Time-variant Reliability Problems Research Articles

A Novel Time-Variant Reliability Analysis Method Based on Failure Processes Decomposition for Dynamic Uncertain Structures

AbstractDue to the uncertainties and the dynamic parameters from design, manufacturing, and working conditions, many engineering structures usually show uncertain and dynamic properties. This paper proposes a novel time-variant reliability analysis method using failure processes decomposition to transform the time-variant reliability problems to the time-invariant problems for dynamic structures under uncertainties. The transformation is achieved via a two-stage failure processes decomposition. First, the limit state function with high dimensional input variables and high order temporal parameters is transformed to a quadratic function of time based on the optimized time point in the first-stage failure processes decomposition. Second, based on the characteristics of the quadratic function and reliability criterion, the time-variant reliability problem is then transformed to a time-invariant system reliability problem in the second-stage failure processes decomposition. Then, the kernel density estimation (KDE) method is finally employed for the system reliability evaluation. Several examples are used to verify the effectiveness of the proposed method to demonstrate its efficiency and accuracy.

Principal component analysis and polynomial chaos expansion for time-variant reliability problems

Time-variant reliability analysis aims to evaluate the probability that an engineering system successfully performs its intended function throughout its service life. The model describing the system behavior should then include the time-variant uncertainties. This results in an even more computationally expensive model what makes the surrogate modeling a promising tool. While Kriging has been widely investigated in the literature, this work explores the Polynomial Chaos Expansion (PCE) in the time-variant reliability field. First, the time interval of study is discretized and an instantaneous performance function is associated to each time node. Afterwards, a principal component analysis is performed in order to represent all these functions with a reduced number of components that are then approximated using PCE. An adaptive algorithm is proposed to automatically enrich the experimental design until the target accuracy is reached. Finally, a global surrogate model of the time-variant response is obtained on which the Monte–Carlo simulation method can be easily applied. This allows to obtain the complete evolution in time of the probability of failure with a unique reliability analysis. Results show that the proposed method is suitable for high dimensional time-dependent problems considering non-stationary and non-Gaussian processes.

The effect of copulas on time-variant reliability involving time-continuous stochastic processes

In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated. This paper is focusing on time-variant reliability problems with continuous stochastic processes, which are collection of dependent random variables and to our knowledge are not modeled by other than Gauss copula in structural reliability. Therefore, the aim of this contribution is to qualitatively and quantitatively analyze the impact of this copula assumption on failure probability. Three illustrative examples are studied considering bivariate Gauss, t, rotated Clayton, Gumbel, and rotated Gumbel copulas. Time-variant actions are modeled as stationary, ergodic, continuous stochastic processes, and the PHI2 method is adopted for the analyses. The calculations show that the copula function has significant effect on failure probability. In the studied examples, application of Gauss copula can four times underestimate or even 10 times overestimate failure probabilities obtained by other copulas. For normal structures agreement on copula type is recommended, while for safety critical ones inference of copula type from observations is advocated. If data are scare, multiple copula functions and model averaging could be used to explore this uncertainty.

Confidence-based adaptive extreme response surface for time-variant reliability analysis under random excitation

Time-variant reliability analysis aims at revealing the time evolution of the reliability of an engineered system under time-dependent uncertainties that are best described by random processes. In practice, it is still a grand challenge to handle random process in time-variant reliability analysis due to the extremely high computational cost. In this work, a new adaptive extreme response surface (AERS) approach is proposed for time-variant reliability problems. With AERS, the dimensionality of a random process is first reduced to a set of standard normal variables and corresponding deterministic orthogonal functions based on spectral decomposition. As a result, the limit state function is reformulated as a function of only random variables and time. Next, Gaussian process (GP) models are constructed as surrogate models for predicting the value of limit state function at all discretized time nodes to approximate the extreme response surface. The accuracy of GP surrogate models is quantified by a confidence level measure and continuously improved through the sequential adaptive sampling. Using the GP surrogate models, time-dependent reliability is computed via Monte Carlo simulations (MCS). Two case studies are used to demonstrate the effectiveness of the AERS method for time-variant reliability analysis.

A Time-Variant Reliability Analysis Method Based on Stochastic Process Discretization

Time-variant reliability problems caused by deterioration in material properties, dynamic load uncertainty, and other causes are widespread among practical engineering applications. This study proposes a novel time-variant reliability analysis method based on stochastic process discretization (TRPD), which provides an effective analytical tool for assessing design reliability over the whole lifecycle of a complex structure. Using time discretization, a stochastic process can be converted into random variables, thereby transforming a time-variant reliability problem into a conventional time-invariant system reliability problem. By linearizing the limit-state function with the first-order reliability method (FORM) and furthermore, introducing a new random variable, the converted system reliability problem can be efficiently solved. The TRPD avoids the calculation of outcrossing rates, which simplifies the process of solving time-variant reliability problems and produces high computational efficiency. Finally, three numerical examples are used to verify the effectiveness of this approach.

