结构区间模糊随机有限元可靠度分析
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摘要
实际工程中存在着大量的不确定性,对工程结构进行可靠性分析,需要充分了解和认识分析过程中存在的各种不确定性,以建立合理的可靠度分析模型。概率模型是处理不确定性中应用最广泛的方法,但是概率模型的确定需要大量的样本。另外,概率模型只考虑了随机性这一种不确定性,没有考虑实际工程中大量存在的其它不确定性,如模糊性、未确知性等。因此需要根据实际情况建立更能满足工程需要、考虑各种不确定性的广义可靠度模型。此外,还需结合现有的有限元计算分析平台,实现基于有限元可靠度方法的结构广义可靠度分析,以解决实际复杂工程结构的安全分析与风险评定问题。
本文以基于MATLAB的面向对象有限元程序OpenFEM工具箱和大型商业有限元软件ANSYS为计算平台,综合考虑概率模型参数为区间数、概率模型参数为模糊数、基本变量为模糊随机变量三种复杂不确定性情况,分别针对极限状态的虑随机性和模糊随机性,统一运用蒙特卡洛模拟法(MCS)和一次可靠度方法(FORM),将现有只考虑随机性的随机有限元可靠度方法拓广到考虑模糊性和未确知性的区间模糊随机有限元可靠度方法。论文的主要工作如下:
(1)研究了概率模型分布参数为区间数的结构区间有限元可靠度分析方法,改进了现有的区间MCS方法,在不考虑状态变量模糊性的区间MCS基础上,提出了考虑状态变量模糊性的区间MCS计算方法;将区间分析与FORM相结合,提出了区间FORM的组合法。分别用所提出的区间MCS和区间FORM方法,计算了不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构和刚架结构的失效概率区间。
(2)研究了概率模型分布参数为模糊数的结构区间有限元可靠度分析方法,分别用所提出的区间MCS和区间FORM方法,计算了不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构失效概率的可能性分布。
(3)建立了基本变量为模糊随机变量的广义可靠性分析模型,提出了模糊随机变量的等效随机变换法和模糊FORM方法。分别用MCS和模糊FORM方法,对不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构模糊随机可靠度进行了分析。
There exists much uncertainty in real-life engineering structures. In order to build rational reliability analysis models, one should well understand all kinds of uncertainties. Probability models are the most widely used ones, but it needs large samples to determine their distributions. Furthermore, probabilistic models can only considers randomness, without consideration of other uncertainties, such as fuzziness, incompleteness, etc. Therefore, some generalized reliability models considering all kinds of uncertainties should be investigated to satisfy practical needs in engineering. In addition, there is a need to realize generalized reliability analysis of engineering structures based on existing finite element analysis platforms, to solve the problems of safety analysis and risk evaluation of complex engineering structures in real life.
In this thesis, the object-oriented finite element analysis toolbox OpenFEM for MATLAB as well as the large-scale commercial FEM software ANSYS are chosen as computation platforms. Three cases of complex uncertainty for basic variables in structural reliability analysis are considered, namely, interval variables of distribution parameters of probability models, fuzzy variables of distribution parameters of probability models, and fuzzy random variables of basic variables. Two computational reliability methods, i.e., Monte Carlo Simulation (MCS), and First Order Reliability Method (FORM), are implemented and improved to solve the generalized reliability problems under the above uncertainty cases, corresponding to only-randomness and fuzzy-randomness in limit states, respectively. The existing finite element reliability methods (FERM) only considering randomness are then extended to interval fuzzy random FERMs. The main contents are as follows:
(1) The interval finite element reliability method in the case of interval variables of distribution parameters of probability models is investigated. The existing interval MCS method without considering the fuzziness is improved to consider the fuzziness in limit states of structures. Combining interval analysis and FORM, a hybrid interval FORM is proposed. The intervals of failure probabilities of truss and frame structures corresponding to only-randomness and fuzzy-randomness are analyzed.
(2) The interval finite element reliability method in the case of fuzzy variables of distribution parameters of probability models is studied. Use of the proposed interval MCS and interval FORM is made to calculate the possibility distributions of failure probabilities of truss structures corresponding to only-randomness and fuzzy-randomness respectively.
