基于功率谱矩阵开方分解的空间相关多点地震动合成方法
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摘要
基于前人对多点地震动合成方法的研究,提出了一种多点地震动的合成方法。该方法基于复功率谱矩阵的开方分解,将复功率谱矩阵分解为一个Hermite矩阵与其共轭转置矩阵的乘积,并给出了基于此分解的考虑空间相关性并满足功率谱矩阵的幅值谱和相位谱的求解公式。此外,在合成功率谱矩阵时还考虑了空间各点地震动自功率谱谱强度的变化,以使得合成的多点地震动更好地适用于大跨度结构物的多点输入分析。
Based on previous studies,a simulation method for multi-point ground motion time histories is developed.Employing the square root operation of complex power spectral density matrix,the paper defined the complex power spectral density matrix as the product of Hermite matrix and its conjugate transposed matrix.Meanwhile,the relationship between amplitudes,phase angels of simulated ground motions and the element of the Hermite matrix was given.In addition,the variation of the amplitude of white-noise bedrock acceleration was taken in account,which makes this method more suitable for the simulation of the multi-point ground motions of long-span structures.
Based on previous studies,a simulation method for multi-point ground motion time histories is developed.Employing the square root operation of complex power spectral density matrix,the paper defined the complex power spectral density matrix as the product of Hermite matrix and its conjugate transposed matrix.Meanwhile,the relationship between amplitudes,phase angels of simulated ground motions and the element of the Hermite matrix was given.In addition,the variation of the amplitude of white-noise bedrock acceleration was taken in account,which makes this method more suitable for the simulation of the multi-point ground motions of long-span structures.
引文
[1]Feng Q M,Hu Y X.Spatial correlation of earthquakemotion and its effect on structural response[C]//HuYuxian.Proceeding of US-PRC,Bilateral Workshop onEarthquake Engineering,Beijing:1982.
[2]Bogdanoff J,Goldber J,Shiff J.The effect of groundtransmission time on the response of long structures[J].The Bulletion of Seismological Society of America,1965,55:627―640.
[3]Chen M T,Harichandran R S.Response of an earth damto spatially varying earthquake ground motion[J].Journal of Engineering Mechanics,2001,127(9):932―939.
[4]Yasemin B,Alemdar B,Kurtulus S.Stochastic dynamicresponse of dam-reservoir-foundation systems tospatially varying earthquake ground motions[J].SoilDynamics and Earthquake Engineering,2009,29(3):444―458.
[5]Zhang Y H,Li Q S,Lin J H,Williams F W.Randomvibration analysis of long-span structures subjected tospatially varying ground motions[J].Soil Dynamics andEarthquake Engineering,2009,29(4):620―629.
[6]Hao H,Oliveira C S,Penzien J.Multiple-station groundmotion processing and simulation based on SMART-1array data[J].Nuclear Engineering and Design,1989,111(3):293―310.
[7]屈铁军,王前信.空间相关的多点地震动合成(I):基本公式[J].地震工程与工程振动,1998,18(1):8―15.Qu Tiejun,Wang Qianxin.Simulation of spatiallycorrelative time histories of multi-point ground motion,Part I:Fundamental formulas[J].EarthquakeEngineering and Engineering Vibration,1998,18(1):8―15.(in Chinese)
[8]屈铁军,王前信.空间相关的多点地震动合成(II):合成实例[J].地震工程与工程振动,1998,18(2):25―32.Qu Tiejun,Wang Qianxin.Simulation of spatiallycorrelative time histories of multi-point ground motion,Part II:Application of fundamental formulas[J].Earthquake Engineering and Engineering Vibration,1998,18(2):25―32.(in Chinese)
[9]刘先明,叶继红,李爱群.空间相关多点地震动合成的简化方法[J].工程抗震,2003(1):30―36.Liu Xianming,Ye Jihong,Li Aiqun.A simplified methodfor the simulation of the time histories of spatiallycorrelative multi-point ground motion[J].EarthquakeResistant Engineering,2003(1):30―36.(in Chinese)
[10]夏友柏,王年桥,张尚根.一种合成多点地震动时程的方法[J].世界地震工程,2002,18(1):119―122.Xia Youbai,Wang Nianqiao,Zhang Shanggen.Asimulation method for spatial correlative time histories ofmulti-point ground motion[J].World Information onEarthquake Engineering,2002,18(1):119―122.(inChinese)
[11]姜海鹏,王航,杨庆山.空间变异地震作用下大型连续刚构桥的延性性能分析[J].铁道工程学报,2001(2):37―45.Jiang Haipeng,Wang Hang,Yang Qingshan.Behavior oflong rigid frame rc bridges subjected to spatial varyingground motion:(1)Generation of spatial varying groundmotion based on phase difference spectrum[J].Journalof Railway Engineering Society,2001(2):37―45.(inChinese)
[12]董汝博,周晶,冯新.非平稳空间相关多点地震动合成方法研究[J].地震工程与工程振动,2007,27(3):10―14.Dong Rubo,Zhou Jing,Feng Xin.Simulation ofnon-stationary correlative time histories of multi-pointground motion[J].Journal of Engineering andEngineering Vibration,2007,27(3):10―14.(in Chinese)
[13]Ramadan O,Novak M.Simulation of spatiallyinconherent random ground motions[J].Journal ofEngineering Mechanics,1993,119(5):997―1016.