Modeling random corrosion processes via polynomial chaos expansion

Polynomial Chaos Expansion (PCE) is widely recognized as a flexible tool to represent different types of random variables/processes. However, applications to real, experimental data are still limited. In this article, PCE is used to represent the random time-evolution of metal corrosion growth in marine environments. The PCE coefficients are determined in order to represent data of 45 corrosion coupons tested by Jeffrey and Melchers (2001) at Taylors Beach, Australia. Accuracy of the representation and possibilities for model extrapolation are considered in the study. Results show that reasonably accurate smooth representations of the corrosion process can be obtained. The representation is not better because a smooth model is used to represent non-smooth corrosion data. Random corrosion leads to time-variant reliability problems, due to resistance degradation over time. Time variant reliability problems are not trivial to solve, especially under random process loading. Two example problems are solved herein, showing how the developed PCE representations can be employed in reliability analysis of structures subject to marine corrosion. Monte Carlo Simulation is used to solve the resulting time-variant reliability problems. However, an accurate and more computationally efficient solution is also presented.

Analytical derivation of the outcrossing rate in time-variant reliability problems

The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows one to compute the outcrossing rate by solving a two-component parallel system reliability problem using the First Order Reliability Method (FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then, it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load.

Aleatory or epistemic? Does it matter?

The sources and characters of uncertainties in engineering modeling for risk and reliability analyses are discussed. While many sources of uncertainty may exist, they are generally categorized as either aleatory or epistemic. Uncertainties are characterized as epistemic, if the modeler sees a possibility to reduce them by gathering more data or by refining models. Uncertainties are categorized as aleatory if the modeler does not foresee the possibility of reducing them. From a pragmatic standpoint, it is useful to thus categorize the uncertainties within a model, since it then becomes clear as to which uncertainties have the potential of being reduced. More importantly, epistemic uncertainties may introduce dependence among random events, which may not be properly noted if the character of uncertainties is not correctly modeled. Influences of the two types of uncertainties in reliability assessment, codified design, performance-based engineering and risk-based decision-making are discussed. Two simple examples demonstrate the influence of statistical dependence arising from epistemic uncertainties on systems and time-variant reliability problems.

Mean points of failure and safety domains: Estimation and application

A conceptual representation of the mean points of failure and safety domains in reliability problems is presented, and a practical approach for approximating the mean points based on the first-order reliability method (FORM) is proposed. The effectiveness of the approach is demonstrated with two nonlinear examples. The mean points of failure and safety domains and the approach for approximating them are then used to develop an innovative method for estimating transition probabilities between failure and safety domains in discrete time-variant reliability problems. The performance of the new method is demonstrated in example problems. It is shown that, from a practical viewpoint, the method is quite effective. Although the example problems are stationary univariate systems, the method shows promise for being applicable for more complex cases, such as non-stationary multivariate systems.

Estimation of failure probabilities of linear dynamic systems by importance sampling

An iterative method for estimating the failure probability for certain time-variant reliability problems has been developed. In the paper, the focus is on the displacement response of a linear oscillator driven by white noise. Failure is then assumed to occur when the displacement response exceeds a critical threshold. The iteration procedure is a two-step method. On the first iteration, a simple control function promoting failure is constructed using the design point weighting principle. After time discretization, two points are chosen to construct a compound deterministic control function. It is based on the time point when the first maximum of the homogenous solution has occurred and on the point at the end of the considered time interval. An importance sampling technique is used in order to estimate the failure probability functional on a set of initial values of state space variables and time. On the second iteration, the concept of optimal control function can be implemented to construct a Markov control which allows much better accuracy in the failure probability estimate than the simple control function. On both iterations, the concept of changing the probability measure by the Girsanov transformation is utilized. As a result the CPU time is substantially reduced compared with the crude Monte Carlo procedure.