(3) A generalized reliability analysis model with fuzzy random basic variables is built up, a new fuzzy FORM with equivalent random transformation of fuzzy random variables is put forward. Using MCS and the proposed fuzzy FORM, the fuzzy random reliabilities of truss structures corresponding to only-randomness and fuzzy-randomness are analyzed respectively.
本文以基于MATLAB的面向对象有限元程序OpenFEM工具箱和大型商业有限元软件ANSYS为计算平台,综合考虑概率模型参数为区间数、概率模型参数为模糊数、基本变量为模糊随机变量三种复杂不确定性情况,分别针对极限状态的虑随机性和模糊随机性,统一运用蒙特卡洛模拟法(MCS)和一次可靠度方法(FORM),将现有只考虑随机性的随机有限元可靠度方法拓广到考虑模糊性和未确知性的区间模糊随机有限元可靠度方法。论文的主要工作如下:
(1)研究了概率模型分布参数为区间数的结构区间有限元可靠度分析方法,改进了现有的区间MCS方法,在不考虑状态变量模糊性的区间MCS基础上,提出了考虑状态变量模糊性的区间MCS计算方法;将区间分析与FORM相结合,提出了区间FORM的组合法。分别用所提出的区间MCS和区间FORM方法,计算了不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构和刚架结构的失效概率区间。
(2)研究了概率模型分布参数为模糊数的结构区间有限元可靠度分析方法,分别用所提出的区间MCS和区间FORM方法,计算了不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构失效概率的可能性分布。
(3)建立了基本变量为模糊随机变量的广义可靠性分析模型,提出了模糊随机变量的等效随机变换法和模糊FORM方法。分别用MCS和模糊FORM方法,对不考虑状态变量模糊性和考虑状态变量模糊性的桁架结构模糊随机可靠度进行了分析。
There exists much uncertainty in real-life engineering structures. In order to build rational reliability analysis models, one should well understand all kinds of uncertainties. Probability models are the most widely used ones, but it needs large samples to determine their distributions. Furthermore, probabilistic models can only considers randomness, without consideration of other uncertainties, such as fuzziness, incompleteness, etc. Therefore, some generalized reliability models considering all kinds of uncertainties should be investigated to satisfy practical needs in engineering. In addition, there is a need to realize generalized reliability analysis of engineering structures based on existing finite element analysis platforms, to solve the problems of safety analysis and risk evaluation of complex engineering structures in real life.
In this thesis, the object-oriented finite element analysis toolbox OpenFEM for MATLAB as well as the large-scale commercial FEM software ANSYS are chosen as computation platforms. Three cases of complex uncertainty for basic variables in structural reliability analysis are considered, namely, interval variables of distribution parameters of probability models, fuzzy variables of distribution parameters of probability models, and fuzzy random variables of basic variables. Two computational reliability methods, i.e., Monte Carlo Simulation (MCS), and First Order Reliability Method (FORM), are implemented and improved to solve the generalized reliability problems under the above uncertainty cases, corresponding to only-randomness and fuzzy-randomness in limit states, respectively. The existing finite element reliability methods (FERM) only considering randomness are then extended to interval fuzzy random FERMs. The main contents are as follows:
(1) The interval finite element reliability method in the case of interval variables of distribution parameters of probability models is investigated. The existing interval MCS method without considering the fuzziness is improved to consider the fuzziness in limit states of structures. Combining interval analysis and FORM, a hybrid interval FORM is proposed. The intervals of failure probabilities of truss and frame structures corresponding to only-randomness and fuzzy-randomness are analyzed.
(2) The interval finite element reliability method in the case of fuzzy variables of distribution parameters of probability models is studied. Use of the proposed interval MCS and interval FORM is made to calculate the possibility distributions of failure probabilities of truss structures corresponding to only-randomness and fuzzy-randomness respectively.
(3) A generalized reliability analysis model with fuzzy random basic variables is built up, a new fuzzy FORM with equivalent random transformation of fuzzy random variables is put forward. Using MCS and the proposed fuzzy FORM, the fuzzy random reliabilities of truss structures corresponding to only-randomness and fuzzy-randomness are analyzed respectively.