[14]Ayman A Shama.Simplified procedure for simulatingspatially correlated earthquake ground motions[J].Engineering Structures,2007,29:248―258.
[15]Alves S W,Hall J F.Generation of spatially nonuniformground motion for nonlinear analysis of a concrete archdam[J].Earthquake Engineering and StructuralDynamics,2006,35(11):1339―1357.
[16]屈铁军,王前信.SMART-1台阵记录的自功率谱特性研究[J].世界地震工程,1995,11(1):35―43.Qu Tiejun,Wang Qianxin.Study on characteristics ofspectra of ground motion record at SMART-1 array[J].World Information on Earthquake Engineering,1995,11(1):35―43.(in Chinese)
[17]屈铁军,王君杰,王前信.空间变化的地震动功率谱的实用模型[J].地震学报,1996,18(1):55―62.Qu Tiejun,Wang Junjie,Wang Qianxin.A practicalmodel for the power spectrum of spatially varyingground motion[J].Acta Seismologica Sinica,1996,18(1):55―62.(in Chinese)
[18]Feng Q M,Hu Y X.Spatial correlation of earthquakemotion and its effect on structural response[C].Proceeding USPRC Bilateral Workshop on EarthquakeEngineering,1982,1:A-5-1―A-5-14.
[19]Harichandran R S,Vanmavcke E H.Stochastic variationof earthquake ground motion in space and time[J].Journal of Engineering Mechanics,1986,112(2):154―174.
[20]Ohsaki Y.On the significance of phase content inearthquake ground motions[J].Earthquake Engineeringand Structual Dynamics,1979,7(5):427―439.
[21]朱昱,冯启明.相位差谱的分布特征和人造地震动[J].地震工程与工程振动,1992,12(1):37―44.Zhu Yu,Feng Qiming.Digital characteristics of phasedifference spectrum distribution of earthquakeaccelerogram[J].Earthquake Engineering andEngineering Vibration,1992,12(1):37―44.(in Chinese)
[22]朱昱,冯启明.地震加速度相位差谱分布的数字特征[J].地震工程与工程振动,1993,13(2):30―37.Zhu Yu,Feng Qiming.Distribution characteristic ofphase difference spectrum and artificial accelerogram[J].Earthquake Engineering and Engineering Vibration,1993,13(2):30―37.(in Chinese)
[23]Vanmarcke E H.Structural response to earthquakes[M].Seismic Risk and Engineering Decision,McGraw-Hill,New York,NY,1977:287―337.
[2]Bogdanoff J,Goldber J,Shiff J.The effect of groundtransmission time on the response of long structures[J].The Bulletion of Seismological Society of America,1965,55:627―640.
[3]Chen M T,Harichandran R S.Response of an earth damto spatially varying earthquake ground motion[J].Journal of Engineering Mechanics,2001,127(9):932―939.
[4]Yasemin B,Alemdar B,Kurtulus S.Stochastic dynamicresponse of dam-reservoir-foundation systems tospatially varying earthquake ground motions[J].SoilDynamics and Earthquake Engineering,2009,29(3):444―458.
[5]Zhang Y H,Li Q S,Lin J H,Williams F W.Randomvibration analysis of long-span structures subjected tospatially varying ground motions[J].Soil Dynamics andEarthquake Engineering,2009,29(4):620―629.
[6]Hao H,Oliveira C S,Penzien J.Multiple-station groundmotion processing and simulation based on SMART-1array data[J].Nuclear Engineering and Design,1989,111(3):293―310.
