结构地震反应DQ解法的两种数值格式
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摘要
通过对时间计算域的离散化将结构地震时程分析问题转化为一系列初值问题的求解,建立了一种基于DQ原理的结构地震反应高精度数值分析方法.为了使DQ原理能够应用于任意变化的地震地面加速度激励下的结构动力分析问题,提出了两种数值方案———单时步格式和多时步格式.单时步格式要求假定时步内地面加速度的分布模式,并且需要对时步进行网格剖分;多时步格式直接利用地面加速度的离散信息构造数值格式,一次运算可获得多个时刻的反应值.数值试验表明:两种数值格式均可以获得较高的数值精度.对于单时步格式,可以通过采用较大步距的方式降低结构动力分析的计算工作量;对于多时步格式,可以通过时步数量的合理选择达到提高计算效率的目的.
Based on the step-by-step procedure,a high order numerical approach was presented for structural dynamic analysis subjected to earthquake induced ground motion using differential quadrature(DQ)rule in time domain.Total two numerical schemes,the single-step scheme and the multi-step scheme,were proposed for the DQ-based analysis procedure to achieve the seismic response of structures.For the single-step scheme,the DQ rule was used during each time step,and the dynamic response at the final point of the step was calculated from the initial conditions existing at the beginning of the step and from the history of earthquake ground acceleration over the step,but the distribution function of the ground acceleration during the step must be assumed and at least one additional sampling point had to be interpolated within the discrete time step to fit in with the needs of the DQ method.In another way,the DQ rule was used directly among a time interval consisting of a few time steps due to multi-step scheme,so the response values at more than one discrete time points may be obtained simultaneously.Results from numerical analysis showed that both the DQ-based analysis procedures could be used to achieve the time history of response of the seismic structures with a high level of accuracy.The time steps employed in the single-step procedure could usually be made a little longer to reduce the computational efforts,and the efficiency will be improved due to the multi-step procedure because several time steps have been contained within each interval in which the DQ rule is substituted to give a equivalent static equation for the solution of the seismic response.
Based on the step-by-step procedure,a high order numerical approach was presented for structural dynamic analysis subjected to earthquake induced ground motion using differential quadrature(DQ)rule in time domain.Total two numerical schemes,the single-step scheme and the multi-step scheme,were proposed for the DQ-based analysis procedure to achieve the seismic response of structures.For the single-step scheme,the DQ rule was used during each time step,and the dynamic response at the final point of the step was calculated from the initial conditions existing at the beginning of the step and from the history of earthquake ground acceleration over the step,but the distribution function of the ground acceleration during the step must be assumed and at least one additional sampling point had to be interpolated within the discrete time step to fit in with the needs of the DQ method.In another way,the DQ rule was used directly among a time interval consisting of a few time steps due to multi-step scheme,so the response values at more than one discrete time points may be obtained simultaneously.Results from numerical analysis showed that both the DQ-based analysis procedures could be used to achieve the time history of response of the seismic structures with a high level of accuracy.The time steps employed in the single-step procedure could usually be made a little longer to reduce the computational efforts,and the efficiency will be improved due to the multi-step procedure because several time steps have been contained within each interval in which the DQ rule is substituted to give a equivalent static equation for the solution of the seismic response.
