结构可靠性若干专题研究
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摘要
结构可靠性的基本理论和方法逐渐趋于完善，但可靠性相关计算方法在实际工程中遇到了许多问题，如非正态随机变量处理不准确，相关可靠性分析方法计算量偏大，计算精度不高等，本文针对若干问题，结合近年来最新的数学理论方法，如模拟退火全局优化算法，统计学习理论，马尔可夫链模拟，凸集模型等，发展了一系列计算效率高，计算精度好的适于工程运用的可靠性分析方法。具体研究内容如下：
(1) 提出了基于等价正态三参数变换和模拟退火全局优化的失效概率计算方法，阐明了所提方法的理论基础和实施过程。由于采用了基于模拟退火的全局优化使得等价正态的三个参数更加准确，从而降低了计算失效概率的误差；与Monte-Carlo法相比，所提算法计算效率高且误差小，适于工程应用。
(2) 针对结构隐式极限状态函数的可靠性分析，提出了一种支持向量机分类迭代算法。该算法以支持向量机分类来替代隐式极限状态方程，通过构造合理的迭代格式，使得分类支持向量机在对失效概率贡献大的区域收敛于真实的极限状态方程，从而提高了可靠性分析的精度。
(3) 提出了两种快速高效的可靠性参数灵敏度分析方法——半解析法和快速数值模拟法。两种方法均基于重要抽样马尔可夫链模拟，半解析法只要在重要抽样法计算失效概率的结果中增加非常小的工作量，即可以快速得到可靠性参数灵敏度；快速数值模拟法未引入线性假设，可以较好地考虑极限状态方程非线性对可靠性灵敏度的影响。
(4) 提出了一种基于支持向量机回归近似极限状态方程的多馍式结构系统可靠性和可靠性灵敏度分析方法，所提方法首先由支持向量机拟合系统各失效模式的极限状态方程，将复杂或隐式极限状态方程近似等价为显式极限状态方程，然后根据系统各个失效模式的逻辑结构，由高精度的显式极限状态方程方法计算系统的失效概率和可靠性参数灵敏度。由于支持向量机回归具有良好的小样本学习能力和推广泛化性能，因而其在拟合非线性极限状态方程上表现优越，计算精
Although the structure reliability theory and analysis methods are gradually well established, the application of reliability methods encounters many problems in engineering, such as imprecision of transforming non-normal randon variables into normal randon variables, unacceptable computional cost and low accuracy, etc. For the problems, the thesis presented a series of reliability methods combining with new mathematic theory, such as simulated annealing optimization, statistical learning theory, Markov chain simulation, convex model, etc. These presented methods listed as follows possess high precision and efficiency in the application.
(1) On the basis of the equivalent normal distribution with three parameters and the simulated annealing optimization, an algorithm to calculate the failure probability is proposed for the structure with non-normal variables. The relative formulae are derived, and the implementation of the presented algorithm is demonstrated. The error of the failure probability calculation by this method is decreased due to the precise three parameters sought by the simulated annealing, which has global optimization property. Comparing with Monte-Carlo simulation, the computational effort of the presented algorithm is smaller, therefore it is more suitable for engineering application than Monte-Carlo simulation.
(2) For reliability analysis of structure with implicit limit state function, an iterative algorithm is presented on the basis of support vector classification machine. In the presented method, the support vector classification machine is employed to construct surrogate of the implicit limit state function. By use of the rational iteration, the constructed support vector classification machine can converge to the actual limit state function in the important region, which contributes to the failure probability significantly, thus the precision of the reliability analysis is improved.
