非线性随机结构动力可靠度的密度演化方法
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摘要
建议了一类新的非线性随机结构动力可靠度分析方法。基于非线性随机结构反应分析的概率密度演化方法,根据首次超越破坏准则对概率密度演化方程施加相应的边界条件,求解带有初、边值条件的概率密度演化方程,可以给出非线性随机结构的动力可靠度。研究了数值计算技术,建议了具有自适应功能的TVD差分格式。以具有双线型恢复力性质的8层框架结构为例进行了地震作用下的动力可靠度分析,与随机模拟结果的比较表明,所建议的方法具有较高的精度和效率。
An original approach for dynamic reliability assessment of nonlinear stochastic structures is proposed.In the past few years,a new method,named the probability density evolution method,has been developed,showing versatile capability in engineering stochastic mechanics such as the stochastic response analysis of either linear or nonlinear structures in either static or dynamic occasions.In the method,the probability density evolution equation,a first order quasi-linear partial differential equation in terms of the joint probability density,is deduced and uncoupled to a one-dimensional partial differential equation,which is easy to be numerically solved combining the deterministic dynamic response analysis of structures,such as the precise integration method or Newmark-Beta time integration method,and the finite difference method.Therefore,the instantaneous probability density function,rather than the second order statistical characteristics such as the mean,the covariance function and the power spectrum density and so on which are focused on by traditional stochastic finite element methods,of the response quantities of interest can be obtained.In the present paper,the probability density evolution method is applied to assess the dynamic reliability of stochastic structures.To achieve the purpose,an absorbing boundary condition corresponding to the failure criterion of the first passage problem is imposed on the probability density evolution equation.Solving the initial-boundary-value partial differential equation problem with a numerical algorithm will give the "remaining" probability density function of the response.The dynamic reliability can then be assessed through integrating the "remaining" probability density function over the safe domain.The numerical algorithm is studied in detail where an adaptive TVD difference scheme is presented.In contrast to the widely used level-crossing process based reliability assessment approach,in the proposed method the mean out-crossing rate,which is usually obtained through the Rice formula,is not needed,nor the properties of the level-crossing process such as the Poisson and Markovian assumption.Therefore the proposed method is expected to have a high accuracy.An 8-story frame structure with bilinear hysteretic restoring force,which is subjected to seismic excitation,is investigated.The results of dynamic reliability assessment are compared with those evaluated by the Monte Carlo simulation.The investigation shows that the proposed approach is of high accuracy and efficiency.
An original approach for dynamic reliability assessment of nonlinear stochastic structures is proposed.In the past few years,a new method,named the probability density evolution method,has been developed,showing versatile capability in engineering stochastic mechanics such as the stochastic response analysis of either linear or nonlinear structures in either static or dynamic occasions.In the method,the probability density evolution equation,a first order quasi-linear partial differential equation in terms of the joint probability density,is deduced and uncoupled to a one-dimensional partial differential equation,which is easy to be numerically solved combining the deterministic dynamic response analysis of structures,such as the precise integration method or Newmark-Beta time integration method,and the finite difference method.Therefore,the instantaneous probability density function,rather than the second order statistical characteristics such as the mean,the covariance function and the power spectrum density and so on which are focused on by traditional stochastic finite element methods,of the response quantities of interest can be obtained.In the present paper,the probability density evolution method is applied to assess the dynamic reliability of stochastic structures.To achieve the purpose,an absorbing boundary condition corresponding to the failure criterion of the first passage problem is imposed on the probability density evolution equation.Solving the initial-boundary-value partial differential equation problem with a numerical algorithm will give the "remaining" probability density function of the response.The dynamic reliability can then be assessed through integrating the "remaining" probability density function over the safe domain.The numerical algorithm is studied in detail where an adaptive TVD difference scheme is presented.In contrast to the widely used level-crossing process based reliability assessment approach,in the proposed method the mean out-crossing rate,which is usually obtained through the Rice formula,is not needed,nor the properties of the level-crossing process such as the Poisson and Markovian assumption.Therefore the proposed method is expected to have a high accuracy.An 8-story frame structure with bilinear hysteretic restoring force,which is subjected to seismic excitation,is investigated.The results of dynamic reliability assessment are compared with those evaluated by the Monte Carlo simulation.The investigation shows that the proposed approach is of high accuracy and efficiency.
