Time-Variant Reliability Formulated as First Passage Problem

The paper describes a new practical method of updating the stochastic remnant life of pipelines with defects using ILI data. The paper describes a comprehensive algorithm for assessing pipeline remnant life taking into account the stochastic results of in-line inspection (ILI). It is assumed that the pipeline segment wall has a longitudinal external crack of semi-elliptical form and is described by the J-integral. The limit state function (LSF) is described as the difference of the critical and current value of the J-integral. The latter is calculated for the current time of pipe performance and is assumed known due to monitoring of the pumping equipment. The critical crack depth is defined using the notion of fracture toughness and the J-integral approach. The algorithm contains solutions of three sequentially interconnected problems. First, the deterministic problem of fatigue crack growth (FCG) is analyzed. Then the stochastic FCG is analyzed. The probability of failure assessment algorithm is designed on the basis of the authors’ version of the adaptive important sampling (AIS) procedure. The main steps of the AIS algorithm are described in detail. The samples are generated in such a way, that at all times a majority of samples belong to the fracture region. Finally, the results of the latest ILI are fused into the algorithm, providing best possible assessment of pipeline remnant life as a random variable. The remnant life update for pipeline segment with crack-like defects using ILI data takes into account three possible outcomes: defect not discovered: defect is discovered but not measured; defect is discovered and measured. This result permits solving most important problems of pipeline maintenance: prioritization of pipeline segments for repair/rehabilitation; optimization of the time between ILI; minimization of pipe operational risk. Two real cases are described of assessing the probability of fracture/leak of a pipeline section with an external crack at different periods of its performance. The described approach currently is being generalized for the case of multiple stress corrosion SC cracks.

In this paper, an efficient adaptive nonlinear algorithm for estimation and identification, the so-called adaptive Lainiotis filter (ALF), is applied to the problem of fatigue crack growth (FCG) estimation, identification, and prediction of the final crack (failure). A suitable nonlinear state-space FCG model is introduced for both ALF and extended Kalman filter (EKF). Both algorithms are tested in order to compare their efficiency. Through extensive analysis and simulation, it is demonstrated that the ALF has superior performance both in FCG estimation, as well as in predicting the remaining lifetime to failure. Furthermore, it is shown that the ALF is faster and easier to implement in a parallel/distributed processing mode, and much more robust than the classic EKF.

Formulations are presented for estimating probability of failure considering uncertainty of distribution parameters in time invariant and time variant reliability analyses. Based on the formulations the probability of failure can be calculated by recursively using the first order reliability method. Also, a more efficient approximate analysis procedure by using the point estimate method to estimate the probability of failure is given. In this analysis procedure, the point estimate method is used to discretize the uncertain distribution parameters in the time invariant reliability analysis, and to discretize the time-independent random variables and uncertain distribution parameters in the time variant reliability analysis. The probability of failure is then obtained by weighting the probability of failure conditioned at each of discrete points. The conditional probability of failure can be calculated by using first order reliability method, second order reliability method or any other convenient ...

Fatigue crack growth prediction method based on machine learning model correction

The crack energy density was proposed by the author as one of the most important crack parameters, of which the physical meaning is clear throughout the life of a crack. In this paper, this crack energy density is applied to the problem of fatigue crack growth and it is tried to establish it as the parameter which is available from the first to the last of the life of fatigue crack. The results are as follows ; (1) The role of crack energy density as a crack parameter is made clear. (2) The relationship between the range of crack energy density, Δε and the load-displacement curves of specimens is obtained and it is shown that Δε is useful as a fatigue crack parameter throught the life. (3) It is shown that the growth rate of fatigue crack is determined by the increment of crack energy density per one cycle, Δεt<cycle> and the gradient of the distribution of crack energy density.

Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium

Continuous Crack Growth Monitoring and Residual Life Prediction under Variable-amplitude Loading Conditions

The results of analytical and experimental researches concerning predicting of fatigue crack growth in the operating conditions are presented. First of all the main factors causing a fatigue damage initiation and growth are analyzed and divided to two groups. Common conditions of fatigue damage precise predicting are established. The problem of fatigue crack growth at the stresses of variable amplitude was analyzed and an approach of description of this process is performed. Two examples present the efficiency of this approach. Theory of fatigue crack growth indication and the crack growth indicator (CGI) are developed. There is planned and executed a flight experiment using CGI located on two aircraft An-24 and An-26. Results of crack growth in CGI at operational load allowed to evaluate the parameters of generalized Paris-Erdogan law and statistical properties of crack increment per flight.

In this paper a torque estimation model for induction machines is studied. The model requires several machine parameters as input. Since the input parameters are deduced from measurements burdened with inaccuracy, it is necessary to take into account uncertainty in the input of the model. A first order and second order reliability method are performed to evaluate the probability of failure of the system under the uncertain inputs. The system failure is defined through the limit state function and occurs when the calculated value of the torque differs significantly from the rated torque at rated conditions. Monte Carlo simulations are performed for validation and importance sampling is discussed to reduce the Monte Carlo simulation error. The probability of failure is also derived from the torque probability density function by means of a stochastic response surface method. The analysis includes current and speed measurement uncertainties as well.

