Reliability Analysis by Mean-Value Second-Order Expansion

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Deshun Liu ◽  
Yehui Peng
Keyword(s):  
Saddle Point ◽  
Limit State ◽  
Quadratic Function ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Limit State Function ◽  
Chi Square ◽  
Normal Distributions ◽  
Independent Variables

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).

Second-Order Reliability Method With First-Order Efficiency

2010 ◽  
Author(s):  
Xiaoping Du ◽  
Junfu Zhang
Keyword(s):  
Limit State ◽  
Saddlepoint Approximation ◽  
Quadratic Function ◽  
Second Order ◽  
State Function ◽  
Limit State Function ◽  
First Order ◽  
Reliability Method ◽  
Order Of Magnitude ◽  
Univariate Function

The widely used First Order Reliability Method (FORM) is efficient, but may not be accurate for nonlinear limit-state functions. The Second Order Reliability Method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second order reliability analysis method with first order efficiency. The method first performs the FORM and identifies the Most Probable Point (MPP). Then the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated as a quadratic function, which is created with the gradient information at the MPP and one more point near the MPP. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied for computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

A Second-Order Reliability Method With First-Order Efficiency

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Junfu Zhang ◽  
Xiaoping Du
Keyword(s):  
Limit State ◽  
Saddlepoint Approximation ◽  
Quadratic Function ◽  
Second Order ◽  
State Function ◽  
Limit State Function ◽  
First Order ◽  
Reliability Method ◽  
Order Of Magnitude ◽  
Univariate Function

The first-order reliability method (FORM) is efficient but may not be accurate for nonlinear limit-state functions. The second-order reliability method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second-order reliability analysis method with first-order efficiency. The method first performs the FORM to identify the most probable point (MPP). Then, the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated by a quadratic function. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied in computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

Risk Based Inspection Methodology for Components Subject to High-Temperature Creep

2017 ◽  
Author(s):  
Carl E. Jaske ◽  
Panos Topalis ◽  
Wong Sin Loong ◽  
Azura Sharina Md Sidek
Keyword(s):  
Limit State ◽  
Creep Damage ◽  
Mean Value ◽  
Pressure Vessels ◽  
State Function ◽  
Limit State Function ◽  
First Order ◽  
First Order Second Moment ◽  
Time In Service

Risk-based inspection (RBI) methodologies are widely used by industry to develop effective inspection programs for pressure vessels and piping. The RBI approach use data on equipment design, maintenance, and operation along with inspection history to evaluate both the likelihood and consequences of failure. RBI results provide a basis for selecting inspection methods and establishing inspection intervals and coverage. API RP 580 provides guidance on developing a RBI program for fixed equipment and piping, while API RP 581 provides quantitative procedures for establishing RBI methodology. Appendix J of the first edition (2000) of API RP 581 contained procedures for application to creep damage of furnace tubes. However, the second (2008) and third (2016) did not contain any procedures for application to creep damage of equipment, including furnace tubes. DNV GL undertook a RBI project for a coal-fired power plant in Malaysia that required evaluation of components subject to creep damage. As part of this project, a detailed likelihood of failure (LoF) model for creep was developed. This paper reviews the creep LoF model that was developed and discusses a case study of its application. The LoF is estimated using a limit state function where the resistance is characterized using Larson-Miller parameter creep-rupture expressions for the materials of interest and the load is characterized by the time in service. A mean value first order second moment (MVFOSM) method is employed to numerically compute LoF. Guidelines for including metallurgical replication results in the LoF estimate and for assigning inspection effectiveness for creep damage also are discussed.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2017 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer is based on a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first and second-order sensitivity derivatives of the limit state cumulant generating function with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are computed using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. The results are compared with the available mean value first-order saddlepoint approximation (MVFOSA) method and Monte Carlo simulation. Finally, the proposed RBTO-MVSOSA method for minimizing compliance-based probability of failure, is demonstrated using two 2D beam structures under random loading.

scholarly journals Second Order Reliability Method for Time-Dependent Reliability Analysis Using Sequential Efficient Global Optimization

2019 ◽  
Author(s):  
Zhangli Hu ◽  
Xiaoping Du
Keyword(s):  
Global Optimization ◽  
Reliability Analysis ◽  
Limit State ◽  
Second Order ◽  
Time Dependent ◽  
Efficient Global Optimization ◽  
State Function ◽  
Limit State Function ◽  
Reliability Problem ◽  
Reliability Method

