各态历经的地震动空间场的简化模拟
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摘要
基于地震动空间场模拟中运用较广泛的原型谱表示法,给出了一种各态历经地震动空间场的模拟公式.首先给出原型谱表示法中体现任意两点间相关性的相位角的具体表达式,使其具有明确的物理意义,从而使得对复功率谱矩阵的Cholesky分解转化为对实数域内的相干函数矩阵的Cholesky分解,提高了合成效率.同时还在模拟公式中引入了双索引频率,以使得模拟公式具有均值与相关函数(自/互功率谱密度函数)的各态历经性,还对其各态历经性进行了理论上的证明,证明了当所模拟的各点地震动时间序列取一个周期时,其均值与相关函数的各态历经性.为进一步提高模拟效率,还给出了特定条件下,相干函数矩阵Cholesky分解后下三角矩阵中各元素的解析解,使得在合成中可以避免反复地在各频率下对相干函数矩阵进行Cholesky分解,从而实现了对模拟模型的简化.最后,给出一个简单的模拟实例,对改进前后模拟的地震动场进行比较,以说明改进后公式的模拟效果.
A simplified method for the simulation of ergodic spatially correlated seismic ground motions was proposed,based on the commonly used original spectral representation method.Firstly,the phase angles,to represent the correlation among ground motions, were given by explicit items with a clear physical.By using these explicit items,computational efficiency can be increased by changing the decomposition of complex cross-spectral matrix into the decomposition of real incoherence coefficient matrix.Double-indexing frequencies were introduced to simulate ergodic seismic ground motions,and the ergodic feature of the improved method was demonstrated theoretically.Subsequently,an explicit solution of the elements of the lower triangular matrix under Cholesky decomposition was given.By using this explicit solution,the improved method had been simplified,and the computational efficiency can be increased greatly,by avoiding repetitive Cholesky decomposition of cross-spectral matrix in every frequency step.At last,a numerical example was employed to illustrate the good character of the improved method.
A simplified method for the simulation of ergodic spatially correlated seismic ground motions was proposed,based on the commonly used original spectral representation method.Firstly,the phase angles,to represent the correlation among ground motions, were given by explicit items with a clear physical.By using these explicit items,computational efficiency can be increased by changing the decomposition of complex cross-spectral matrix into the decomposition of real incoherence coefficient matrix.Double-indexing frequencies were introduced to simulate ergodic seismic ground motions,and the ergodic feature of the improved method was demonstrated theoretically.Subsequently,an explicit solution of the elements of the lower triangular matrix under Cholesky decomposition was given.By using this explicit solution,the improved method had been simplified,and the computational efficiency can be increased greatly,by avoiding repetitive Cholesky decomposition of cross-spectral matrix in every frequency step.At last,a numerical example was employed to illustrate the good character of the improved method.
引文
[1]Shinozuka M,Deodatis G.Stochastic process models for earthquake ground motion[J].Probabilistic Engineering Mechanics,1988,3(3):114-123.
[2]Hao H,Oliveira C S,Penzien J.Multiple-station ground motion processing and simulationbased on SMART-1 array date[J].Nuclear Engineering and Design,1989,111(3):293-310.
[3]LI You-sun,Kareem A.Simulation of multivariate nonstationary random processes by FFT[J].Journal of Engineering Mechanics,ASCE,1991,117(5):1037-1058.
[4]Zerva A.Seismic ground motion simulations from a class of spatial variability models[J].Earthquake Engineering and Structural Dynam ics,1992,21(4):351-361.
[5]Ramadan O,Novak M.Simulation of spatially incoherent random ground motions[J].Jour-nal of Engineering Mechanics,ASCE,1993,119(5):997-1016.
[6]Ramadan O,Novak M.Simulation of multidimensional,anisotropic ground motions[J].Journal of Engineering Mechanics,1994,120(8):1173-1785.
[7]Deodatis George.Non-stationary stochastic vector processes:seisimic ground motion appli-cations[J].Probabilistic Engineering Mechanics,1996,11(3):149-168.
[8]Shrikhande M,Gupta Vinay K.Synthesizing ensembles of spatially correlated accelerograms[J].Journal of Engineering Mechanics,ASCE,1998,124(11):1185-1192.
[9]Paola M Di,Zingales M.Digital simulation of multivariate earthquake ground motions[J].Earthquake Engineering and Structural Dynam ics,2000,17(2):1011-1027.
[10]Jankowski R,Wilde K.A simple method of conditional random field simulation of ground mo-tions for long structures[J].Engineering Structures,2000,22(5):552-561.
[11]Shama A A.Simplified procedure for simulating spatially correlated earthquake ground mo-tions[J].Engineering Structures,2007,29(2):248-258.