Time-Variant Reliability Assessment of FPSO Hull Girder With Long Cracks

An efficient time-variant reliability formulation for the safety assessment of an aging floating production storage and offloading (FPSO) vessels with the presence of through-thickness cracks (i.e., long cracks), is presented in this paper. Often in ship structures, cracks are detected by means of close visual inspection when they have already propagated through the thickness. The propagation of long cracks in stiffened panels is therefore considered, as they may be present in critical details of the deck and/or bottom plating of the vessel. Although it has been found that stiffened panels are tolerant to fatigue cracking, the safety of such structural components with the presence of long cracks may be threatened when exposed to overload extreme conditions, i.e., brittle or ductile fracture may occur. The probability of brittle fracture of an aging hull structure, i.e., a stiffened panel at the bottom plating with the presence of long cracks is studied in this paper. The mean stress effect due to the continuously varying still-water loading as well as residual stresses is explicitly accounted for in the crack growth calculation procedure presented herein. An analytical model is established for determining the equivalent long-term stress range including the mean stress effect. The continuously varying still-water load effects due to the operational nature of FPSOs introduce additional uncertainties in the estimation of fatigue damage as well as in the likelihood of fracture failure mode. In the present case study it is found that the time-invariant approach is a good approximation when dealing with the time-variant reliability problem. One of the main conclusions drawn from this study is that the still-water mean stress has a significant effect on the failure probabilities of stiffened panels with long cracks under brittle fracture mode.

Barrier failure dominance in time variant reliability analysis

In this paper, an important feature of time variant reliability problems is identified. The concept of barrier failure dominance (BFD) is introduced to characterize those problems where an out-crossing or overload failure is more likely to be caused by a very small realization of the barrier (resistance) than by an exceptionally large realization of the load process. This result arises from a study of the ensemble out-crossing rate approximation, reported in a previous paper and which is reviewed herein. It is shown how the notion of BFD can be used to understand the limits of application of the ensemble out-crossing rate approximation. It is also suggested that the acknowledgement of BFD can support the choice of the appropriate simplified solution method for particular problems. A three-dimensional load combination and a resistance degradation problem are studied as examples. The ensemble up-crossing rate approximation is compared with load process-based approximations, using the time-integrated approach, the load combination rule of Turkstra and the point-crossing formula.

Reliability of Uncertain Inelastic Structures under Earthquake Excitation

The assessment of safety for structures subjected to earthquake excitation is a time-variant reliability problem whose practical solution, when the structure behaves inelastically and uncertainty in the mechanical parameters is taken into account, is traditionally sought by means of simulation. In this paper the feasibility of an alternative solution based on the first order reliability method is explored. The problem is expressed in terms of first-excursion event within a conceptual framework, established in recent years, in which random vibration problems are discretized to make them tractable by the well-established methods of time-invariant structural reliability. The necessary extensions to apply the methodology to reinforced concrete frame structures are presented. Applications to a simple spring-mass system and to a complete three-dimensional bridge structure are presented.

On the ensemble crossing rate approach to time variant reliability analysis of uncertain structures

The ‘ensemble’ up-crossing rate technique consists of averaging the rate at which a random load process up-crosses a deterministic barrier level over the resistance distribution at successive time points. Averaging over the resistance makes the assumption of independent up-crossings less appropriate. As a result, first passage failure probabilities may become excessively conservative in problems with other than extremely low failure probabilities. The ensemble up-crossing rate technique has a significant potential in simplifying the solution of time variant reliability problems under resistance degradation. However, little is known about the quality of this approximation or its limits of application. In the paper, a Monte Carlo simulation-based methodology is developed to predict the error in the approximation. An error parameter is identified and error functions are constructed. The methodology is applied to a range of time-invariant and time-variant random barriers, and it is shown that the error in the original ensemble up-crossing rate approximation is largely reduced. The study provides unprecedented insight into characteristics of the ensemble up-crossing rate approximation.

The PHI2 method: a way to compute time-variant reliability

Time-variant reliability problems appear in the engineering practice when (a) the material properties of the structure deteriorate in time or (b) random loading modelled as random processes is involved. This paper presents a method called PHI2 which is based on the outcrossing approach and allows to solve such problems using classical time-invariant reliability tools such as FORM/SORM methods. The PHI2 method is first presented. Then it is benchmarked with the well-established ‘asymptotic methods’ [Stochast. Process. Appl. 13 (1988) 195; J. Offshore Mech. Arctic Engng 113 (1991) 241; Probab. Engng Mech. 10 (1995) 53; J. Struct. Engng 25 (1998) 1] on three examples dealing with scalar or vector processes and linear or non-linear limit state functions. The PHI2 method appears more accurate in all cases. As an application example, the method is finally applied on a case representing a mechanical system (a beam) placed in an environment that can have exceptional configuration.

On the failure probability of pipings

Various methods for determining the structural reliability analysis of piping systems of NPP's are discussed in view of their accuracy, efficiency and possibility of practical application. Ultimate load as well as fatigue failure modes are considered in the analysis. The time variant reliability problem, e.g. due to fatigue and/or corrosion is solved by utilizing advanced simulation procedures.