Abstract Reliability depends on time if the associated limit-state function includes time. A time-dependent reliability problem can be converted into a time-independent reliability problem by using the extreme value of the limit-state function. Then the first order reliability method can be used but it may produce a large error since the extreme limit-state function is usually highly nonlinear. This study proposes a new reliability method so that the second order reliability method can be applied to time-dependent reliability analysis for higher accuracy while maintaining high efficiency. The method employs sequential efficient global optimization to transform the time-dependent reliability analysis into the time-independent problem. The Hessian approximation and envelope theorem are used to obtain the second order information of the extreme limit-state function. Then the second order saddlepoint approximation is use to evaluate the reliability. The accuracy and efficiency of the proposed method are verified through numerical examples.

Reliability-Based Design Using Simulation for Aluminum Elements

2007 ◽  
Author(s):  
Malek Brahimi ◽  
Sidi Berri ◽  
Joel Lopez
Keyword(s):  
Current Practice ◽  
Limit State ◽  
Good Method ◽  
Probability Of Failure ◽  
State Function ◽  
Limit State Function ◽  
Direct Simulation ◽  
Safety Factors ◽  
Normal Distributions ◽  
Reliability Based Design

Studies of reliability in current practice indicate that reliability based on conventional methods requires a nonlinear transformation to a set of normal distributions, which effectively changes the shape of limit state function. In this paper, the general formulation of safety for aluminum elements and the associated methods of analysis are reviewed. Direct simulation is used to find the probability of failure. It is concluded that direct simulations of safety of aluminum elements of Pr (probability of failure) by failure counting is a good method to achieve acceptable safety factors.

An Efficient Response Surface Method Using Givens Transformation for Structural Reliability Analysis

2012 ◽  
Vol 532-533 ◽  
pp. 408-411
Author(s):  
Wei Tao Zhao ◽  
Yi Yang ◽  
Tian Jun Yu
Keyword(s):  
Response Surface ◽  
Response Surface Method ◽  
Structural Reliability ◽  
Limit State ◽  
State Function ◽  
Limit State Function ◽  
Surface Method ◽  
Mathematical Techniques ◽  
Input Variables ◽  
Improved Accuracy

The response surface method was proposed as a collection of statistical and mathematical techniques that are useful for modeling and analyzing a system which is influenced by several input variables. This method gives an explicit approximation of the implicit limit state function of the structure through a number of deterministic structural analyses. However, the position of the experimental points is very important to improve the accuracy of the evaluation of failure probability. In the paper, the experimental points are obtained by using Givens transformation in such way these experimental points nearly close to limit state function. A Numerical example is presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the classical response surface method. As seen from the result of the example, the proposed method leads to a better approximation of the limit state function over a large region of the design space, and the number of experimental points using the proposed method is less than that of classical response surface method.

Development of Ultimate Strength Evaluation Method for Piping Subjected to Seismic Loads

2013 ◽  
Author(s):  
Hideo Machida ◽  
Hiromasa Chitose ◽  
Tatsuhiro Yamazaki
Keyword(s):  
Power Plants ◽  
Evaluation Method ◽  
Failure Modes ◽  
Limit State ◽  
Seismic Loading ◽  
Earthquake Resistance ◽  
State Function ◽  
Limit State Function ◽  
Input Acceleration ◽  
Resistance Tests

This paper reports the results of the study on the failure modes and limit loads of piping in nuclear power plants subjected to cyclic seismic loading. By investigating the past fracture tests and earthquake resistance tests, it became clear that dominant failure mode of piping was fatigue, and the effect of ratchet strain was negligible. Until now, the stress generated with the acceleration of an earthquake was classified into the primary stress. However, the relationship between the input acceleration and the seismic response displacement of the pipe observed from earthquake resistance tests is non-linear, and increasing rate of displacement is lower than that of input acceleration in elastic-plastic stress condition. Therefore, the seismic loading can be treated as displacement controlled loading. To evaluate the reliability-based critical acceleration, a limit state function was defined taking the variations in the fatigue strength or some parameters into consideration. By using the limit state function, the reliability was evaluated for the typical piping of boiling water reactor (BWR) plants subjected to cyclic seismic loading, and a partial safety factors were calculated. Based on these results, a fatigue curve corresponding to the target reliability was proposed.

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