(3) Two reliability parameter sensitivity analysis methods are presented, which are semi-analytical method and fast numerical simulation method. The two methods are both based on importance sampling Markov Chain simulation. The remarkable advantage of the semi-analytical method is its high efficiency. By increasing very limited computational effort in the presented semi-analytical method, the reliability
(1) 提出了基于等价正态三参数变换和模拟退火全局优化的失效概率计算方法，阐明了所提方法的理论基础和实施过程。由于采用了基于模拟退火的全局优化使得等价正态的三个参数更加准确，从而降低了计算失效概率的误差；与Monte-Carlo法相比，所提算法计算效率高且误差小，适于工程应用。
(2) 针对结构隐式极限状态函数的可靠性分析，提出了一种支持向量机分类迭代算法。该算法以支持向量机分类来替代隐式极限状态方程，通过构造合理的迭代格式，使得分类支持向量机在对失效概率贡献大的区域收敛于真实的极限状态方程，从而提高了可靠性分析的精度。
(3) 提出了两种快速高效的可靠性参数灵敏度分析方法——半解析法和快速数值模拟法。两种方法均基于重要抽样马尔可夫链模拟，半解析法只要在重要抽样法计算失效概率的结果中增加非常小的工作量，即可以快速得到可靠性参数灵敏度；快速数值模拟法未引入线性假设，可以较好地考虑极限状态方程非线性对可靠性灵敏度的影响。
(4) 提出了一种基于支持向量机回归近似极限状态方程的多馍式结构系统可靠性和可靠性灵敏度分析方法，所提方法首先由支持向量机拟合系统各失效模式的极限状态方程，将复杂或隐式极限状态方程近似等价为显式极限状态方程，然后根据系统各个失效模式的逻辑结构，由高精度的显式极限状态方程方法计算系统的失效概率和可靠性参数灵敏度。由于支持向量机回归具有良好的小样本学习能力和推广泛化性能，因而其在拟合非线性极限状态方程上表现优越，计算精
Although the structure reliability theory and analysis methods are gradually well established, the application of reliability methods encounters many problems in engineering, such as imprecision of transforming non-normal randon variables into normal randon variables, unacceptable computional cost and low accuracy, etc. For the problems, the thesis presented a series of reliability methods combining with new mathematic theory, such as simulated annealing optimization, statistical learning theory, Markov chain simulation, convex model, etc. These presented methods listed as follows possess high precision and efficiency in the application.
(1) On the basis of the equivalent normal distribution with three parameters and the simulated annealing optimization, an algorithm to calculate the failure probability is proposed for the structure with non-normal variables. The relative formulae are derived, and the implementation of the presented algorithm is demonstrated. The error of the failure probability calculation by this method is decreased due to the precise three parameters sought by the simulated annealing, which has global optimization property. Comparing with Monte-Carlo simulation, the computational effort of the presented algorithm is smaller, therefore it is more suitable for engineering application than Monte-Carlo simulation.
(2) For reliability analysis of structure with implicit limit state function, an iterative algorithm is presented on the basis of support vector classification machine. In the presented method, the support vector classification machine is employed to construct surrogate of the implicit limit state function. By use of the rational iteration, the constructed support vector classification machine can converge to the actual limit state function in the important region, which contributes to the failure probability significantly, thus the precision of the reliability analysis is improved.
(3) Two reliability parameter sensitivity analysis methods are presented, which are semi-analytical method and fast numerical simulation method. The two methods are both based on importance sampling Markov Chain simulation. The remarkable advantage of the semi-analytical method is its high efficiency. By increasing very limited computational effort in the presented semi-analytical method, the reliability
引文
[1] Au S K, Beck J L. Importance sampling in high dimension. Structural Safety. 2003, 25:139-163.
[2] Au S K, Beck J L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 2001, 16: 263-277.
[3] Au S K. Reliability-based design sensitivity by efficient simulation. Computers and Structures, 2005, 83:1048-1061.
[4] Gomes H M, Awruch A M. Comparison of response surface and neural network with other methods for structural reliability analysis. Structural Safety, 2004, 26:49-67.
[5] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[6] Rajashekhar M R, Ellingwood B R. A new look at the response surface approach for reliability analysis. Structural Safety, 1993, 12:205-220.
[7] Guan X L, Melchers R E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety, 2001, 23:429-444.
[8] Schueremans L, Gemert D V. Benefit of splines and neural networks in simulation based structural reliability analysis. Structural Safety, 2005, 27:246-261.
[9] Hurtado J E, Alvarez D A. Neural-network-based reliability analysis: a comparative study. Computer methods in applied mechanics and engineering, 2001, 191: 113-132.
[10] Papadrakakis M, Lagaros N D. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer methods in applied mechanics and engineering, 2002, 191:3491-3507.