An investigation was performed to study the effect of uncertainty in applied loads, helix angles, and failure stresses on the probability of failure of composite cylinders. Also, the effect of the factor of safety on the probability of failure of the cylinder was studied. The probability of failure for a composite cylinder subject to axisymmetric loading was obtained using the First Order Reliability method (FORM). The failure of the cylinder was studied based on first ply failure. Ply failure was predicted using the non-smooth Hashin criterion. Hashin criterion is based on writing a different interaction equation for separate failure modes. Four failure modes are considered associated with matrix or fiber failure, in either tension or compression. The accuracy of FORM as applied to this problem was verified by comparing the FORM results to those obtained by Monte-Carlo Simulation method with importance sampling. i x µ = mean values of the design parameters i x

The paper presents an efficient importance sampling approach to the Monte Carlo solution of first-passage problems. Based on Girsanov’s theorem it is shown that an appropriate importance sampling density can be constructed utilizing “design point excitations” which are obtained by a procedure analogous to the first order reliability method. The presented approach can efficiently handle non-white and non-stationary excitations can be applied to a wide range of nonlinear problems, and yields high levels of confidence at very low numbers of samples.

Methodologies to calculate failure probability and to estimate the reliability of fatigue loaded structures are developed. The applicability of the methodologies is evaluated with the help of the fatigue crack growth models suggested by Paris and Walker. The probability theories such as the FORM (first order reliability method), the SORM (second order reliability method) and the MCS (Monte Carlo simulation) are utilized. It is found that the failure probability decreases with the increase of the design fatigue life and the applied minimum stress, the decrease of the initial edge crack size, the applied maximum stress and the slope of Paris equation. Furthermore, according to the sensitivity analysis of random variables, the slope of Pairs equation affects the failure probability dominantly among other random variables in the Paris and the Walker models.

The importance of design robustness and reliability is a well-established notion and practice in today’s industry. Manufacturing companies strive to achieve Six-Sigma quality measures. While Virtual Prototyping is a key-factor in accelerating the product development process while reducing development costs; it has not contributed in the quest for improved product reliability and robustness performance since it is based on deterministic approaches. This paper provides a systematic approach to design for sixsigma simultaneously addressing variability and uncertainty present in real life on most design parameters. The first step is the ability to flexibly integrate and federate best-of-class virtual prototyping simulation software programs -most based on Finite Element Modeling techniques- thus, capturing existing engineering processes. These engineering processes provide the deterministic calculation engines for assessing design performance outputs of interest, such as -but not limited to- stress, structural impact, eigenfrequencies, forced response functions, fatigue life. Second is the assessment of design parameter variability on the design performance outputs. Analytical methods for the efficient calculation of dynamic response variability examined in this work include reliability-based methods such as First and Second Order Reliability methods, the Importance Sampling Monte Carlo (ISMC) method and the Transformed Importance Latin Hypercube Sampling (TILHS) method. They have been assessed for their ability to accurately predict the failure probability in a reasonable amount of time. The latter criterion is paramount for successful application to large industrial FE models, where the computation time is the main bottleneck. The accuracy has been examined with the comparison of the aforementioned methods with classical and advanced Monte-Carlo methodologies. Uncertainty is treated with the use of Transformation Method based on Fuzzy arithmetic concepts. Fuzzy arithmetic is an attractive mathematical concept to solve problems with uncertain parameters. The Transformation Method (TM) follows a fuzzy approach based on the concept of α-cuts, i.e. discretization of the parameter’s magnitude into a number of intervals. Uncertainty is then assessed by solving the problem in a systematic way: for each combination of the values attained by the uncertain parameters a deterministic computation of the response is performed. In this paper, the TM is demonstrated on an airplane FE model, to assess the effect of material stiffness uncertainty on the FRF from wing tip to pilot seat. A second demonstration example on this model aims to ensure a certain frequency difference between two free-free modes that may result in flutter at certain air speed; several reliability methods are applied for this purpose, namely Monte Carlo simulations, FORM and Importance Sampling.

Tether system is a critical component for the TLPs, since its failure may lead to the collapse of the whole structure involving human lives, economic losses and damages to the environment. Due to this fact, reliability methods have been proposed to design TLP tethers and new codes are being developed to increase their safety level. The objective of this paper is to compare the probability of failure for TLP tethers considering the maximum tension limit state obtained with three methods, which are: a methodology based on Evolutionary Strategies and the Monte Carlo Importance Sampling, the First Order Reliability Method, and the Second Order Reliability Method. Von-Mises failure criterion is used as limit state function for the most loaded tether of a TLP submitted to different sea states. Efficiency of the ES algorithm to find design points and probabilities of failure obtained with the reliability methods are discussed.

Reliability analysis of laminated composite structures using finite elements and neural networks