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Structural reliability analysis based on dimensionality reduction and Edgeworth series
MENG Guangwei, FENG Xinyu, LI Feng , ZHOU Liming
School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China
Received: 2015-03-30; Accepted: 2015-06-26; Published online: 2015-09-17 16:50
Foundation items: Foundation of Jilin Provincial Science & Technology Department (201205001,201215048); National Key Scientific Instrument and Equipment Development Projects of China (2012YQ030075)
Corresponding author: Tel.:0431-85095843 E-mail:fengli@jlu.edu.cn
Abstract: A reliability analysis method based on the dimension reduction algorithm and the Edgeworth series was proposed to treat the complicate structures with implicit and high dimensional nonlinear limit state functions in practical engineering. By utilizing the dimension reduction method, the n-dimensional function was expanded to n unidimensional functions and the random variable were made to subject to the independent normal distribution with mean value being zero and variance deviation being 0.5 by means of the variable transformation. The origin moments of the unidimensional functions were obtained after the Gauss-Hermite integration. In this case, the central moments of the limit state function of the structure were achieved successfully and applied to the Edgeworth series expanding expressions, from which the cumulative distribution function of the limit state function could be generated and finally the probability of failure could be obtained. Avoiding gradient computation, the proposed method requires less definite reanalysis and is proved to be effective and correct via numerical examples.
Key words: structural reliability     dimension reduction method     Gauss-Hermite numerical integration     Edgeworth series     moment method

Rahman[8]和Cho[9]等提出的降维算法(Dimension Reduction Method,DRM)避免了对功能函数梯度与矩阵的逆以及迭代最可能失效点(Most Probable failure Point,MPP)的求解,大大降低了计算工作量;Youn等[10]指出在基于矩的积分法时,随积分点的增多,导致线性方程组的系数矩阵出现奇异性、条件数和数值结果不稳定等。

1 降维算法

2 变量转换

 图 1 结构失效概率计算流程图 Fig. 1 Calculation flowchart of structural failure probability
4 数值算例 4.1 算例1

 方法 失效概率 相对误差/% 样本数 MCS 0.028 0 107 FORM 0.023 0 20.1 39 SORM 0.025 2 12.5 15 本文方法 0.026 7 7.3 9
4.2 算例2

 图 2 平面十杆桁架结构 Fig. 2 Structure of ten plane truss

 随机变量 参数统计特征 分布类型 均值/kN 变异系数 偏态系数 峰度系数 P1 对数正态 750 0.10 0.301 4 3.163 1 P2 对数正态 950 0.12 0.362 5 3.235 1 P3 正态 950 0.11 0 3

 图 3 功能函数的累积分布 Fig. 3 Cumulative distribution of performance function

 方法 失效概率Pf/10-4 相对误差/% 样本 耗时/s MCS 6.000 0 104 310 440 000 FORM 6.526 2 8.77 15 6.338 395 SORM 6.779 9 12.99 15 4.697 708 本文方法 6.046 3 0.77 9 2.891 358
5 结 论

1) 运用降维算法使得高维积分运算的工作量大大减少,利用Edgeworth级数方法拟合功能函数的累积分布函数。

2) 解决了功能函数为隐式或高维非线性的复杂结构失效概率的计算问题。

3) 高效且稳定地降低了计算成本,算法流程简单,易于编程。具有收敛速度快、计算次数少和可以处理复杂结构可靠性分析问题等优点。

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#### 文章信息

MENG Guangwei, FENG Xinyu, LI Feng, ZHOU Liming

Structural reliability analysis based on dimensionality reduction and Edgeworth series

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(3): 421-425.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0181