[11] Deng J, Gu D S, et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[12] Kaymaz I. Application of kriging method to structural reliability problems. Structural Safety, 2005, 27:133-151.
[13] Romero V J, Swiler L P, Giunta A A. Construction of response surfaces based on progressive lattice sampling experimental designs with application to uncertainty propagation. Structural Safety, 2004,26:201-219.
[14] Vapnik V N. The nature of statistical learning theory. New York: Springer-Verlag, 1995.
[15] Vapnik V N. An overiew of statistical learning theory. IEEE Transaction on Neural Networks, 1999, 10:988-998.
[16] 邓乃扬，田英杰．数据挖掘中的新方法—支持向量机．北京：科学出版社，2004．
[17] Rackwitz R, Fiessler B. Structural Reliability under Combined Random Load Sequences. Computers and Structures, 1978, 9:489～494.
[18] 李良巧 机械可靠性设计与分析，北京：国防工业出版社，1998.
[19] Chen X, Lind N C. Fast Probability Integration by Three-parameter Normal Tail Approximation. Structural Safety, 1983, 1:269～2766.
[20] Wu Y T, Wirsching P H. New Algorithm for Structural Reliability Estimation. Journal of The Engineering Mechanics, 1987, 113(9):1319～1336.
[21] Zhao Yan-Gang, Tetsuro Ono. Third-moment standardization for structural reliability analysis. Journal of structural engineering, 2000:724-732.
[22] Chen Xingyuan, Yeou-Koung Tung. Investigation of polynomial normal transform. Structural Safety, 2003, 25:423-445.
[23] Au S K, Beck J L. A new adaptive importance sampling scheme for reliability calculations. Structural Safety, 1999,21(2): 135-158.
[24] Au S K. Probabitistic failure analysis by importance sampling markov chain simulation. Journal of Engineering Mechanics, ASCE, 2004,130(3):303-311.
[25] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[26] Hurtado J E, Alvarez D A. Classification approach for reliability analysis with stochastic finite- element modeling. Journal of Structural Engineering, 2003, 129:1141-1149.
[27] Hurtado J E. An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293.
[28] Ben-Haim Y, Elishakoff I. Convex Models of Uncertainty in Applied Mechanics. Amesterdam: Elsevier Science, 1990.
[29] Ben-Haim Y. A non-probabilistic concept of reliability. Structural Safety, 1994, 14(4):227～245.
[30] Elishakoff I. Discussion on: a non-probabilistic concept of reliability. Structural Safety, 1995, 17(3): 195～199.
[31] Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Structural Safety, 1995, 17(2):91～109.
[32] Ben-Haim Y. Robust Reliability in the Mechanical Sciences. Berlin: Springer-Verlag, 1996.
[33] 郭书祥．非随机不确定结构的可靠性方法和优化设计研究．博士论文，西北工业大学，西安，2002．
[1] Rackwitz R, Fiessler B. Structural Reliability under Combined Random Load Sequences. Computers and Structures, 1978, 9:489～494
[2] 李良巧．机械可靠性设计与分析，北京：国防工业出版社，1998
[3] Chen X, Lind N C. Fast Probability Integration by Three-parameter Normal Tail Approximation. Structural Safety, 1983, 1:269～2766
[4] Wu Y T, Wirsching P H. New Algorithm for Structural Reliability Estimation. Journal of The Engineering Mechanics, 1987, 113(9):1319～1336
[5] 何水清，王善等．结构可靠性分析与设计，北京：国防工业出版社，1993
[6] 康立山，谢云等．非数值并行算法——模拟退火算法，北京：科学出版社，1994
[1] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[2] Rajashekhar M R, Ellingwood B R. A new look at the response surface approach for reliability analysis. Structural Safety, 1993, 12:205-220.
[3] Kim S, Na S. Response surface method using vector projected sampling points. Structural Safety, 1997, 19(1): 3-19.
[4] Guan X L, Melchers R E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety, 2001,23:429-444.
[5] Schueremans L, Gemert D V. Benefit of splines and neural networks in simulation based structural reliability analysis. Structural Safety, 2005, 27:246-261.
