基于DQ原理的结构弹塑性地震反应分析
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摘要
微分求积法(DQ法)是一种高精度,高计算效率的微分方程数值求解方法。该文采用微分求积法进行了结构弹塑性地震反应分析。针对结构弹塑性地震反应分析的特点,对于采样周期内地震加速度记录的增量分布做了线性假定,采用了一种新型的不等距的时间网格并在新的网格上实施DQ算法。对于拐点的处理则提出了一种简单的迂回处理方法,可以方便地利用现有研究成果来完成DQ方案中拐点的处理。数值算例的结果表明:采用微分求积法求解结构弹塑性地震反应可以达到较高的计算精度并且可以从两个方面提高计算效率。
Differential quadrature method(DQM) is a numerical approach to solve differential equations with high accuracy and high computation efficiency.And it is applied to finish structural elasto-plastic seismic response analysis.According to the properties of a structural elasto-plastic seismic response analysis,the increment distribution of a seismic acceleration in a sampling period is assumed as a liner distribution.DQM is applied by using a new non-equal time grid.A simple and non-direct procedure is used for the treatment of critical points;it can utilize the normal methods for treating critical points conveniently.Numerical experiments are carried out.The results show that it can gather high accuracy by using DQM to analyze the structural elasto-plastic seismic responses.And the efficiency can be improved in two ways.
Differential quadrature method(DQM) is a numerical approach to solve differential equations with high accuracy and high computation efficiency.And it is applied to finish structural elasto-plastic seismic response analysis.According to the properties of a structural elasto-plastic seismic response analysis,the increment distribution of a seismic acceleration in a sampling period is assumed as a liner distribution.DQM is applied by using a new non-equal time grid.A simple and non-direct procedure is used for the treatment of critical points;it can utilize the normal methods for treating critical points conveniently.Numerical experiments are carried out.The results show that it can gather high accuracy by using DQM to analyze the structural elasto-plastic seismic responses.And the efficiency can be improved in two ways.
引文
[1]胡聿贤.地震工程学[M]北京：地震出版社，2006Hu Yuxian. Earthquake engineering[M]. Beijing:Earthquake Press of China,2006.(in Chinese)
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[3]李鸿晶,王通.结构地震反应分析的逐步微分积分方法[J].力学学报,2011,43(2):430―435.Li Hongjing,Wang Tong.A time-stepping method of seismic response analysis for structures using differential quadrature rule[J].Chinese Journal of Theoretical and Applied Mechnics,2011,43(2):430―435.(in Chinese)
[4]李鸿晶,廖旭,王通.结构地震反应DQ解法的两种数值格式[J].应用基础与工程科学学报,2011,19(5):758―766.Li Hongjing,Liao Xu,Wang Tong.Two numerical schemes of differential quadrature analysis procedure for response of the earthquake-resistant structures[J].Journal of Basic Science and Engineering,2011,19(5):758―766.(in Chinese)
[5]Bellman R E,Casti J.Differential quadrature and long-term integration[J].Journal of Mathematical Analysis and Applications,1971,34:235―238.
[6]Bert C W,Malik M.Differential quadrature method in computational mechanics:A review[J].Applied Mechanics Reviews,1996,49(1):1―28.
[7]Nie G J,Zhong Z.Elasto-plastic analysis of beams by differential quadrature method[J].Journal of Engineering Mechanics,2005,22(1):59―62.
[8]王鑫伟.微分求积法在结构力学中的应用[J].力学进展,1995,25(2):232―240.Wang Xinwei.Differential quadrature in the analysis of structural components[J].Advances in Mechanics,1995,25(2):232―240.(in Chinese)
[9]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,2005,22(1):59―62.Nie Guojun,Zhong Zheng.Elasto-plastic analysis of beams by differential quadrature method[J].Engineering Mechanics,2005,22(1):59―62.
[10]Rose M A De,Franciosi C.Exact and approximate dynamic analysis of circular arches using DQM[J].International Journal of Solids and Structures,2000,37:1103―1117.