引文
1 李杰.随机结构系统--分析与建模.北京:科学出版社,1996 (Li Jie. Stochastic Structural Systems: Analysis and Modeling. Beijing: Science Press, 1996 (in Chinese))
2 Schueller GI ed. A state-of-the-art report on computational stochastic mechanics. Probability Engineering Mechanics,1997, 12(4) : 198～321
3 Spencer BF Jr, Elishakoff I. Reliability of uncertain linear and nonlinear systems. Journal of Engineering Mechanics,1988, 114(1) , 135～149
4 Brenner CE, Bucher C. A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading. Probabilistic Engineering Mechanics, 1995, 10:265～273
5 Pradlwarter HJ, Schueller GI. Reliability of MDOF--systems with hysterotic devices. Engineering Structures,1998, 20(8) : 685～691
6 张义民,刘巧伶,闻邦椿.非线性随机系统的独立失效模式可靠性灵敏度.力学学报, 2003, 35(1) : 117～120 (Zhang Yimin,Liu Qiaoling, Wen Bangchun. Sensitivity of reliability in nonlinear random systems with independent failure modes.Acta Mechanica Sinica, 2003, 35(1) : 117～120 (in Chinese))
7 Li J, Chen JB. Probability density evolution method for dynamic response analysis of stochastic structures. In:Zhu WQ, Cai GQ, Zhang RC, eds. Advances in Stochastic Structural Dynamics. Proceedings of the 5th International Conference on Stochastic Structural Dynamics-SSD03, May 26～28, Hangzhou, China. Boca Raton: CRC Press, 2003． 309～316
8 李杰.陈建兵.随机结构非线性动力反应的概率密度演化分析.力学学报, 2003, 35(6) : 716～722 (Li Jie, Chen Jianbing.Probability density evolution analysis of nonlinear dynamic ??response of stochastic structures. Acta Mechanica Sinica,2003, 35(6) : 716～722 (in Chinese))
9 Chen JB, Li J. Dynamic reliability analysis of stochastic structures. In: Wu ZS, Abe M, eds. Proceedings of First International Conference on Structural Health Monitoring and Intelligent Infrastructure, Nov.13～15, Tokyo,Japan. Lisse/ Abingdon / Exton / Tokyo: A. A. Balkema Publishers, 2003． 771～776
10 Syski R. Stochastic differential equations. In: Saaty TL, ed.Modern Nonlinear Equations. McGraw-Hill: New Yok,1967
11 Clough RW, Penzien J. Dynamics of Structures. 2nd edition. New York: McGraw-Hill, 1993
12 Newmark NM. A method of computation for structural dynamics. ASCE Transactions, 1962, 127: 1406～1435
13 Anderson JD Jr. Computational Fluid Dynamics. New York: McGraw-Hill, 1995
14 Shen MY, Zhang ZB, Niu XL. Some advances in study of high order accuracy and high resolution finite difference schemes. In: Dubois F, Wu HM eds. New Advance. in Computational Fluid Dynamics. Beiling: Higher Education Press, 2001
15 朱自强,吴子牛等.应用计算流体力学.北京:北京航空航天大学出版社, 1998 (Zhu Ziqiang, Wu Ziniu, et al. Applied Computational Fluid Dynamics. Beijing: Beijing University of Aeronautics and Astronautics Press, 1998 (in Chinese))
2 Schueller GI ed. A state-of-the-art report on computational stochastic mechanics. Probability Engineering Mechanics,1997, 12(4) : 198～321
3 Spencer BF Jr, Elishakoff I. Reliability of uncertain linear and nonlinear systems. Journal of Engineering Mechanics,1988, 114(1) , 135～149
4 Brenner CE, Bucher C. A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading. Probabilistic Engineering Mechanics, 1995, 10:265～273
5 Pradlwarter HJ, Schueller GI. Reliability of MDOF--systems with hysterotic devices. Engineering Structures,1998, 20(8) : 685～691
6 张义民,刘巧伶,闻邦椿.非线性随机系统的独立失效模式可靠性灵敏度.力学学报, 2003, 35(1) : 117～120 (Zhang Yimin,Liu Qiaoling, Wen Bangchun. Sensitivity of reliability in nonlinear random systems with independent failure modes.Acta Mechanica Sinica, 2003, 35(1) : 117～120 (in Chinese))
7 Li J, Chen JB. Probability density evolution method for dynamic response analysis of stochastic structures. In:Zhu WQ, Cai GQ, Zhang RC, eds. Advances in Stochastic Structural Dynamics. Proceedings of the 5th International Conference on Stochastic Structural Dynamics-SSD03, May 26～28, Hangzhou, China. Boca Raton: CRC Press, 2003． 309～316
8 李杰.陈建兵.随机结构非线性动力反应的概率密度演化分析.力学学报, 2003, 35(6) : 716～722 (Li Jie, Chen Jianbing.Probability density evolution analysis of nonlinear dynamic ??response of stochastic structures. Acta Mechanica Sinica,2003, 35(6) : 716～722 (in Chinese))
9 Chen JB, Li J. Dynamic reliability analysis of stochastic structures. In: Wu ZS, Abe M, eds. Proceedings of First International Conference on Structural Health Monitoring and Intelligent Infrastructure, Nov.13～15, Tokyo,Japan. Lisse/ Abingdon / Exton / Tokyo: A. A. Balkema Publishers, 2003． 771～776
10 Syski R. Stochastic differential equations. In: Saaty TL, ed.Modern Nonlinear Equations. McGraw-Hill: New Yok,1967
11 Clough RW, Penzien J. Dynamics of Structures. 2nd edition. New York: McGraw-Hill, 1993
12 Newmark NM. A method of computation for structural dynamics. ASCE Transactions, 1962, 127: 1406～1435
13 Anderson JD Jr. Computational Fluid Dynamics. New York: McGraw-Hill, 1995
14 Shen MY, Zhang ZB, Niu XL. Some advances in study of high order accuracy and high resolution finite difference schemes. In: Dubois F, Wu HM eds. New Advance. in Computational Fluid Dynamics. Beiling: Higher Education Press, 2001
15 朱自强,吴子牛等.应用计算流体力学.北京:北京航空航天大学出版社, 1998 (Zhu Ziqiang, Wu Ziniu, et al. Applied Computational Fluid Dynamics. Beijing: Beijing University of Aeronautics and Astronautics Press, 1998 (in Chinese))
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