[6] Hurtado J E, Alvarez D A. Neural-network-based reliability analysis: a comparative study. Computer methods in applied mechanics and engineering, 2001, 191:113-132.
[7] Papadrakakis M, Lagaros N D. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer methods in applied mechanics and engineering, 2002, 191:3491-3507.
[8] Deng Jian, Gu Desheng, Li Xibing, et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[9] Kaymaz I. Application of kriging method to structural reliability problems. Structural Safety, 2005, 27:133-151.
[10] Romero V J, Swiler L P, Giunta A A. Construction of response surfaces based on progressive lattice sampling experimental designs with application to uncertainty propagation. Structural Safety, 2004,26:201-219.
[11] Hurtado J E. An examination of method for approximating implicit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293.
[12] Cortes C, Vapnik V N. Support vector Networks. Machine Learning. 1995, 20:273-297.
[13] Vapnik V N. An overiew of statistical learning theory. IEEE Transaction on Neural Networks, 1999, 10:988-998.
[14] Vapnik V N. The nature of statistical learning theory. New York: Springer-Verlag, 1995.
[15] Cristianini N, Shawe-Tayler J. An introduction to support vector machines. Cambridge: Cambridge University Press, 2000.
[16] 邓乃扬，田英杰．数据挖掘中的新方法—支持向量机．北京：科学出版社，2004．
[17] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[18] Hurtado J E, Alvarez D A. Classification approach for reliability analysis with stochastic finite-element modeling. Journal of Structural Engineering, 2003, 129:1141-1149.
[19] 李洪双．小子样可靠性分析方法及应用研究．硕士论文．西安：西北工业大学，2006．
[20] Kiureghian A D, Dakessian T. Multiple design point in structural reliability. Structural safety and reliability. Rotterdam: Balkema. ISBN 9054109785.
[21] Gayton N, Bourinet J M, Lemaire M. CQ2RS: a new statistical approach to the response surface method for reliability analysis. Structural Safety, 2003, 25(1):99-121.
[22] Thacker B H, Riha D S. Probalilistic engineering analysis using the NESSUS software. Structural Safety, 2006, 28:83-107.
[1] Bjerager P, Krenk S. Parametric Sensitivity in First Order Reliability Analysis. Journal of Engineering Mechanics, ASCE, 1989, 115(7): 1577-1582.
[2] Karamchandani A, Cornell G A. Sensitivity Estimator within First and Second Order Reliability Methods. Structural Safety, 1992, 11(2):95-107.
[3] Lataillade A D, Blanco S, Clergent Y etc. Monte Carlo Method and Sensitivity Estimations. Journal of Quantitative Spectroscopy and Radiative Transfer, 2002, 75(5):529-538
[4] Melchers R E, Ahammed M. A Fast Approximate Method for Parameter Sensitivity Estimation in Monte- Carlo Structural Reliability. Computers and Structures, 2004, 82:55-61.
[5] Au S K. Probabilistic Failure Analysis by Importance Sampling Markov Chain Simulation. Journal of Engineering Mechanics, ASCE, 2004, 130(3):303-311.
[6] Nie Jinsuo, Ellingwood Bruce R. Directional methods for structural reliability analysis. Structural Safety, 2000, 22(3): 233～249.
[7] Au S K, Beck J L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 2001, 16: 263～277.
[8] Schueller G I, Pradlwarter H J, Koutsourelakis P S. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics, 2004, 19: 463～474.
[9] Wu Y T, Sitakanta Monhanty. Variable screening and ranking using sampling-based sensitivity measures. Reliability Engineering and System Safety, 2006, 91(6): 634-647.
[10]Kaymaz I. Application of Kriging method to structural reliability problems. Structural Safety, 2005,27(2): 133-151.
[11] Au S K, Beck J L. A new adaptive importance sampling scheme for reliability calculations. Structural Safety, 1999, 21(2): 135-138.
[1] Hunter D. An Upper Bounds for the Probability of A Union. Journal of Applied Probability, 1976, 3(3):597-603.
[2] Ditlevsen O. Narrow Reliabiltiy Bounds for Structural System. Journal of Structural Mechanics, 1979, 7(4):453-472.
