能力谱方法的混沌动力学分析
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摘要
能力谱方法是一种简化的结构抗震非线性分析方法,ATC-40等文件采用它来预测结构的力和位移需求。然而,ATC-40能力谱方法对一些非弹性系统的求解出现不收敛现象。现利用混沌动力学理论对ATC-40能力谱方法的收敛问题进行分析。ATC-40能力谱方法等效线性化迭代求解非弹性体系的位移,形成一个离散非线性映射或差分方程。对于这一离散动力系统,随着控制参数的变化,给出了动力系统迭代解的分岔图,展示了解的周期振荡、倍周期分岔、混沌等复杂动力学现象,阐明了能力谱方法迭代过程周期振荡解产生的原因。计算了动力系统的Lyapunov指数,从而判定系统解的运动状态和稳定性。最后指出了实现能力谱法等迭代算法收敛振荡控制的途径———混沌控制。
The capacity spectrum method is a simplified nonlinear analysis procedure for seismic design of structures,the demands of forces and deformations of inelastic system can be predicated,and the methool has been incorporated into the ATC-40 document.However,the iterative procedure of the capacity spectrum method in the ATC-40 procedure A do not converge for some inelastic systems.The convergence analysis of the capacity spectrum method in ATC-40 is studied by using the theory of chaotic dynamics.The deformation of inelastic system is solved iteratively by using the equivalent linearization for ATC-40 capacity spectrum method,which forms a discrete nonlinear map or difference equation.For discrete dynamical system,the bifurcation plots of iterative solution of the system are presented with the change of controlling parameters.And the bifurcation plots show that there are the complicated dynamics phenomena for the solutions such as the periodic oscillation,period-doubling bifurcation,chaos and so on.The essential factors of periodic solution from iterative calculation of the capacity spectrum method are studied.The Lyapunov exponent of the dynamic system is computed,which is used to identify the motion state and stability of solutions.Finally,the chaos controlling on the convergent oscillation of iterative algorithms including capacity spectrum method is suggested.
The capacity spectrum method is a simplified nonlinear analysis procedure for seismic design of structures,the demands of forces and deformations of inelastic system can be predicated,and the methool has been incorporated into the ATC-40 document.However,the iterative procedure of the capacity spectrum method in the ATC-40 procedure A do not converge for some inelastic systems.The convergence analysis of the capacity spectrum method in ATC-40 is studied by using the theory of chaotic dynamics.The deformation of inelastic system is solved iteratively by using the equivalent linearization for ATC-40 capacity spectrum method,which forms a discrete nonlinear map or difference equation.For discrete dynamical system,the bifurcation plots of iterative solution of the system are presented with the change of controlling parameters.And the bifurcation plots show that there are the complicated dynamics phenomena for the solutions such as the periodic oscillation,period-doubling bifurcation,chaos and so on.The essential factors of periodic solution from iterative calculation of the capacity spectrum method are studied.The Lyapunov exponent of the dynamic system is computed,which is used to identify the motion state and stability of solutions.Finally,the chaos controlling on the convergent oscillation of iterative algorithms including capacity spectrum method is suggested.
引文
[1]ATC.Seismic evaluation and retrofit of concrete buildings[R].California:Report ATC-40,Applied Technology Council,Redwood City,1996.
[2]Chopra A K,Goel R K.Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures:SDF systems[R].Califor-nia:Report No.PEER-1999/02,Pacific Earthquake Engineering Research Center,University of California,Berkeley,1999.
[3]Chopra A K,Goel R K.Evaluation of NSP to estimate seismic deformation:SDF system[J].ASCE Journal of Structural Engineering,2000,126(4):482-490.
[4]Fajfar P.Capacity spectrum method based on inelastic demand spectrum[J].Earthquake Engineering and Structural Dynamics,1999,28(9):979-993.
[5]Chopra A K,Goel R K.Capacity-demand-diagram methods based on inelastic design spectrum[C]//Proceedings of 12th World Congress ofEarthquake Engineering,2000,1612.
[6]Chopra AK,Goel R K.Amodal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings[J].Earthquake Engi-neering and Structural Dynamics,2004,33(9):903-927.
[7]欧进萍,何政,龙旭,等.振戎中学食堂楼耗能减震分析与设计(Ⅱ)——能力谱法与地震损伤性能控制设计[J].地震工程与工程振动,2001,21(1):115-122.
[8]李宏男,何浩祥.利用能力谱法对结构地震损伤评估简化方法[J].大连理工大学学报,2004,44(2):267-270.
[9]Lin Y Y,Chuang TF,Chang KC.Amathematical basis for the convergence of the capacity spectrum method[J].Earthquake Engineering andStructural Dynamics,2004,33(9):1059-1066.
[10]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2004.
[11]Robinson R C.An Introduction to Dynamical System:Continuous and Discrete[M].Pearson Education:Inc,2004.
[12]Hurley M,Martin C.Newton’s algorithm and chaotic dynamic system[J].SIAMJournal of Mathematical Analysis,1984,15(2):238-252.
[13]杨迪雄,许林,李刚.结构可靠度FORM方法的混沌动力学分析[J].力学学报,2005,37(6):799-804.
[14]Yang D X,Li G,Cheng G D.Convergence analysis of first order reliability method using chaos theory[J].Computers and Structures,2006,84(8/9):563-571.
[15]张化光,王智良,黄伟.混沌系统的控制理论[M].沈阳:东北大学出版社,2003.
[2]Chopra A K,Goel R K.Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures:SDF systems[R].Califor-nia:Report No.PEER-1999/02,Pacific Earthquake Engineering Research Center,University of California,Berkeley,1999.
[3]Chopra A K,Goel R K.Evaluation of NSP to estimate seismic deformation:SDF system[J].ASCE Journal of Structural Engineering,2000,126(4):482-490.
[4]Fajfar P.Capacity spectrum method based on inelastic demand spectrum[J].Earthquake Engineering and Structural Dynamics,1999,28(9):979-993.
[5]Chopra A K,Goel R K.Capacity-demand-diagram methods based on inelastic design spectrum[C]//Proceedings of 12th World Congress ofEarthquake Engineering,2000,1612.
[6]Chopra AK,Goel R K.Amodal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings[J].Earthquake Engi-neering and Structural Dynamics,2004,33(9):903-927.
[7]欧进萍,何政,龙旭,等.振戎中学食堂楼耗能减震分析与设计(Ⅱ)——能力谱法与地震损伤性能控制设计[J].地震工程与工程振动,2001,21(1):115-122.
[8]李宏男,何浩祥.利用能力谱法对结构地震损伤评估简化方法[J].大连理工大学学报,2004,44(2):267-270.
[9]Lin Y Y,Chuang TF,Chang KC.Amathematical basis for the convergence of the capacity spectrum method[J].Earthquake Engineering andStructural Dynamics,2004,33(9):1059-1066.
[10]刘秉正,彭建华.非线性动力学[M].北京:高等教育出版社,2004.
[11]Robinson R C.An Introduction to Dynamical System:Continuous and Discrete[M].Pearson Education:Inc,2004.
[12]Hurley M,Martin C.Newton’s algorithm and chaotic dynamic system[J].SIAMJournal of Mathematical Analysis,1984,15(2):238-252.
[13]杨迪雄,许林,李刚.结构可靠度FORM方法的混沌动力学分析[J].力学学报,2005,37(6):799-804.
[14]Yang D X,Li G,Cheng G D.Convergence analysis of first order reliability method using chaos theory[J].Computers and Structures,2006,84(8/9):563-571.
[15]张化光,王智良,黄伟.混沌系统的控制理论[M].沈阳:东北大学出版社,2003.
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