基于凸集模型的非概率可靠性灵敏度分析
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摘要
可靠性灵敏度能够确定可靠度和不确定变量分布参数的内在联系,对可靠性分析和设计具有指导作用。本文将非概率可靠性灵敏度指标定义为非概率可靠性指标对变量分布参数的偏导数,提出了一种基于椭球凸集模型的非概率可靠性灵敏度分析方法。推导了线性极限状态函数的非概率可靠性灵敏度的解析公式。对于非线性极限状态函数,通过将极限状态函数在两种不同线性化位置进行泰勒一阶展开,分别给出了非概率可靠性灵敏度分析的近似解析解。算例分析结果表明文中所提方法有效可行。
Reliability sensitivity determines the relationship between reliability and the distribution parameters of uncertain variables, and guides reliability analysis and reliability design. A novel non-probabilistic reliability sensitivity analysis method based on ellipsoidal convex model was presented, in which non-probabilistic reliability sensitivity was defined as a partial derivative of non-probabilistic reliability index with respect to the distribution parameters of uncertain variables. The analytical equations of non-probabilistic reliability sensitivity for linear limit state functions were first derived. Through-two different choices of the linearization point, the approximate functions of non-linear limit state functions can be then obtained by using the first-order Taylor expansion, then the approximate analytical equations of non-probabilistic reliability sensitivity were given. Several numerical examples will be used to demonstrate the effectiveness and feasibility of the proposed methods.
Reliability sensitivity determines the relationship between reliability and the distribution parameters of uncertain variables, and guides reliability analysis and reliability design. A novel non-probabilistic reliability sensitivity analysis method based on ellipsoidal convex model was presented, in which non-probabilistic reliability sensitivity was defined as a partial derivative of non-probabilistic reliability index with respect to the distribution parameters of uncertain variables. The analytical equations of non-probabilistic reliability sensitivity for linear limit state functions were first derived. Through-two different choices of the linearization point, the approximate functions of non-linear limit state functions can be then obtained by using the first-order Taylor expansion, then the approximate analytical equations of non-probabilistic reliability sensitivity were given. Several numerical examples will be used to demonstrate the effectiveness and feasibility of the proposed methods.
引文
[1] 吕震宙,宋述芳,李洪双,等.结构机构可靠性及可靠性灵敏度分析[M].北京:科学出版社,2009:6.LV ZhenZhou,SONG ShuFang,LI HongShuang,et al.Reliability and reliability sensitivity analysis of structures and mechanism [M].Beijing:Science Press,2009:6 (In Chinese)
[2] Bjerager P,Krenk S.Parametric sensitivity in first order reliability analysis[J].Journal of Engineering Mechanics,1989,115(7):1577-1582.
[3] 张艳林,张义民,金雅娟,等.基于均值一阶Esscher’s 近似的可靠性灵敏度分析[J].机械工程学报,2011,47(6):168-172.ZHANG YanLin.ZHANG YiMin.JIN YaJuan,et al.Reliability sensitivity analysis based on mean value first order Esscher’s Approximation [J].Journal of Mechanical Engineering,2011,47(6):168-172 (In Chinese).
[4] Karamchandani A,Cornell C A.Sensitivity estimation within first and second order reliability methods[J].Structural Safety,1992,11(2):95-102.
[5] 宋军,吕震宙.非正态变量可靠性灵敏度分析方法[J].机械强度,2008,30(1):052-057.SONG Jun,LV ZhenZhou.Reliability sensitivity method for non-normal variable [J].Journal of Mechanical Strength,2008,30(1):052-057 (In Chinese).
[6] Melchers R E,Ahammed M.A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability[J].Probabilistic Engineering Mechanics,2004(82):55-61.
[7] Lv Zhenzhou,Song Shufang,Yue Zhufeng,et al.Reliability sensitivity method by line sampling[J].Structural Safety,2008,30(6):517-532.
[8] Song Shufang,LV Zhenzhou,Qiao Hongwei.Subset simulation for structural reliability sensitivity analysis[J].Reliability Engineering and System Safety,2009,94(2):658-665.
[9] Elishakoff I.Essay on uncertainties in elastic and viscoelastic structures:from A.M.Freudenthal’s criticisms to modern convex modeling[J].Computers & Structures,1995,56(6):871-895.
[10] Bae H R,Grandi R V,Canfield R A.Sensitivity analysis of structural response uncertainty propagation using evidence theory[J].Structural and Multidisciplinary Optimization,2006(31):270-291.
[11] Helton J C,Johnson J D,Oberkampf W L et al.Sensitivity analysis in conjunction with evidence theory representations of epistemic uncertainty[J].Reliability Engineering and System Safety,2006,91(10-11):1414-1434.
[12] 李贵杰,吕震宙,王攀.结构非概率可靠性灵敏度分析方法[J].航空学报.2012,33(3):501-507.LI GuiJie.,LV ZhenZhou,WANG Pan.Sensitivity analysis of non-probabilistic reliability of uncertain structures[J].Acta Aeronautica et Astronautica Sinica,2012,33(3):501-507 (In Chinese).
[13] Xiao Ning-Cong,Huang Hong-zhong,Li Yan-Feng,et al.Non-probabilistic reliability sensitivity analysis of the model of structural systems which interval variables whose state of dependence is determined by constraints[J].Proceedings of the Institution of Mechanical Engineers.Part O:Journal of Risk and Reliability.2013,227(5):491-498.
