Structural reliability analysis method based on second order parabolic approximation

doi:10.3785/j.issn.1006-754X.2024.03.305

Chinese Journal of Engineering Design

2024, Vol. 31

Issue (1): 50-58 DOI: 10.3785/j.issn.1006-754X.2024.03.305

Reliability and Quality Design

Structural reliability analysis method based on second order parabolic approximation

Zhenzhong CHEN¹(

),Dongyu HUANG¹,Jiao TIAN¹(

),Xiaoke LI²,Zihao WU³

^1.College of Mechanical Engineering, Donghua University, Shanghai 201620, China
^2.College of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China
^3.School of Intelligent Manufacturing and Control Engineering, Shanghai Polytechnic University, Shanghai 200135, China

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Abstract

Aiming at the problem that the first order reliability analysis method commonly used in engineering has insufficient accuracy in solving the reliability of limit state function with high nonlinearity, a structural reliability analysis method based on parabolic approximation is proposed on the basis of second order reliability analysis method. Firstly, the first order reliability analysis method was used to iteratively solve the most probable point in standard normal space. Then, the vector composed of the most probable point and the coordinate origin was taken as the new coordinate axis, and the approximate parabola was constructed based on the curvature of the most probable point in each direction of the limit state function to improve the approximate accuracy of the boundary region. Finally, the approximate parabola on the new axis was integrated according to the standard normal distribution probability density to solve the structural reliability probability. The first order reliability method, the second order reliability method and the reliability analysis method based on the second order parabola approximation were compared by four examples to verify the feasibility of the proposed method. The results showed that when facing the reliability problems with high nonlinearity, the accuracy of the first order reliability method was low and the second order reliability method might make solving errors in special cases, while the parabolic approximate integral method could effectively improve the accuracy of structural reliability analysis and ensure the stability of the solution. The research results can provide reference for reliability analysis of complex structures.

Key words： first order reliability second order reliability parabolic approximation reliability analysis

Received: 20 October 2023 Published: 04 March 2024

CLC:

TH 122

Corresponding Authors: Jiao TIAN E-mail: zhenzh.chen@dhu.edu.cn;tianjiao@dhu.edu.cn

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Cite this article:

Zhenzhong CHEN,Dongyu HUANG,Jiao TIAN,Xiaoke LI,Zihao WU. Structural reliability analysis method based on second order parabolic approximation. Chinese Journal of Engineering Design, 2024, 31(1): 50-58.

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https://www.zjujournals.com/gcsjxb/10.3785/j.issn.1006-754X.2024.03.305 OR https://www.zjujournals.com/gcsjxb/Y2024/V31/I1/50

基于二阶抛物线近似的结构可靠性分析方法

针对工程中常用的一阶可靠性分析方法在求解非线性程度较高的极限状态函数的可靠性时精度不足的问题，在二阶可靠性分析方法的基础上提出了一种基于抛物线近似的结构可靠性分析方法。首先，采用一阶可靠性分析方法迭代求解标准正态空间下的最大可能点。然后，以最大可能点与坐标原点构成的向量作为新坐标轴，基于极限状态函数最大可能点各方向上的曲率构建近似抛物线，以提高边界区域的近似精度。最后，根据标准正态分布概率密度对新坐标轴上的近似抛物线进行积分，以求解结构的可靠概率。通过4个算例来比较一阶可靠性方法、二阶可靠性方法与基于二阶抛物线近似的可靠性分析方法，以验证所提出方法的可行性。结果表明，当面对非线性程度较高的可靠性问题时，一阶可靠性方法的求解精度较低，二阶可靠性方法在特殊情况下会发生求解错误，而通过抛物线近似积分的方法可有效提高结构可靠性分析的精度并保证求解的稳定性。研究结果可为复杂结构的可靠性分析提供参考。

关键词： 一阶可靠性, 二阶可靠性, 抛物线近似, 可靠性分析

Fig.1 Reliability index and MPP in two-dimensional reliability analysis

Fig.2 Parabola fitting and rotation of limit state functions in two-dimensional reliability analysis

Fig.3 Schematic of feasible and failure domain division in two-dimensional reliability analysis

Fig.4 Limit state function of example 1

Table 1 Reliability analysis results of example 1

Fig.5 Schematic of circular axis under random torque

Fig.6 Reliability analysis results of example 2

变量	均值	标准差	分布类型
$x 1$	60	6.0	正态
$x 2$	2 000	74.0	对数正态
$x 3$	24	1.2	对数正态
$x 4$	50	10.0	极值Ⅰ型

Table 2 Random variables distribution of example 3

方法	可靠概率	可靠度指标 $β$	相对误差/%
MCS法	0.956 4	1.709 8
FORM	0.961 0	1.761 8	0.481 0
SORM	0.957 1	1.717 7	0.073 2
本文方法	0.956 5	1.711 7	0.001 0

Table 3 Reliability analysis results of example 3

Fig.7 Schematic of roof truss structure

变量	单位	均值	标准偏差	分布类型
$q$	N/m	20 000	1 400	正态
L	m	12	0.12	正态
$A s$	m²	$9.82 × 10 - 4$	$5.985 ? 2 × 10 - 5$	正态
$A c$	m²	0.04	0.004 8	正态
$E s$	Pa	$1.0 × 1011$	$6.0 × 109$	正态
$E c$	Pa	$2.0 × 1010$	$1.2 × 109$	正态

Table 4 Random variables distribution of example 4

方法	可靠概率	可靠度指标 $β$	相对误差/%
MCS法	0.990 5	2.344 5
FORM	0.992 3	2.421 2	0.181 0
SORM	0.991 7	2.397 3	0.128 1
本文方法	0.991 6	2.392 0	0.111 1

Table 5 Reliability analysis results of example 4


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