引文
[1]李廷杰,高和.模糊可靠性[J].模糊系统与数学,1988,2(2):11-18.
[2]黄洪钟.对常规可靠性理论的批判性评述——兼论模糊可靠性理论的产生、发展及应用前景[J].机械设计,1994(3):1-5,47.
[3] Cai K Y, Wen C Y, Zhang M L. Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context [J]. Fuzzy Sets and Systems, 1991(42):145-172.
[4] Cai K Y, Wen C Y, Zhang, M L. Fuzzy states as a basis for a theory of fuzzy reliability [J]. Microelectronics and Reliability, 1993(33):2253-2263.
[5]黄洪钟.关于模糊可靠性若干基本问题的讨论——将清晰工作时间拓广到模糊工作时间.机械工程学报[J]. 1994,9(4):1-8.
[6] Shiraishi N, Furuta H. Reliability analysis based on fuzzy probability [J]. Journal of Engineering Mechanics, 1983,109(6):32-38.
[7] Onisawa T. Fuzzy theory in reliability analysis [J]. Fuzzy Sets and Systems, 1989(29):250-257.
[8]王亚军,张我华,金伟良.一次逼近随机有限元对堤坝模糊失效概率的分析[J].浙江大学学报(工学版),2007,41(1):52-56.
[9]李胡生,叶黔元.陡峭岩石边坡随机——模糊可靠度算法[J].岩土工程学报,2006,28(8):1019-1022.
[10]董玉革,陈心昭,赵显德,等.基于模糊事件概率理论的模糊可靠性分析通用方法[J].计算力学学报,2005,27(3):281-286.
[11] Jiang Q, Chen C H. A numerical algorithm of fuzzy reliability [J]. Reliability Engineering and System Safety, 2003, 80: 229-307.
[12] Adduri P R, Penmetsa R C. Confidence bounds on component reliability in the presence of mixed uncertain variables [J]. International Journal of Mechanical Science, 2008:481-489.
[13]郭书详,吕震宙.概率模型含模糊分布参数时的模糊失效概率计算方法[J].机械强度,2003,25(5):527-529.
[14]吕震宙,岳珠峰,模糊随机可靠性分析的统一模型[J].力学学报,2004,36(5): 533-539.
[15] Moore R E. Interval Analysis [M]. Englewood Cliffs, NJ: Prentice Hall, 1966.
[16] Elishakoff I. Discussion on a non-probabilistic concept of reliability [J]. Structural Safety, 1995, 17(3): 195-199.
[17] Wang X J, Qiu Z P, Elishakoff I. Non-probabilistic set-theoretic model for structural safety measure [J]. Acta Mechanica Sinaca, 2008, 198(1-2): 51-64.
[18]郭书详,吕震宙,冯元生.基于区间分析的结构非概率可靠性模型[J].计算力学学报,2001,18(1):56-60.
[19]郭书详,吕震宙.基于非概率模型的结构稳健可靠性优化设计[J].航空学报,2001,22(5):451-453.
[20] Penmetsa R C, Grandhi RV. Efficient estimation of structural reliability for problems with uncertain intervals [J]. Computers & Structures, 2002; 80: 1103-12.
[21] Qiu Z, Yang D, Elishakoff I. Probabilistic interval reliability of structural systems [J]. Internatinal Journal of Solids and Structures, 2008; 45: 2850-2860.
[22] Zhang H, Mullen R.L., and Muhanna R.L. Interval Monte Carlo methods for structural reliability [J]. Structural Safety, 2010(32): 183-190.
[23]周道成,段忠东,欧进萍.具有隐式功能函数的结构可靠指标计算的改进矩法[J].东南大学学报(自然科学版),2005,35(Ⅰ):139-143.
[24]刘天云.随机有限元可靠度分析方法[D].大连:大连理工大学,1998.
[25] Wong F S. Slope reliability and response surface method [J]. Journal of Geotechnical Engineering, 1984, 111: 32-53.
[26] Faravelli L. A response surface approach for reliability analysis [J]. Journal of Engineering Mechanics, 1989, 115(12): 2763-2781.