[7]屈铁军,王前信.空间相关的多点地震动合成(I):基本公式[J].地震工程与工程振动,1998,18(1):8―15.Qu Tiejun,Wang Qianxin.Simulation of spatiallycorrelative time histories of multi-point ground motion,Part I:Fundamental formulas[J].EarthquakeEngineering and Engineering Vibration,1998,18(1):8―15.(in Chinese)
[8]屈铁军,王前信.空间相关的多点地震动合成(II):合成实例[J].地震工程与工程振动,1998,18(2):25―32.Qu Tiejun,Wang Qianxin.Simulation of spatiallycorrelative time histories of multi-point ground motion,Part II:Application of fundamental formulas[J].Earthquake Engineering and Engineering Vibration,1998,18(2):25―32.(in Chinese)
[9]刘先明,叶继红,李爱群.空间相关多点地震动合成的简化方法[J].工程抗震,2003(1):30―36.Liu Xianming,Ye Jihong,Li Aiqun.A simplified methodfor the simulation of the time histories of spatiallycorrelative multi-point ground motion[J].EarthquakeResistant Engineering,2003(1):30―36.(in Chinese)
[10]夏友柏,王年桥,张尚根.一种合成多点地震动时程的方法[J].世界地震工程,2002,18(1):119―122.Xia Youbai,Wang Nianqiao,Zhang Shanggen.Asimulation method for spatial correlative time histories ofmulti-point ground motion[J].World Information onEarthquake Engineering,2002,18(1):119―122.(inChinese)
[11]姜海鹏,王航,杨庆山.空间变异地震作用下大型连续刚构桥的延性性能分析[J].铁道工程学报,2001(2):37―45.Jiang Haipeng,Wang Hang,Yang Qingshan.Behavior oflong rigid frame rc bridges subjected to spatial varyingground motion:(1)Generation of spatial varying groundmotion based on phase difference spectrum[J].Journalof Railway Engineering Society,2001(2):37―45.(inChinese)
[12]董汝博,周晶,冯新.非平稳空间相关多点地震动合成方法研究[J].地震工程与工程振动,2007,27(3):10―14.Dong Rubo,Zhou Jing,Feng Xin.Simulation ofnon-stationary correlative time histories of multi-pointground motion[J].Journal of Engineering andEngineering Vibration,2007,27(3):10―14.(in Chinese)
[13]Ramadan O,Novak M.Simulation of spatiallyinconherent random ground motions[J].Journal ofEngineering Mechanics,1993,119(5):997―1016.
[14]Ayman A Shama.Simplified procedure for simulatingspatially correlated earthquake ground motions[J].Engineering Structures,2007,29:248―258.
[15]Alves S W,Hall J F.Generation of spatially nonuniformground motion for nonlinear analysis of a concrete archdam[J].Earthquake Engineering and StructuralDynamics,2006,35(11):1339―1357.
[16]屈铁军,王前信.SMART-1台阵记录的自功率谱特性研究[J].世界地震工程,1995,11(1):35―43.Qu Tiejun,Wang Qianxin.Study on characteristics ofspectra of ground motion record at SMART-1 array[J].World Information on Earthquake Engineering,1995,11(1):35―43.(in Chinese)
[17]屈铁军,王君杰,王前信.空间变化的地震动功率谱的实用模型[J].地震学报,1996,18(1):55―62.Qu Tiejun,Wang Junjie,Wang Qianxin.A practicalmodel for the power spectrum of spatially varyingground motion[J].Acta Seismologica Sinica,1996,18(1):55―62.(in Chinese)
[18]Feng Q M,Hu Y X.Spatial correlation of earthquakemotion and its effect on structural response[C].Proceeding USPRC Bilateral Workshop on EarthquakeEngineering,1982,1:A-5-1―A-5-14.
[19]Harichandran R S,Vanmavcke E H.Stochastic variationof earthquake ground motion in space and time[J].Journal of Engineering Mechanics,1986,112(2):154―174.
[20]Ohsaki Y.On the significance of phase content inearthquake ground motions[J].Earthquake Engineeringand Structual Dynamics,1979,7(5):427―439.
[21]朱昱,冯启明.相位差谱的分布特征和人造地震动[J].地震工程与工程振动,1992,12(1):37―44.Zhu Yu,Feng Qiming.Digital characteristics of phasedifference spectrum distribution of earthquakeaccelerogram[J].Earthquake Engineering andEngineering Vibration,1992,12(1):37―44.(in Chinese)
[22]朱昱,冯启明.地震加速度相位差谱分布的数字特征[J].地震工程与工程振动,1993,13(2):30―37.Zhu Yu,Feng Qiming.Distribution characteristic ofphase difference spectrum and artificial accelerogram[J].Earthquake Engineering and Engineering Vibration,1993,13(2):30―37.(in Chinese)
[23]Vanmarcke E H.Structural response to earthquakes[M].Seismic Risk and Engineering Decision,McGraw-Hill,New York,NY,1977:287―337.
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