引文
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[7]Kang K,Bert C W,Striz A G.Vibration analysis of horizontally curved beams with warping using DQM[J].Journal ofStructural Engineering,19961,22(6):657-662
[8]Kang K,Bert C W,Striz A G.Vibration analysis of shear deformable circular arches by the differential quadraturemethod[J].Journal of Sound and Vibration1,995,181(2):353-360
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[11]De Rosa M A,Franciosi C.Exact and approximate dynamic analysis of circular arches using DQM[J].InternationalJournal of Solids and Structures,2000,37:1103-1117
[12]聂国隽,仲政.用微分求积法求解梁的弹塑性问题[J].工程力学2,005,22(1):59-62Nie Guojun,Zhong Zheng.Elasto-plastic analysis of beams by differential quadrature method[J].EngineeringMechanics,20052,2(1):59-62
[13]任永亮.谱方法在工程波动分析中的应用研究[D].南京:南京工业大学硕士学位论文,2008Ren Yongliang.The application of spectral method in Engineering wave analysis[D].Nanjing:Master’s Thesis ofNanjing University of Technology,2008
[14]Fung T C.Solving initial value problems by differential quadrature method—Part 1:first-order equations[J].International Journal for Numerical Methods in Engineering2,001,50:1411-1427
[15]Fung T C.Solving initial value problems by differential quadrature method—Part 2:second-and higher-order equations[J].International Journal for Numerical Methods in Engineering,20015,0:1429-1454
[16]王通.曲线梁桥地震时程反应的微分求积分析方法[D].南京:南京工业大学硕士学位论文,2009Wang Tong.Differential quadrature analysis method for seismic time history response of curved girder bridges[D].Nanjing:Master’s Thesis of Nanjing University of Technology,2009
[17]Chopra A K.Dynamics of structures:theory and application to earthquake engineering(Second edition)[M].TsinghuaUniversity Press2,004
[2]Bellman R E,Casti J.Differential quadrature and long-term intergration[J].Journal of Mathematical Analysis andApplications,19713,4:235-238
[3]Bert C W,Malik M.Differential quadrature method in computational mechanics:A review[J].Applied MechanicsReviews,19964,9(1):1-28
[4]Mingle J O.The method of differential quadrature for transient nonlinear diffusion[J].Journal of Mathematical Analysisand Applications,19776,0:559-569
[5]Civan F,Sliicpcevich C M.Application of differential quadrature to transport processes[J].Journal of MathematicalAnalysis and Applications,19839,3(1):206-221
[6]Shu C,Richards B E.Parallel simulation of incompressible viscous flows by generalized differential quadrature[J].Computing Systems in Engineering,19923,:271-281
[7]Kang K,Bert C W,Striz A G.Vibration analysis of horizontally curved beams with warping using DQM[J].Journal ofStructural Engineering,19961,22(6):657-662
[8]Kang K,Bert C W,Striz A G.Vibration analysis of shear deformable circular arches by the differential quadraturemethod[J].Journal of Sound and Vibration1,995,181(2):353-360
[9]Wu T Y,Liu G R.The generalized differential quadrature rule for fourth-order differential equations[J].InternationalJournal for Numerical Methods in Engineering,20015,0:1907-1929
[10]Wu T Y,Liu G R,Wang Y Y.Application of the generalized differential quadrature rule to initial-boundary-valueproblems[J].Journal of Sound and Vibrarion,20032,64(4):883-891
[11]De Rosa M A,Franciosi C.Exact and approximate dynamic analysis of circular arches using DQM[J].InternationalJournal of Solids and Structures,2000,37:1103-1117
[12]聂国隽,仲政.用微分求积法求解梁的弹塑性问题[J].工程力学2,005,22(1):59-62Nie Guojun,Zhong Zheng.Elasto-plastic analysis of beams by differential quadrature method[J].EngineeringMechanics,20052,2(1):59-62
[13]任永亮.谱方法在工程波动分析中的应用研究[D].南京:南京工业大学硕士学位论文,2008Ren Yongliang.The application of spectral method in Engineering wave analysis[D].Nanjing:Master’s Thesis ofNanjing University of Technology,2008
[14]Fung T C.Solving initial value problems by differential quadrature method—Part 1:first-order equations[J].International Journal for Numerical Methods in Engineering2,001,50:1411-1427
[15]Fung T C.Solving initial value problems by differential quadrature method—Part 2:second-and higher-order equations[J].International Journal for Numerical Methods in Engineering,20015,0:1429-1454
[16]王通.曲线梁桥地震时程反应的微分求积分析方法[D].南京:南京工业大学硕士学位论文,2009Wang Tong.Differential quadrature analysis method for seismic time history response of curved girder bridges[D].Nanjing:Master’s Thesis of Nanjing University of Technology,2009
[17]Chopra A K.Dynamics of structures:theory and application to earthquake engineering(Second edition)[M].TsinghuaUniversity Press2,004
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