[3] Robert H S, Mark A C. System Reliability and Sensitivity Factors via the MPPSS Method. Probabilistic Engineering Mechanics, 2005, 20(2): 148-157.
[4] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995.
[5] Cristianini N, Shawe-Tayler J. An Introduction to Support Vector Machines and Other Kernel-based Learing Methods. Cambridge: Cambridge University Press, 2000.
[6] Wu Y T, Sitakanta Monhanty. Variable Screening and Ranking Using Sampling-based Sensitivity Measures. Reliability Engineering and System Safety. 2006, 91(6):634-64
[1] Sexsmith R G, Probability-Based Safety Analysis- Value and Drawbacks. Structural Safety, 1999, 21(4): 303-310.
[2] Ben-Haim Y. A Non-probabilistic Concept of Reliability. Structural Safety, 1994, 14(4):227-245.
[3] Elishakoff I. Discussion on: A Non-probabilistic Concept of Reliability. Structural Safety, 1995, 17(3):195-199.
[4] Ben-Haim Y. A Non-probabilistic Measure of Reliability of Linear Systems Based on Expansion of Convex Models. Structural Safety, 1995, 17(2):91-109.
[5] 郭书祥，吕震宙，冯元生．基于区间分析的结构非概率可靠性模型．计算力学学报，2001，18(1)：56-60．
[6] 刘成立．复杂结构可靠性分析及设计研究．博士论文．西安：西北工业大学，2007．
[7] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[8] Deng J, Gu D S et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[9] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[10] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995.
[2] Au S K, Beck J L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 2001, 16: 263-277.
[3] Au S K. Reliability-based design sensitivity by efficient simulation. Computers and Structures, 2005, 83:1048-1061.
[4] Gomes H M, Awruch A M. Comparison of response surface and neural network with other methods for structural reliability analysis. Structural Safety, 2004, 26:49-67.
[5] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[6] Rajashekhar M R, Ellingwood B R. A new look at the response surface approach for reliability analysis. Structural Safety, 1993, 12:205-220.
[7] Guan X L, Melchers R E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety, 2001, 23:429-444.
[8] Schueremans L, Gemert D V. Benefit of splines and neural networks in simulation based structural reliability analysis. Structural Safety, 2005, 27:246-261.
[9] Hurtado J E, Alvarez D A. Neural-network-based reliability analysis: a comparative study. Computer methods in applied mechanics and engineering, 2001, 191: 113-132.
[10] Papadrakakis M, Lagaros N D. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer methods in applied mechanics and engineering, 2002, 191:3491-3507.
[11] Deng J, Gu D S, et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[12] Kaymaz I. Application of kriging method to structural reliability problems. Structural Safety, 2005, 27:133-151.
[13] Romero V J, Swiler L P, Giunta A A. Construction of response surfaces based on progressive lattice sampling experimental designs with application to uncertainty propagation. Structural Safety, 2004,26:201-219.
[14] Vapnik V N. The nature of statistical learning theory. New York: Springer-Verlag, 1995.
[15] Vapnik V N. An overiew of statistical learning theory. IEEE Transaction on Neural Networks, 1999, 10:988-998.
[16] 邓乃扬，田英杰．数据挖掘中的新方法—支持向量机．北京：科学出版社，2004．
[17] Rackwitz R, Fiessler B. Structural Reliability under Combined Random Load Sequences. Computers and Structures, 1978, 9:489～494.
[18] 李良巧 机械可靠性设计与分析，北京：国防工业出版社，1998.
[19] Chen X, Lind N C. Fast Probability Integration by Three-parameter Normal Tail Approximation. Structural Safety, 1983, 1:269～2766.
[20] Wu Y T, Wirsching P H. New Algorithm for Structural Reliability Estimation. Journal of The Engineering Mechanics, 1987, 113(9):1319～1336.
[21] Zhao Yan-Gang, Tetsuro Ono. Third-moment standardization for structural reliability analysis. Journal of structural engineering, 2000:724-732.
[22] Chen Xingyuan, Yeou-Koung Tung. Investigation of polynomial normal transform. Structural Safety, 2003, 25:423-445.
[23] Au S K, Beck J L. A new adaptive importance sampling scheme for reliability calculations. Structural Safety, 1999,21(2): 135-158.
