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Research Papers

# Reliability Analysis by Mean-Value Second-Order Expansion

[+] Author and Article Information
Deshun Liu1

Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment,  Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. Chinaliudeshun@hnust.edu.cn

Yehui Peng

School of Mathematics and Computational Science,  Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. Chinapengyehui@yahoo.com.cn

1

Corresponding author.

J. Mech. Des 134(6), 061005 (Apr 27, 2012) (8 pages) doi:10.1115/1.4006528 History: Received April 30, 2011; Revised March 28, 2012; Published April 27, 2012; Online April 27, 2012

## Abstract

In this paper, two second-order methods are proposed for reliability analysis. First, general random variables are transformed to standard normal random variables. Then, the limit-state function is additively decomposed into one-dimensional functions, which are then expanded at the mean-value point to second-order terms. The approximated limit-state function becomes the sum of independent variables following noncentral chi-square distributions or normal distributions. The first method computes the probability of failure by the saddle-point approximation. If a saddle-point does not exist, the second method is then used. The second method approximates the limit-state function by a quadratic function with independent variables following normal distributions with the same variances. This treatment leads to a quadratic function that follows a noncentral chi-square distribution. These methods generally produce more accurate reliability approximations than the first-order reliability method (FORM) with 2n + 1 function evaluations, where n is the dimension of the problem. The effectiveness of the proposed methods is demonstrated with three examples, and the proposed methods are compared with the first- and second-order reliability methods (SROMs).

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## Figures

Figure 1

Flowchart of SMSA

Figure 2

K'Z(t): (a) C = 0 and (b) C = 100

Figure 3

Flowchart of SMNC method

Figure 4

A cantilever beam

Figure 5

CDF of y (Example 1)

Figure 6

CDF of y (Example 2)

Figure 7

Cantilever beam

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