[12]屈铁军,王前信.空间相关的多点地震动合成?基本公式[J].地震工程与工程振动,1998,18(1):8-15.(QU Tie-jun,WANG Qian-xin.Simulation of spatially correlative time histories ofmulti point ground motion partⅠ:fundamental formulas[J].Journal of Earthquake Engi-neering and Engineering Vibration,1998,18(1):8-15.(in Chinese))
[13]屈铁军,王前信.空间相关的多点地震动合成?合成实例[J].地震工程与工程振动,1998,18(2):25-32.(QU Tie-jun,WANG Qian-xin.Simulation of spatially correlative time histories ofmulti point ground motion partⅡ:application of fundamental formulas[J].Journal ofEarthquake Engineering and Engineering Vibration,1998,18(2):25-32.(in Chinese))
[14]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 You-bo,WANG Nian-qiao,ZHANG Shang-gen.A simulation method for spatialcorrelative time histories of multi-point ground motion[J].World Inform ation on EarthquakeEngineering,2002,18(1):119-122.(in Chinese))
[15]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 Xian-ming,YE Ji-hong,LI Ai-qun.A simplified method for the simulation of thetime histories of spatial correlative multi-point ground motion[J].Earthquake Resistant En-gineering,2003,(1):30-36.(in Chinese))
[16]梁建文.非平稳地震动过程模拟方法?[J].地震学报,2005,27(2):213-228.(LIANG Jian-w en.Simulation of non-stationary ground motion processes?[J].Acta Seism ologica Sinca,2005,27(2):213-228.(in Chinese))
[17]梁建文.非平稳地震动过程模拟方法?[J].地震学报,2005,27(3):346-351.(LIANG Jian-w en.Simulation of non-stationary ground motion processes?[J].Acta Seism ologica Sinca,2005,27(3):346-351.(in Chinese))
[18]胡亮,李黎,樊剑.基于特征正交分解的空间变异地震动模拟[J].西南交通大学学报,2006,41(6):685-689.(HU Liang,LI Li,FAN Jian.Proper orthogonal decomposition-based simula-tion of spatially correlated seismic ground motions[J].Journal of Southw est Jiaotong Uni-versity,2006,41(6):685-689.(in Chinese))
[19]董汝博,周晶,冯新.一种考虑局部场地收敛性的多点地震动合成方法[J].振动与冲击,2007,26(4):5-9.(DONG Ru-bo,ZHOU Jing,FENG Xin.Simulation of non-stationary spatiallycorrelative time histories of multi-point ground motion[J].Journal of Earthquake Engineer-ing and Engineering Vibration,2007,26(4):5-9.(in Chinese))
[20]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 Ru-bo,ZHOU Jing,FENG Xin.A local convergent method forsimulation multi-point earthquake ground motion[J].Journal of Vibration and Shock,2007,27(3):10-14.(in Chinese))
[21]全伟,李宏男.基于小波变换的拟合规范反应谱多维地震动模拟[J].地震工程与工程振动,2007,27(4):103-108.(QUAN Wei,LI Hong-nan.Generation of spectrum-compatible multi-dimensional ground motions via w avelet transform[J].Journal of Earthquake Engineeringand Engineering Vibration,2007,27(4):103-108.(in Chinese))
[22]Shinozuka M.Simulation of multivariate and multidimensional random processes[J].Journalof the Acoustical Society of Am erica,1971,49(1):357-368.
[23]Shinozuka M.Digital simulation of random processes and its applications[J].Journal ofSound and Vibration,1972,25(1):111-128.
[24]Yang J N.Simulation of random envelope processes[J].Journal of Sound and Vibration,1972,21(1):73-85.
[25]Vaicaitis R,Takeno M,Shinozuka M.Response analysis of tall buildings to wind loadings[J].Journal of Structure Engineering,ASCE,1975,101(3):585-600.
[26]Shinozuka M,Ynu C B,Seya H.Stochastic methods in wind engineering[J].Journal ofWind Engineering and Industrial Aerodynam ics,1990,36(1/3):829-843.
[27]Shinozuka M,Deodatis G.Simulation of stochastic processes by spectral representation[J].Applied Mechanics Review,1991,44(4):191-204.
[28]Deodatis G.Simulation of ergodic multivariate stochastic processes[J].Journal of Engineer-ing Mechanics,ASCE,1996,122(8):778-787.
[29]Der Kiureghian A,Keshishian P,Hakobian A.Multiple support response spectrum analysis ofbridges including the site-response effect and the MSRS code[R].Earthquake EngineeringResearch Center,College of Engineering,University of California,1997.
[30]Yang W W,Chang T Y P,Chang C C.An efficient wind field simulation technique for bridges[J].Journal of Wind Engineering and Industrial Aerodynamics,1997,19(67/68):697-708.