[24] Au S K. Probabitistic failure analysis by importance sampling markov chain simulation. Journal of Engineering Mechanics, ASCE, 2004,130(3):303-311.
[25] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[26] Hurtado J E, Alvarez D A. Classification approach for reliability analysis with stochastic finite- element modeling. Journal of Structural Engineering, 2003, 129:1141-1149.
[27] Hurtado J E. An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293.
[28] Ben-Haim Y, Elishakoff I. Convex Models of Uncertainty in Applied Mechanics. Amesterdam: Elsevier Science, 1990.
[29] Ben-Haim Y. A non-probabilistic concept of reliability. Structural Safety, 1994, 14(4):227～245.
[30] Elishakoff I. Discussion on: a non-probabilistic concept of reliability. Structural Safety, 1995, 17(3): 195～199.
[31] Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Structural Safety, 1995, 17(2):91～109.
[32] Ben-Haim Y. Robust Reliability in the Mechanical Sciences. Berlin: Springer-Verlag, 1996.
[33] 郭书祥．非随机不确定结构的可靠性方法和优化设计研究．博士论文，西北工业大学，西安，2002．
[1] Rackwitz R, Fiessler B. Structural Reliability under Combined Random Load Sequences. Computers and Structures, 1978, 9:489～494
[2] 李良巧．机械可靠性设计与分析，北京：国防工业出版社，1998
[3] Chen X, Lind N C. Fast Probability Integration by Three-parameter Normal Tail Approximation. Structural Safety, 1983, 1:269～2766
[4] Wu Y T, Wirsching P H. New Algorithm for Structural Reliability Estimation. Journal of The Engineering Mechanics, 1987, 113(9):1319～1336
[5] 何水清，王善等．结构可靠性分析与设计，北京：国防工业出版社，1993
[6] 康立山，谢云等．非数值并行算法——模拟退火算法，北京：科学出版社，1994
[1] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[2] Rajashekhar M R, Ellingwood B R. A new look at the response surface approach for reliability analysis. Structural Safety, 1993, 12:205-220.
[3] Kim S, Na S. Response surface method using vector projected sampling points. Structural Safety, 1997, 19(1): 3-19.
[4] Guan X L, Melchers R E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety, 2001,23:429-444.
[5] Schueremans L, Gemert D V. Benefit of splines and neural networks in simulation based structural reliability analysis. Structural Safety, 2005, 27:246-261.
[6] Hurtado J E, Alvarez D A. Neural-network-based reliability analysis: a comparative study. Computer methods in applied mechanics and engineering, 2001, 191:113-132.
[7] Papadrakakis M, Lagaros N D. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer methods in applied mechanics and engineering, 2002, 191:3491-3507.
[8] Deng Jian, Gu Desheng, Li Xibing, et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[9] Kaymaz I. Application of kriging method to structural reliability problems. Structural Safety, 2005, 27:133-151.
[10] Romero V J, Swiler L P, Giunta A A. Construction of response surfaces based on progressive lattice sampling experimental designs with application to uncertainty propagation. Structural Safety, 2004,26:201-219.
[11] Hurtado J E. An examination of method for approximating implicit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293.
[12] Cortes C, Vapnik V N. Support vector Networks. Machine Learning. 1995, 20:273-297.
[13] Vapnik V N. An overiew of statistical learning theory. IEEE Transaction on Neural Networks, 1999, 10:988-998.
[14] Vapnik V N. The nature of statistical learning theory. New York: Springer-Verlag, 1995.
[15] Cristianini N, Shawe-Tayler J. An introduction to support vector machines. Cambridge: Cambridge University Press, 2000.
[16] 邓乃扬，田英杰．数据挖掘中的新方法—支持向量机．北京：科学出版社，2004．
[17] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[18] Hurtado J E, Alvarez D A. Classification approach for reliability analysis with stochastic finite-element modeling. Journal of Structural Engineering, 2003, 129:1141-1149.
[19] 李洪双．小子样可靠性分析方法及应用研究．硕士论文．西安：西北工业大学，2006．
[20] Kiureghian A D, Dakessian T. Multiple design point in structural reliability. Structural safety and reliability. Rotterdam: Balkema. ISBN 9054109785.
