Non-probabilistic Reliability Sensitivity Analysis for Multi-source Uncertain Variables
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摘要: 采用多椭球模型描述结构多源不确定性, 提出了一种非概率可靠性灵敏度分析方法。首先, 将非概率可靠性指标作为多椭球模型非概率可靠性的度量标准, 分别给出线性和非线性极限状态函数的非概率可靠性指标求解方法。其次, 将非概率可靠性灵敏度定义为非概率可靠性指标对不确定变量均值、区间半径及相关系数的偏导数, 提出一种非概率可靠性灵敏度分析的近似解析方法。再次, 讨论了多椭球模型可靠性灵敏度分析方法较之于区间模型和椭球模型可靠性灵敏度分析方法的一般性和广泛适用性。最后通过3个算例验证本文所提方法的可行性和优越性。Abstract: This paper proposes a novel non-probabilistic reliability sensitivity analysis method and uses the multi-ellipsoid model to describe the multi-source uncertainties of a structure. It takes the non-probabilistic reliability index as the measure of non-probabilistic reliability and gives the solutions of non-probabilistic reliability indices of a linear limit state function and a nonlinear limit state function respectively. It develops an approximate analytical method for non-probabilistic reliability sensitivity analysis and defines reliability sensitivity as the partial derivative of the non-probabilistic reliability index with respect to uncertain variable's mean value, interval radius and correlation coefficient. The generality and wide applicability of the multi-ellipsoid model's reliability sensitivity analysis method compared with those for interval model and ellipsoid model are discussed. Three numerical examples are utilized to verify the feasibility and superiority of the proposed method.
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表 1 算例1可靠性指标及参数灵敏度计算结果
名称 线性解析法 有限差分法(步长0.1) 有限差分法(步长0.01) 有限差分法(步长0.001) βm 1.939 1.939 1.939 1.939 ∂βm/∂x1c 0.033 687 0.033 722 0.033 721 0.033 722 ∂βm/∂xc2 0.016 843 0.016 861 0.016 861 0.016 859 ∂βm/∂x3c -0.008 421 7 -0.008 430 5 -0.008 430 5 -0.008 430 5 ∂βm/∂x1r -0.065 259 -0.064 676 -0.064 868 -0.064 888 ∂βm/∂x2r -0.007 254 7 -0.005 68 -0.005 646 3 -0.005 643 ∂βm/∂x3r -0.016 347 -0.016 333 -0.016 345 -0.016 347 ∂βm/∂ρ1x1x2 -0.080 608 -0.080 033 -0.080 55 -0.080 602 表 2 悬臂梁结构的变量不确定区间
变量名称 均值 上界 下界 Py/N 40 000 38 000 42 000 Pz/N 15 000 12 500 17 500 L/mm 1 000 900 1 100 b/mm 100 90 110 h/mm 200 180 220 表 3 算例2可靠性指标及参数灵敏度计算结果
名称 非线性近似解析法 基于区间模型的解析法 基于椭球模型的解析法 有限差分法 βm 2.308 5 1.460 7 2.750 3 2.308 5 ∂βm/∂Pyc -0.000 111 58 -0.000 071 203 -0.000 084 624 -0.000 111 58 ∂βm/∂Pzc -0.000 053 542 -0.000 035 602 -0.000 042 312 -0.000 053 542 ∂βm/∂Lc -0.005 252 6 -0.003 283 7 -0.004 019 6 -0.005 252 6 ∂βm/∂bc 0.114 82 0.071 493 0.074 046 0.114 82 ∂βm/∂hc 0.351 37 0.022 863 0.232 72 0.351 37 ∂βm/∂Pyr -0.000 143 39 -0.000 065 648 -0.000 048 008 -0.000 143 39 ∂βm/∂Pzr -0.000 051 7 -0.000 032 824 -0.000 016 741 -0.000 051 7 ∂βm/∂Lr -0.004 297 9 -0.003 118 3 -0.007 472 6 -0.004 297 9 ∂βm/∂br -0.105 -0.057 442 -0.196 25 -0.105 ∂βm/∂hr -0.020 634 -0.018 053 -0.055 381 -0.020 634 ∂βm/∂ρPypz1 -0.080 782 - -0.024 619 -0.080 823 ∂βm/∂ρLb2 0.295 24 - 0.409 3 0.295 25 ∂βm/∂ρbh2 0.185 58 - 0.257 27 0.185 55 ∂βm/∂ρhL2 -0.341 86 - -0.473 92 -0.341 83 表 3 算例3可靠性指标及参数灵敏度计算结果
名称 非线性近似解析法 有限差分法 βm 0.924 46 0.924 46 ∂βm/∂R1c 0.139 04 0.139 04 ∂βm/∂R2c 0.007 505 0.007 544 7 ∂βm/∂R3c -0.094 501 -0.094 494 ∂βm/∂R4c -0.000 852 86 -0.000 835 15 ∂βm/∂Hc 0.900 7 0.901 67 ∂βm/∂R1r -0.018 023 -0.018 023 ∂βm/∂R2r -0.006 585 3 -0.006 586 4 ∂βm/∂R3r -0.020 635 -0.020 635 ∂βm/∂R4r -0.000 003 204 9 -0.000 003 205 7 ∂βm/∂Hr -0.051 377 -0.051 374 ∂βm/∂ρR1R21 -0.007 632 2 -0.007 632 ∂βm/∂ρR1R31 0.050 142 0.050 15 ∂βm/∂ρR2R31 0.007 606 8 0.007 606 9 -
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