[31]Yang W W,Chang T Y P,Chang C C.Numerical simulation of turbulent fluctuations along theaxis of a bridge[J].Engineering Structures,1998,20(9):837-848.
[32]Cao Y H,Xiang H F,Zhou Y.Simulation of stochastic wind velocity field on long-span bridg-es[J].Journal of Engineering Mechanics,2000,126(1):1-6.
[33]Li Y L,Liao H L,Qiang S Z.Simplifying the simulation of stochastic wind velocity field forlong cable-stayed bridges[J].Com putes and Structures,2004,82(20/21):1591-1598.
[34]Loh C H,Lin S G.Directionality and simulation in spatial variation of seismic waves[J].En-gineering Structures,1990,12(2):134-143.
[35]Harichandran R S,Vanmarcke Erik H.Stochastic variation of earthquake ground motion inspace and time[J].Journal of Engineering Mechanics,ASCE,1986,112(2):154-174.
[36]Feng Q M,Hu Y X.Spatial correlation of earthquake motion and its effect on structural re-sponse[C]//Proc of US-PRC,Bilateral Workshop on Earthquake Engineering.Vol 1,A-5,Beijing:1982.
[37]Loh C H,Yeh Y T.Spatial variation and stochastic modeling of seismic differential groundmovement[J].Earthquake Engineering and Structural Dynam ics,1988,16(5):583-596.
[38]Clough R W,Penzien J.Dynamics of Structures[M].2nd ed.Singapore:McGraw Hill,Inc,1993.
[2]Hao H,Oliveira C S,Penzien J.Multiple-station ground motion processing and simulationbased on SMART-1 array date[J].Nuclear Engineering and Design,1989,111(3):293-310.
[3]LI You-sun,Kareem A.Simulation of multivariate nonstationary random processes by FFT[J].Journal of Engineering Mechanics,ASCE,1991,117(5):1037-1058.
[4]Zerva A.Seismic ground motion simulations from a class of spatial variability models[J].Earthquake Engineering and Structural Dynam ics,1992,21(4):351-361.
[5]Ramadan O,Novak M.Simulation of spatially incoherent random ground motions[J].Jour-nal of Engineering Mechanics,ASCE,1993,119(5):997-1016.
[6]Ramadan O,Novak M.Simulation of multidimensional,anisotropic ground motions[J].Journal of Engineering Mechanics,1994,120(8):1173-1785.
[7]Deodatis George.Non-stationary stochastic vector processes:seisimic ground motion appli-cations[J].Probabilistic Engineering Mechanics,1996,11(3):149-168.
[8]Shrikhande M,Gupta Vinay K.Synthesizing ensembles of spatially correlated accelerograms[J].Journal of Engineering Mechanics,ASCE,1998,124(11):1185-1192.
[9]Paola M Di,Zingales M.Digital simulation of multivariate earthquake ground motions[J].Earthquake Engineering and Structural Dynam ics,2000,17(2):1011-1027.
[10]Jankowski R,Wilde K.A simple method of conditional random field simulation of ground mo-tions for long structures[J].Engineering Structures,2000,22(5):552-561.
[11]Shama A A.Simplified procedure for simulating spatially correlated earthquake ground mo-tions[J].Engineering Structures,2007,29(2):248-258.
[12]屈铁军,王前信.空间相关的多点地震动合成?基本公式[J].地震工程与工程振动,1998,18(1):8-15.(QU Tie-jun,WANG Qian-xin.Simulation of spatially correlative time histories ofmulti point ground motion partⅠ:fundamental formulas[J].Journal of Earthquake Engi-neering and Engineering Vibration,1998,18(1):8-15.(in Chinese))
[13]屈铁军,王前信.空间相关的多点地震动合成?合成实例[J].地震工程与工程振动,1998,18(2):25-32.(QU Tie-jun,WANG Qian-xin.Simulation of spatially correlative time histories ofmulti point ground motion partⅡ:application of fundamental formulas[J].Journal ofEarthquake Engineering and Engineering Vibration,1998,18(2):25-32.(in Chinese))
[14]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 You-bo,WANG Nian-qiao,ZHANG Shang-gen.A simulation method for spatialcorrelative time histories of multi-point ground motion[J].World Inform ation on EarthquakeEngineering,2002,18(1):119-122.(in Chinese))
[15]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 Xian-ming,YE Ji-hong,LI Ai-qun.A simplified method for the simulation of thetime histories of spatial correlative multi-point ground motion[J].Earthquake Resistant En-gineering,2003,(1):30-36.(in Chinese))