[14] Guo J,Du X P.Sensitivity analysis with mixture of epistemic and aleatory uncertainties[J].AIAA Journal,2007,45(9):2337-2349.
[15] Xiao Ningcong,Huang Hongzhong,Wang Zhonglai et al.Reliability sensitivity analysis for structural systems in interval probability form[J].Structural and Multidisciplinary Optimization,2011(44):691-705.
[16] L.P.Zhu,I.Elishakoff.J.H Starnes.Derivation of muti-dimensional ellipsoidal convex model for experimental data[J].Mathematical and Computer Modelling,1996,24(2):103-114.
[17] C.Jiang,X.Han,G.Y.Lu,et al.Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique[J].Computer Methods in Applied Mechanics and Engineering,2011(200):2528-2546.
[18] C.Jiang,G.Y.Lu,X.Han,et al.Some important issues on first-order reliability analysis with non-probabilistic convex model[J].Journal of Mechanical Design,2014,136:034501-1-034501-5.
[19] C.Jiang,R.G.Bi,G.Y.Lu,et al.Structural reliability analysis using non-probabilistic convex model[J].Computer Methods in Applied Mechanics and Engineering,2013(254):83-89.
[2] Bjerager P,Krenk S.Parametric sensitivity in first order reliability analysis[J].Journal of Engineering Mechanics,1989,115(7):1577-1582.
[3] 张艳林,张义民,金雅娟,等.基于均值一阶Esscher’s 近似的可靠性灵敏度分析[J].机械工程学报,2011,47(6):168-172.ZHANG YanLin.ZHANG YiMin.JIN YaJuan,et al.Reliability sensitivity analysis based on mean value first order Esscher’s Approximation [J].Journal of Mechanical Engineering,2011,47(6):168-172 (In Chinese).
[4] Karamchandani A,Cornell C A.Sensitivity estimation within first and second order reliability methods[J].Structural Safety,1992,11(2):95-102.
[5] 宋军,吕震宙.非正态变量可靠性灵敏度分析方法[J].机械强度,2008,30(1):052-057.SONG Jun,LV ZhenZhou.Reliability sensitivity method for non-normal variable [J].Journal of Mechanical Strength,2008,30(1):052-057 (In Chinese).
[6] Melchers R E,Ahammed M.A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability[J].Probabilistic Engineering Mechanics,2004(82):55-61.
[7] Lv Zhenzhou,Song Shufang,Yue Zhufeng,et al.Reliability sensitivity method by line sampling[J].Structural Safety,2008,30(6):517-532.
[8] Song Shufang,LV Zhenzhou,Qiao Hongwei.Subset simulation for structural reliability sensitivity analysis[J].Reliability Engineering and System Safety,2009,94(2):658-665.
[9] Elishakoff I.Essay on uncertainties in elastic and viscoelastic structures:from A.M.Freudenthal’s criticisms to modern convex modeling[J].Computers & Structures,1995,56(6):871-895.
[10] Bae H R,Grandi R V,Canfield R A.Sensitivity analysis of structural response uncertainty propagation using evidence theory[J].Structural and Multidisciplinary Optimization,2006(31):270-291.
[11] Helton J C,Johnson J D,Oberkampf W L et al.Sensitivity analysis in conjunction with evidence theory representations of epistemic uncertainty[J].Reliability Engineering and System Safety,2006,91(10-11):1414-1434.
[12] 李贵杰,吕震宙,王攀.结构非概率可靠性灵敏度分析方法[J].航空学报.2012,33(3):501-507.LI GuiJie.,LV ZhenZhou,WANG Pan.Sensitivity analysis of non-probabilistic reliability of uncertain structures[J].Acta Aeronautica et Astronautica Sinica,2012,33(3):501-507 (In Chinese).
[13] Xiao Ning-Cong,Huang Hong-zhong,Li Yan-Feng,et al.Non-probabilistic reliability sensitivity analysis of the model of structural systems which interval variables whose state of dependence is determined by constraints[J].Proceedings of the Institution of Mechanical Engineers.Part O:Journal of Risk and Reliability.2013,227(5):491-498.
[14] Guo J,Du X P.Sensitivity analysis with mixture of epistemic and aleatory uncertainties[J].AIAA Journal,2007,45(9):2337-2349.
[15] Xiao Ningcong,Huang Hongzhong,Wang Zhonglai et al.Reliability sensitivity analysis for structural systems in interval probability form[J].Structural and Multidisciplinary Optimization,2011(44):691-705.
[16] L.P.Zhu,I.Elishakoff.J.H Starnes.Derivation of muti-dimensional ellipsoidal convex model for experimental data[J].Mathematical and Computer Modelling,1996,24(2):103-114.
[17] C.Jiang,X.Han,G.Y.Lu,et al.Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique[J].Computer Methods in Applied Mechanics and Engineering,2011(200):2528-2546.
[18] C.Jiang,G.Y.Lu,X.Han,et al.Some important issues on first-order reliability analysis with non-probabilistic convex model[J].Journal of Mechanical Design,2014,136:034501-1-034501-5.
[19] C.Jiang,R.G.Bi,G.Y.Lu,et al.Structural reliability analysis using non-probabilistic convex model[J].Computer Methods in Applied Mechanics and Engineering,2013(254):83-89.