[27] Schueler G I, Bucher C G, Bourgund U, et al. On efficient computational schemes to calculate structural failure probabilities [J]. Probabilistic Engineering Mechanics, 1989, 4(1): 10-18.
[28] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems [J]. Structural Safety, 1990, 7(1): 115-127.
[29] Kim S H, Na S W. Response surface method using vector projected sampling points [J]. Structural Safety, 1997,19(1):3-19.
[30] Das P K, Zheng Y. Cumulative formation of response surface and its use in reliability analysis [J]. Probabilistic Engineering Mechanics, 2000, 15(4): 309-315.
[31] Zheng Y, Das P K. Improved response surface method and its application to stiffened plate reliability analysis [J]. Engineering Structures, 2000, 22(5): 544-551.
[32] Rao S S, Berke L. Analysis of uncertain structural systems using interval analysis[C]. AIAA Journal, 1997,35(4):727-35.
[33] Qiu Z, Elishakoff I. Antioptimization of structures with large uncertain-but-non-rand parameters via interval analysis [J]. Computer Methods in Applied Mechanics and Engineering, 1998, 152: 361-372.
[34] Pantelides C P, Granzerli S. Comparison of fuzzy set and convex model theories in structural design [J]. Mechanical Systems and Signal Processing, 2001, 15(3): 499-511.
[35] Chen S H, Lian H D, Yang, X. W. Interval static displacement analysis for structures with interval parameters [J]. International Journal for Numerical Methods in Engineering, 2002, 53: 393-407.
[36] McWilliam S. Anti-optimisation of uncertain structures using interval analysis [J]. Computers&Structures, 2000, 79: 421-430.
[37] Moller B, Graf W, Beer M. Fuzzy structural analysis using level-optimization [J]. Computational Mechanics, 2006,26(6), 547-565.
[38] Dessombz O, Thouverez F, Laine J, et al. Analysis of mechanical systems using interval computations applied to finite elements methods [J]. Journal of Sound and Vibration, 2001, 238(5), 949-968.
[39] Muhanna R L, Mullen R L. Development of interval based methods for fuzziness in continuum mechanics [C]. Proceedings of ISUMA-NAFIPS’95 The Third International Symposium on Uncertainty Modeling and Analysis and Annual Conference of the North American Fuzzy Information Processing Society, 1995, 95, 23-45.
[40] Mullen R L, Muhanna R L. Structural analysis with fuzzy-based load uncertainty[C]. Proceedings of 7th ASCE EMD/STD Joint Specialty Conf on Probabilistic Mechanics and structure Reliability, 1996, 310-313.
[41] Muhanna R L, Mullen R L. Formulation of fuzzy finite element methods for mechanics problems [J]. Computer Aided Civil and Infrastructure Engineering, 1999,14:107-117.
[42] Mullen R L, Muhanna R L. Bounds of structural response for all possible combinations [J]. Journal of Structural Engineering, 1999, 125(1), 98-106.
[43] Muhanna R L, Mullen R L. Uncertainty in mechanics problems-Interval-based approach [J]. Journal of Engineering Mechanics, 2001, 127(6), 557-566.
[44] Pownuk A. Efficient method of solution of large scale engineering problems withinterval parameters [C]. Proceedings of NSF workshop on reliable engineering and computing, Savannah, GA, 2004.
[45] Ferson S, Kreinovich V, Ginzburg L, et al. Constructing probability boxes and Dempster-Shafer structures. SAND2002-4015[R], Sandia National Laboratories, 2003.
[46] Zadeh L A. Fuzzy sets [J]. Information and Control, 1965, 8: 338-353.
[47]张明.结构可靠度分析[M].北京:科学出版社,2009:204-208.
[48]谢启芳,赵鸿铁,薛建阳.结构模糊随机可靠度的计算方法[J].西安建筑科技大学学报(自然科学版),2006,38(4):480-485.
[49]郭书详,吕震宙.概率模型含模糊分布参数时的模糊失效概率计算方法[J].机械强度,2003,25(5):527-529.