[16]梁建文.非平稳地震动过程模拟方法?[J].地震学报,2005,27(2):213-228.(LIANG Jian-w en.Simulation of non-stationary ground motion processes?[J].Acta Seism ologica Sinca,2005,27(2):213-228.(in Chinese))
[17]梁建文.非平稳地震动过程模拟方法?[J].地震学报,2005,27(3):346-351.(LIANG Jian-w en.Simulation of non-stationary ground motion processes?[J].Acta Seism ologica Sinca,2005,27(3):346-351.(in Chinese))
[18]胡亮,李黎,樊剑.基于特征正交分解的空间变异地震动模拟[J].西南交通大学学报,2006,41(6):685-689.(HU Liang,LI Li,FAN Jian.Proper orthogonal decomposition-based simula-tion of spatially correlated seismic ground motions[J].Journal of Southw est Jiaotong Uni-versity,2006,41(6):685-689.(in Chinese))
[19]董汝博,周晶,冯新.一种考虑局部场地收敛性的多点地震动合成方法[J].振动与冲击,2007,26(4):5-9.(DONG Ru-bo,ZHOU Jing,FENG Xin.Simulation of non-stationary spatiallycorrelative time histories of multi-point ground motion[J].Journal of Earthquake Engineer-ing and Engineering Vibration,2007,26(4):5-9.(in Chinese))
[20]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 Ru-bo,ZHOU Jing,FENG Xin.A local convergent method forsimulation multi-point earthquake ground motion[J].Journal of Vibration and Shock,2007,27(3):10-14.(in Chinese))
[21]全伟,李宏男.基于小波变换的拟合规范反应谱多维地震动模拟[J].地震工程与工程振动,2007,27(4):103-108.(QUAN Wei,LI Hong-nan.Generation of spectrum-compatible multi-dimensional ground motions via w avelet transform[J].Journal of Earthquake Engineeringand Engineering Vibration,2007,27(4):103-108.(in Chinese))
[22]Shinozuka M.Simulation of multivariate and multidimensional random processes[J].Journalof the Acoustical Society of Am erica,1971,49(1):357-368.
[23]Shinozuka M.Digital simulation of random processes and its applications[J].Journal ofSound and Vibration,1972,25(1):111-128.
[24]Yang J N.Simulation of random envelope processes[J].Journal of Sound and Vibration,1972,21(1):73-85.
[25]Vaicaitis R,Takeno M,Shinozuka M.Response analysis of tall buildings to wind loadings[J].Journal of Structure Engineering,ASCE,1975,101(3):585-600.
[26]Shinozuka M,Ynu C B,Seya H.Stochastic methods in wind engineering[J].Journal ofWind Engineering and Industrial Aerodynam ics,1990,36(1/3):829-843.
[27]Shinozuka M,Deodatis G.Simulation of stochastic processes by spectral representation[J].Applied Mechanics Review,1991,44(4):191-204.
[28]Deodatis G.Simulation of ergodic multivariate stochastic processes[J].Journal of Engineer-ing Mechanics,ASCE,1996,122(8):778-787.
[29]Der Kiureghian A,Keshishian P,Hakobian A.Multiple support response spectrum analysis ofbridges including the site-response effect and the MSRS code[R].Earthquake EngineeringResearch Center,College of Engineering,University of California,1997.
[30]Yang W W,Chang T Y P,Chang C C.An efficient wind field simulation technique for bridges[J].Journal of Wind Engineering and Industrial Aerodynamics,1997,19(67/68):697-708.
[31]Yang W W,Chang T Y P,Chang C C.Numerical simulation of turbulent fluctuations along theaxis of a bridge[J].Engineering Structures,1998,20(9):837-848.
[32]Cao Y H,Xiang H F,Zhou Y.Simulation of stochastic wind velocity field on long-span bridg-es[J].Journal of Engineering Mechanics,2000,126(1):1-6.
[33]Li Y L,Liao H L,Qiang S Z.Simplifying the simulation of stochastic wind velocity field forlong cable-stayed bridges[J].Com putes and Structures,2004,82(20/21):1591-1598.
[34]Loh C H,Lin S G.Directionality and simulation in spatial variation of seismic waves[J].En-gineering Structures,1990,12(2):134-143.
[35]Harichandran R S,Vanmarcke Erik H.Stochastic variation of earthquake ground motion inspace and time[J].Journal of Engineering Mechanics,ASCE,1986,112(2):154-174.
[36]Feng Q M,Hu Y X.Spatial correlation of earthquake motion and its effect on structural re-sponse[C]//Proc of US-PRC,Bilateral Workshop on Earthquake Engineering.Vol 1,A-5,Beijing:1982.
[37]Loh C H,Yeh Y T.Spatial variation and stochastic modeling of seismic differential groundmovement[J].Earthquake Engineering and Structural Dynam ics,1988,16(5):583-596.
[38]Clough R W,Penzien J.Dynamics of Structures[M].2nd ed.Singapore:McGraw Hill,Inc,1993.
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