结构地震反应分析的逐步微分积分方法
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摘要
将结构地震时程反应分析问题离散化为一系列初值问题的求解,在每个离散时步内应用微分积分(DQ)法则,建立了一种结构地震反应的逐步DQ数值求解方法.根据地震地面运动及结构体系的动态特征和物理本质,假定离散时步内地面加速度为线性分布.推导并给出了该分布模式假定下利用DQ法逐步求解体系地震反应的数值格式,通过一个双自由度体系的数值实验阐述了结构地震反应DQ分析方法的应用.算例表明,体系地震反应DQ解的稳定性和精度比较好,在较大的离散步距条件下仍然可以获得较好的计算结果.
A numerical time-stepping method is developed for the solution of equations governing the motion of a MDOF system subjected to earthquake induced ground acceleration on the basis of the differential quadrature (DQ) rule.Initially the history of the seismic ground motion and the structural dynamic responses are divided into a sequence of time intervals,and the DQ rule is used within each discrete time step,thus the final response of velocity and acceleration during the interval may be expressed as a weighted linear sum of the initial condition and the response of displacement at the sampling grid points.It is assumed that the ground acceleration over the time step increases linearly with time,and the formulae for the quadrature solution of the differential equations of motion are deduced so that the history of the dynamic response could be calculated in the time-stepping procedure using the DQ method.A two-story shear frame structure is employed for numerical illustration, and results from numerical analysis show that the proposed quadrature procedure can still achieve an excellent analysis for the time history of the seismic response even when the length of the time step is selected to be a little larger.
A numerical time-stepping method is developed for the solution of equations governing the motion of a MDOF system subjected to earthquake induced ground acceleration on the basis of the differential quadrature (DQ) rule.Initially the history of the seismic ground motion and the structural dynamic responses are divided into a sequence of time intervals,and the DQ rule is used within each discrete time step,thus the final response of velocity and acceleration during the interval may be expressed as a weighted linear sum of the initial condition and the response of displacement at the sampling grid points.It is assumed that the ground acceleration over the time step increases linearly with time,and the formulae for the quadrature solution of the differential equations of motion are deduced so that the history of the dynamic response could be calculated in the time-stepping procedure using the DQ method.A two-story shear frame structure is employed for numerical illustration, and results from numerical analysis show that the proposed quadrature procedure can still achieve an excellent analysis for the time history of the seismic response even when the length of the time step is selected to be a little larger.
引文
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2 Wilson EL,Farhoomand I,Bathe KJ.Nonlinear dynamic analysis of complex structures.Earthquake Engineering and Structural Dynamics,1973,1:241-252
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4王鑫伟.微分求积法在结构力学中的应用.力学进展,1995, 25(2):232-240(Wang Xinwei.Differential quadrature in the analysis of structural components.Advances in Mechanics, 1995,25(2):232-240(in Chinese))
5 Bert CW,Malik M.Differential quadrature method in computational mechanics:A review.Applied Mechanics Reviews, 1996,49(1):1-28
6 Fung TC.Solving initial value problems by differential quadrature method—Part 1:first-order equations.International Journal for Numerical Methods in Engineering, 2001,50:1411-1427
7 Fung TC.Solving initial value problems by differential quadrature method—Part 2:second- and higher-order equations.International Journal for Numerical Methods in Engineering,2001,50:1429-1454
8 Shen J,Tang T.Spectral and High-order Methods with Applications.Beijing:Science Press,2006
9王通.曲线梁桥地震时程反应的微分求积分析方法.[硕士论文].南京:南京工业大学,2009(Wang Tong.Differential quadrature analysis method for seismic time history response of curved girder bridges.[Master Thesis].Nanjing:Nanjing University of Technology,2009(in Chinese))
2 Wilson EL,Farhoomand I,Bathe KJ.Nonlinear dynamic analysis of complex structures.Earthquake Engineering and Structural Dynamics,1973,1:241-252
3 Bellman RE,Casti J.Differential quadrature and long-term intergration.Journal of Mathematical Analysis and Applications, 1971,34:235-238
4王鑫伟.微分求积法在结构力学中的应用.力学进展,1995, 25(2):232-240(Wang Xinwei.Differential quadrature in the analysis of structural components.Advances in Mechanics, 1995,25(2):232-240(in Chinese))
5 Bert CW,Malik M.Differential quadrature method in computational mechanics:A review.Applied Mechanics Reviews, 1996,49(1):1-28
6 Fung TC.Solving initial value problems by differential quadrature method—Part 1:first-order equations.International Journal for Numerical Methods in Engineering, 2001,50:1411-1427
7 Fung TC.Solving initial value problems by differential quadrature method—Part 2:second- and higher-order equations.International Journal for Numerical Methods in Engineering,2001,50:1429-1454
8 Shen J,Tang T.Spectral and High-order Methods with Applications.Beijing:Science Press,2006
9王通.曲线梁桥地震时程反应的微分求积分析方法.[硕士论文].南京:南京工业大学,2009(Wang Tong.Differential quadrature analysis method for seismic time history response of curved girder bridges.[Master Thesis].Nanjing:Nanjing University of Technology,2009(in Chinese))
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