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Special section: Methods for Uncertainty Computing Either Uncertainty Propagation or Optimization Under Uncertainty

# A Novel Second-Order Reliability Method (SORM) Using Noncentral or Generalized Chi-Squared Distributions

[+] Author and Article Information
Ikjin Lee

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139ilee@engr.uconn.edu

Yoojeong Noh

Department of Mechanical and Automotive Engineering, Keimyung Universiy, Daegu, Korea 704-701romana79@kmu.ac.kr

David Yoo

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139david.yoo@engr.uconn.edu

J. Mech. Des 134(10), 100912 (Sep 28, 2012) (10 pages) doi:10.1115/1.4007391 History: Received January 12, 2012; Revised May 29, 2012; Published September 21, 2012; Online September 28, 2012

## Abstract

This paper proposes a novel second-order reliability method (SORM) using noncentral or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point in standard normal U-space, (2) error due to approximating the quadratic function in U-space by a parabolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only, which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two types of errors by describing the quadratic failure surface with the linear combination of noncentral chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function over the linear failure surface, and the second approach uses the cumulative distribution function of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method, conventional SORM, and Monte Carlo simulation results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

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## Figures

Figure 1

Performance function in original and transformed space

Figure 2

PDF and CDF of linear combination Q for Eq. 40

Figure 3

Comparison of conventional and proposed SORM

Figure 4

Performance function in transformed space

Figure 5

Comparison of distribution function of Q

Figure 6

Comparison of distribution function of G(X) in Eq. 58

Figure 7

Cantilever tube

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