[50]尼早邱志平.概率模型含模糊分布参数的结构体系模糊可靠度分析方法[J].工程力学,2009,26(7):28-34.
[2]黄洪钟.对常规可靠性理论的批判性评述——兼论模糊可靠性理论的产生、发展及应用前景[J].机械设计,1994(3):1-5,47.
[3] Cai K Y, Wen C Y, Zhang M L. Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context [J]. Fuzzy Sets and Systems, 1991(42):145-172.
[4] Cai K Y, Wen C Y, Zhang, M L. Fuzzy states as a basis for a theory of fuzzy reliability [J]. Microelectronics and Reliability, 1993(33):2253-2263.
[5]黄洪钟.关于模糊可靠性若干基本问题的讨论——将清晰工作时间拓广到模糊工作时间.机械工程学报[J]. 1994,9(4):1-8.
[6] Shiraishi N, Furuta H. Reliability analysis based on fuzzy probability [J]. Journal of Engineering Mechanics, 1983,109(6):32-38.
[7] Onisawa T. Fuzzy theory in reliability analysis [J]. Fuzzy Sets and Systems, 1989(29):250-257.
[8]王亚军,张我华,金伟良.一次逼近随机有限元对堤坝模糊失效概率的分析[J].浙江大学学报(工学版),2007,41(1):52-56.
[9]李胡生,叶黔元.陡峭岩石边坡随机——模糊可靠度算法[J].岩土工程学报,2006,28(8):1019-1022.
[10]董玉革,陈心昭,赵显德,等.基于模糊事件概率理论的模糊可靠性分析通用方法[J].计算力学学报,2005,27(3):281-286.
[11] Jiang Q, Chen C H. A numerical algorithm of fuzzy reliability [J]. Reliability Engineering and System Safety, 2003, 80: 229-307.
[12] Adduri P R, Penmetsa R C. Confidence bounds on component reliability in the presence of mixed uncertain variables [J]. International Journal of Mechanical Science, 2008:481-489.
[13]郭书详,吕震宙.概率模型含模糊分布参数时的模糊失效概率计算方法[J].机械强度,2003,25(5):527-529.
[14]吕震宙,岳珠峰,模糊随机可靠性分析的统一模型[J].力学学报,2004,36(5): 533-539.
[15] Moore R E. Interval Analysis [M]. Englewood Cliffs, NJ: Prentice Hall, 1966.
[16] Elishakoff I. Discussion on a non-probabilistic concept of reliability [J]. Structural Safety, 1995, 17(3): 195-199.
[17] Wang X J, Qiu Z P, Elishakoff I. Non-probabilistic set-theoretic model for structural safety measure [J]. Acta Mechanica Sinaca, 2008, 198(1-2): 51-64.
[18]郭书详,吕震宙,冯元生.基于区间分析的结构非概率可靠性模型[J].计算力学学报,2001,18(1):56-60.
[19]郭书详,吕震宙.基于非概率模型的结构稳健可靠性优化设计[J].航空学报,2001,22(5):451-453.
[20] Penmetsa R C, Grandhi RV. Efficient estimation of structural reliability for problems with uncertain intervals [J]. Computers & Structures, 2002; 80: 1103-12.
[21] Qiu Z, Yang D, Elishakoff I. Probabilistic interval reliability of structural systems [J]. Internatinal Journal of Solids and Structures, 2008; 45: 2850-2860.
[22] Zhang H, Mullen R.L., and Muhanna R.L. Interval Monte Carlo methods for structural reliability [J]. Structural Safety, 2010(32): 183-190.
[23]周道成,段忠东,欧进萍.具有隐式功能函数的结构可靠指标计算的改进矩法[J].东南大学学报(自然科学版),2005,35(Ⅰ):139-143.
[24]刘天云.随机有限元可靠度分析方法[D].大连:大连理工大学,1998.
[25] Wong F S. Slope reliability and response surface method [J]. Journal of Geotechnical Engineering, 1984, 111: 32-53.
[26] Faravelli L. A response surface approach for reliability analysis [J]. Journal of Engineering Mechanics, 1989, 115(12): 2763-2781.