[21] Gayton N, Bourinet J M, Lemaire M. CQ2RS: a new statistical approach to the response surface method for reliability analysis. Structural Safety, 2003, 25(1):99-121.
[22] Thacker B H, Riha D S. Probalilistic engineering analysis using the NESSUS software. Structural Safety, 2006, 28:83-107.
[1] Bjerager P, Krenk S. Parametric Sensitivity in First Order Reliability Analysis. Journal of Engineering Mechanics, ASCE, 1989, 115(7): 1577-1582.
[2] Karamchandani A, Cornell G A. Sensitivity Estimator within First and Second Order Reliability Methods. Structural Safety, 1992, 11(2):95-107.
[3] Lataillade A D, Blanco S, Clergent Y etc. Monte Carlo Method and Sensitivity Estimations. Journal of Quantitative Spectroscopy and Radiative Transfer, 2002, 75(5):529-538
[4] Melchers R E, Ahammed M. A Fast Approximate Method for Parameter Sensitivity Estimation in Monte- Carlo Structural Reliability. Computers and Structures, 2004, 82:55-61.
[5] Au S K. Probabilistic Failure Analysis by Importance Sampling Markov Chain Simulation. Journal of Engineering Mechanics, ASCE, 2004, 130(3):303-311.
[6] Nie Jinsuo, Ellingwood Bruce R. Directional methods for structural reliability analysis. Structural Safety, 2000, 22(3): 233～249.
[7] Au S K, Beck J L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 2001, 16: 263～277.
[8] Schueller G I, Pradlwarter H J, Koutsourelakis P S. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics, 2004, 19: 463～474.
[9] Wu Y T, Sitakanta Monhanty. Variable screening and ranking using sampling-based sensitivity measures. Reliability Engineering and System Safety, 2006, 91(6): 634-647.
[10]Kaymaz I. Application of Kriging method to structural reliability problems. Structural Safety, 2005,27(2): 133-151.
[11] Au S K, Beck J L. A new adaptive importance sampling scheme for reliability calculations. Structural Safety, 1999, 21(2): 135-138.
[1] Hunter D. An Upper Bounds for the Probability of A Union. Journal of Applied Probability, 1976, 3(3):597-603.
[2] Ditlevsen O. Narrow Reliabiltiy Bounds for Structural System. Journal of Structural Mechanics, 1979, 7(4):453-472.
[3] Robert H S, Mark A C. System Reliability and Sensitivity Factors via the MPPSS Method. Probabilistic Engineering Mechanics, 2005, 20(2): 148-157.
[4] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995.
[5] Cristianini N, Shawe-Tayler J. An Introduction to Support Vector Machines and Other Kernel-based Learing Methods. Cambridge: Cambridge University Press, 2000.
[6] Wu Y T, Sitakanta Monhanty. Variable Screening and Ranking Using Sampling-based Sensitivity Measures. Reliability Engineering and System Safety. 2006, 91(6):634-64
[1] Sexsmith R G, Probability-Based Safety Analysis- Value and Drawbacks. Structural Safety, 1999, 21(4): 303-310.
[2] Ben-Haim Y. A Non-probabilistic Concept of Reliability. Structural Safety, 1994, 14(4):227-245.
[3] Elishakoff I. Discussion on: A Non-probabilistic Concept of Reliability. Structural Safety, 1995, 17(3):195-199.
[4] Ben-Haim Y. A Non-probabilistic Measure of Reliability of Linear Systems Based on Expansion of Convex Models. Structural Safety, 1995, 17(2):91-109.
[5] 郭书祥，吕震宙，冯元生．基于区间分析的结构非概率可靠性模型．计算力学学报，2001，18(1)：56-60．
[6] 刘成立．复杂结构可靠性分析及设计研究．博士论文．西安：西北工业大学，2007．
[7] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7:57-66.
[8] Deng J, Gu D S et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27:25-48.
[9] Rocco C M, Moreno J A. Fast Monte Carlo reliability evaluation using support vector machine. Reliability Engineering and System Safety, 2002, 76:237-243.
[10] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995.