震反应分析研究现状[J].同济大学学报,2001,29(10):1213-1219.PAN Dan’guang,LOU Menglin,FAN Lichu.Status of seismic response analysis of long-span structures under multiple support excitations[J].Journal of Tongji University,2001,29(10):1213-1219.
[4]杜修力,陈维,李亮,等.斜入射条件下地下结构时域地震反应分析初探[J].震灾防御技术,2007,2(3):290-296.DU Xiuli,CHEN Wei,LI Liang,et al.Preliminary study of time-domain seismic response for underground structures to obliquely incident seismic waves[J].Technology for Earthquake Disaster Prevention,2007,2(3):290-296.
[5]傅淑芳,刘宝诚.地震学教程[M].北京:地震出版社,1991.
[6]廖振鹏.工程波动理论导论[M].2版.北京:科学出版社,2002.
[7]HASKELL N A.The dispersion of surface waves in multilayered media[J].Bulletin of the Seismic Society of American,1953,43:17-34.
[8]BREKHOVSHIKH L M.Waves in layered media[M].New York:Academic Press Inc.,1980.
[9]李山有,王学良,周正华.地震波斜入射情形下水平成层半空间自由场的时域计算[J].吉林大学学报(地球科学版),2003,33(3):372-376.LI Shanyou,WANG Xueliang,ZHOU Zhenghua.The timestep numerical simulation of free field motion of layered halfspace for inclined seismic waves[J].Journal of Jilin University(Earth Science),2003,33(3):372-376.
[10]廖振鹏,黄孔亮,杨柏坡,等.暂态波透射边界[J].中国科学(A辑),1984(6):556-564.LIAO Zhenpeng,HUANG Kongliang,YANG Baipo,et al.A transmitting boundary for transient wave[J].Science in China(Series A),1984(6):556-564.
[11]刘晶波,王艳.成层半空间出平面自由波场的一维化时域算法[J].力学学报,2006,38(2):219-225.LIU Jingbo,WANG Yan.A 1D time-domain method for 2D wave motion in elastic layered half-space by antiplane wave oblique incidence[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):219-225.
[12]刘晶波,王艳.成层介质中平面内自由波场的一维化时域算法[J].工程力学,2007,24(7):16-22.LIU Jingbo,WANG Yan.A 1D time-domain method for inplane wave motion of free field in layered media[J].Engineering Mechanics,2007,24(7):16-22.
[13]LIU Jingbo,WANG Yan.A 1D time-domain method for inplane wave motions in a layered half-space[J].Acta Mechanica Sinica,2007,23(6):673-680.
[14]赵密,杜修力,刘晶波,等.P-SV波斜入射时成层半空间自由场的时域算法[J].地震工程学报,2013,35(1):84-90.ZHAO Mi,DU Xiuli,LIU Jingbo,et al.Time-domain method for free field in layered half space under P-SV waves of oblique incidence[J].China Earthquake Engineering Journal,2013,35(1):84-90.
[15]范留明,赵钦.成层半空间出平面自由波场的界面子波算法[J].地震工程学报,2014,36(4):765-772.FAN Liuming,ZHAO Qin.Interfacial wavelet superposition method for out-of-plane wave motion of a free field in a layered half space[J].China Earthquake Engineering Journal,2014,36(4):765-772.
[16]王笃国,赵成刚.地震波斜入射时二维成层介质自由场求解的等效线性化方法[J].岩土工程学报,2016,38(3):554-561.WANG Duguo,ZHAO Chenggang.Two-dimensional equivalent linear seismic analysis of free field in layered halfspace due to oblique incidence[J].Chinese Journal of Geotechnical Engineering,2016,38(3):554-561.
[17]LYSMER J,KULEMEYER R L.Finite dynamic model for infinite media[J].Journal of Engineering Mechanics,1969,95(4):859-877.
[18]WOLF P J.A comparison of time-domain transmitting boundaries[J].Earthquake Engineering and Structure Dynamics,1986,14(4):655-673.
[19]DEEKS A J,RANDOLPH M F.Axisymmetric time-domain transmitting boundaries[J].Journal of Engineering Mechanics,1994,120(1):25-42.
[20]王振宇,刘晶波.成层地基非线性波动问题人工边界与波动输入研究[J].岩石力学与工程学报,2004,23(7):1169-1173.WANG Zhenyu,LIU Jingbo.Study on wave motion input and artificial boundary for problem of nonlinear wave motion in layered soil[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(7):1169-1173.
[21]刘晶波,王振宇,杜修力,等.波动问题中的三维时域黏弹性人工边界[J].工程力学,2005,22(6):46-51.LIU Jingbo,WANG Zhenyu,DU Xiuli,et al.Threedimensional visco-elastic artificial boundaries in time domain for wave motion problems[J].Engineering Mechanics,2005,22(6):46-51.
[22]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界单元[J].岩土工程学报,2006,28(9):1070-1075.LIU Jingbo,GU Yin,DU Yixin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070-1075.
[23]杜修力,赵密,王进廷.近场波动模拟的人工应力边界条件[J].力学学报,2006,38(1):49-56.DU Xiuli,ZHAO Mi,WANG Jinting.A stress artificial boundary in FEA for near-field wave problem[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(1):49-56.
[24]卢华喜,梁平英,尚守平.地基非线性波动问题中黏-弹性人工边界研究[J].岩土力学,2008,29(7):1911-1916.LU Huaxi,LIANG Pingying,SHANG Shouping.Research on viscoelastic artificial boundary for problem of nonlinear wave motion in soil[J].Rock and Soil Mechanics,2008,29(7):1911-1916.
[25]张波,李术才,杨学英,等.三维黏弹性介质人工边界研究[J].岩土力学,2009,30(11):3469-3475.ZHANG Bo,LI Shucai,YANG Xueying,et al.Study of three dimensional viscoelastic medium artificial boundary[J].Rock and Soil Mechanics,2009,30(11):3469-3475.
[26]赵密,杜修力,刘晶波.一种高阶精度人工边界条件:出平面外域波动问题[J].工程力学,2012,29(4):7-14.ZHAO Mi,DU Xiuli,LIU Jingbo.A high-order accurate artificial boundary condition:out-of-plane exterior waver problem[J].Engineering Mechanics,2012,29(4):7-14.
[27]CHOPRA A K.Dynamics of structures:theory and applications to earthquake engineering[M].3rd ed.New Jersey:Pearson Education Inc.,2007.