[11]Fung T C.Solving initial value problems by differential quadrature method—Part1:First-order equations[J].International Journal for Numerical Methods in Engineering,2001,50:1411―1427.
[12]Fung T C.Solving initial value problems by differential quadrature method—Part2:Second-and higher-order equations[J].International Journal for Numerical Methods in Engineering,2001,50:1429―1454.
[13]朱镜清.非线性动力分析的拐点处理方法[J].地震工程与工程振动,1982,1(3):15―24.Zhu Jingqing.An exact method for dealing with the critical points in the nonlinear dynamic analysis of structures[J].Earthquake Engineering and Engineering Vibration,1982,1(3):15―24.(in Chinese)
[14]徐赵东,郭迎庆.Matlab语言在建筑抗震工程中的应用[M].北京:科学出版社,2004:194―208.Xu Zhaodong,Guo Yingqing.Application of Matlab program in structures seismic resistant engineering[M].Beijing:Science Press,2004:194―208.(in Chinese)
[2]Newmark N M.A method of computation for structural dynamics[J].Journal of the Engineering Mechanics Division,ASCE,1959,85:67―94.
[3]李鸿晶,王通.结构地震反应分析的逐步微分积分方法[J].力学学报,2011,43(2):430―435.Li Hongjing,Wang Tong.A time-stepping method of seismic response analysis for structures using differential quadrature rule[J].Chinese Journal of Theoretical and Applied Mechnics,2011,43(2):430―435.(in Chinese)
[4]李鸿晶,廖旭,王通.结构地震反应DQ解法的两种数值格式[J].应用基础与工程科学学报,2011,19(5):758―766.Li Hongjing,Liao Xu,Wang Tong.Two numerical schemes of differential quadrature analysis procedure for response of the earthquake-resistant structures[J].Journal of Basic Science and Engineering,2011,19(5):758―766.(in Chinese)
[5]Bellman R E,Casti J.Differential quadrature and long-term integration[J].Journal of Mathematical Analysis and Applications,1971,34:235―238.
[6]Bert C W,Malik M.Differential quadrature method in computational mechanics:A review[J].Applied Mechanics Reviews,1996,49(1):1―28.
[7]Nie G J,Zhong Z.Elasto-plastic analysis of beams by differential quadrature method[J].Journal of Engineering Mechanics,2005,22(1):59―62.
[8]王鑫伟.微分求积法在结构力学中的应用[J].力学进展,1995,25(2):232―240.Wang Xinwei.Differential quadrature in the analysis of structural components[J].Advances in Mechanics,1995,25(2):232―240.(in Chinese)
[9]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,2005,22(1):59―62.Nie Guojun,Zhong Zheng.Elasto-plastic analysis of beams by differential quadrature method[J].Engineering Mechanics,2005,22(1):59―62.
[10]Rose M A De,Franciosi C.Exact and approximate dynamic analysis of circular arches using DQM[J].International Journal of Solids and Structures,2000,37:1103―1117.
[11]Fung T C.Solving initial value problems by differential quadrature method—Part1:First-order equations[J].International Journal for Numerical Methods in Engineering,2001,50:1411―1427.
[12]Fung T C.Solving initial value problems by differential quadrature method—Part2:Second-and higher-order equations[J].International Journal for Numerical Methods in Engineering,2001,50:1429―1454.
[13]朱镜清.非线性动力分析的拐点处理方法[J].地震工程与工程振动,1982,1(3):15―24.Zhu Jingqing.An exact method for dealing with the critical points in the nonlinear dynamic analysis of structures[J].Earthquake Engineering and Engineering Vibration,1982,1(3):15―24.(in Chinese)
[14]徐赵东,郭迎庆.Matlab语言在建筑抗震工程中的应用[M].北京:科学出版社,2004:194―208.Xu Zhaodong,Guo Yingqing.Application of Matlab program in structures seismic resistant engineering[M].Beijing:Science Press,2004:194―208.(in Chinese)
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