[27] Schueler G I, Bucher C G, Bourgund U, et al. On efficient computational schemes to calculate structural failure probabilities [J]. Probabilistic Engineering Mechanics, 1989, 4(1): 10-18.
[28] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems [J]. Structural Safety, 1990, 7(1): 115-127.
[29] Kim S H, Na S W. Response surface method using vector projected sampling points [J]. Structural Safety, 1997,19(1):3-19.
[30] Das P K, Zheng Y. Cumulative formation of response surface and its use in reliability analysis [J]. Probabilistic Engineering Mechanics, 2000, 15(4): 309-315.
[31] Zheng Y, Das P K. Improved response surface method and its application to stiffened plate reliability analysis [J]. Engineering Structures, 2000, 22(5): 544-551.
[32] Rao S S, Berke L. Analysis of uncertain structural systems using interval analysis[C]. AIAA Journal, 1997,35(4):727-35.
[33] Qiu Z, Elishakoff I. Antioptimization of structures with large uncertain-but-non-rand parameters via interval analysis [J]. Computer Methods in Applied Mechanics and Engineering, 1998, 152: 361-372.
[34] Pantelides C P, Granzerli S. Comparison of fuzzy set and convex model theories in structural design [J]. Mechanical Systems and Signal Processing, 2001, 15(3): 499-511.
[35] Chen S H, Lian H D, Yang, X. W. Interval static displacement analysis for structures with interval parameters [J]. International Journal for Numerical Methods in Engineering, 2002, 53: 393-407.
[36] McWilliam S. Anti-optimisation of uncertain structures using interval analysis [J]. Computers&Structures, 2000, 79: 421-430.
[37] Moller B, Graf W, Beer M. Fuzzy structural analysis using level-optimization [J]. Computational Mechanics, 2006,26(6), 547-565.
[38] Dessombz O, Thouverez F, Laine J, et al. Analysis of mechanical systems using interval computations applied to finite elements methods [J]. Journal of Sound and Vibration, 2001, 238(5), 949-968.
[39] Muhanna R L, Mullen R L. Development of interval based methods for fuzziness in continuum mechanics [C]. Proceedings of ISUMA-NAFIPS’95 The Third International Symposium on Uncertainty Modeling and Analysis and Annual Conference of the North American Fuzzy Information Processing Society, 1995, 95, 23-45.
[40] Mullen R L, Muhanna R L. Structural analysis with fuzzy-based load uncertainty[C]. Proceedings of 7th ASCE EMD/STD Joint Specialty Conf on Probabilistic Mechanics and structure Reliability, 1996, 310-313.
[41] Muhanna R L, Mullen R L. Formulation of fuzzy finite element methods for mechanics problems [J]. Computer Aided Civil and Infrastructure Engineering, 1999,14:107-117.
[42] Mullen R L, Muhanna R L. Bounds of structural response for all possible combinations [J]. Journal of Structural Engineering, 1999, 125(1), 98-106.
[43] Muhanna R L, Mullen R L. Uncertainty in mechanics problems-Interval-based approach [J]. Journal of Engineering Mechanics, 2001, 127(6), 557-566.
[44] Pownuk A. Efficient method of solution of large scale engineering problems withinterval parameters [C]. Proceedings of NSF workshop on reliable engineering and computing, Savannah, GA, 2004.
[45] Ferson S, Kreinovich V, Ginzburg L, et al. Constructing probability boxes and Dempster-Shafer structures. SAND2002-4015[R], Sandia National Laboratories, 2003.
[46] Zadeh L A. Fuzzy sets [J]. Information and Control, 1965, 8: 338-353.
[47]张明.结构可靠度分析[M].北京:科学出版社,2009:204-208.
[48]谢启芳,赵鸿铁,薛建阳.结构模糊随机可靠度的计算方法[J].西安建筑科技大学学报(自然科学版),2006,38(4):480-485.
[49]郭书详,吕震宙.概率模型含模糊分布参数时的模糊失效概率计算方法[J].机械强度,2003,25(5):527-529.
[50]尼早邱志平.概率模型含模糊分布参数的结构体系模糊可靠度分析方法[J].工程力学,2009,26(7):28-34.