The cross-entropy method for reliability assessment of cracked structures subjected to random Markovian loads
Introduction
The design philosophy of damage tolerant structures is most often based on the application of conservative assumptions and empirical safety factors. The mainly deterministic applied rules and standards are expected to ensure a safe design even though the associated risk is not quantified. Probabilistic approaches in the field of damage tolerance have therefore gained an increasing attention over the past few years in an attempt to get a clearer assessment of the safety margins and to address the limits of such deterministic practices [1], [2], [3]. A probabilistic approach of damage tolerance in its most general form requires efforts at the following major levels for meaningful results:
- 1.
An appropriate fatigue crack growth model needs first to be selected or established in accordance with the level of complexity of the structural problem to be solved and the accuracy expected on the solution.
- 2.
Realistic stochastic models are also required for properly describing the randomness of the input physical parameters. They should be preferably based on observed data in sufficient number, if available, or on experts’ judgment otherwise.
- 3.
The modeled uncertainties are then propagated through the selected fracture model. The random outputs could represent the fatigue lifetime, the final crack size, the stress intensity factor, etc. These results are often the inputs of a given failure criteria and it is therefore of interest to estimate the failure probability, i.e. the probability to unsatisfy the given performance criteria. Such analyses require the development or application of efficient reliability methods.
- 4.
Reliability results of the damage tolerance analysis performed in the random context such as sensitivities and failure probability may finally be incorporated in the design and maintenance processes of structures, with an effort to reduce costs while preserving safety.
The present paper focuses on the third level of such a comprehensive approach. We assume here that the solutions of the first two levels are available based on previous works of the authors [4], [5]. Crack propagation under variable amplitude loads is assessed with the PREFFAS crack closure model proposed by Aliaga et al. in 1985 [6], which accounts for interaction effects between cycles. Fatigue loads are considered as random and they are assumed to be represented by a discrete-time process. In the above cited works, these loads are modeled either with First-order Markov Chains (FMC) or Hidden Markov Models (HMM), whose parameters are identified from in-flight measured loads of a military aircraft fleet.
The first objective of the paper is to set up a general formulation of the reliability problem of a cracked structural component with uncertain material properties and initial crack size, subjected to random amplitude loads. The resulting reliability problem is not trivial to solve: it involves a random number of discrete/continuous random variables due to the use of discrete-time Markov processes of random length and it is shown moreover that it is a time-dependent reliability problem. This therefore precludes a recourse to classical reliability problem solving techniques well known in the structural reliability community. For the sake of tractability, a problem of a lesser complexity is derived. It is proposed here to address the scatter in loads only, modeled by discrete-time Markov processes. Uncertainties in material properties and initial crack size are therefore discarded for this purpose.
The second and main objective of the work is to propose an efficient method to solve the damage tolerance reliability problem in which loads are modeled with random Markov processes. It is assumed that failure is a rare event (case of structures with a high level of safety) and it is expected that the proposed solution is able to accurately assess such low failure probabilities at a computational expense much lower than the one of a crude Monte Carlo simulation. For reaching such a purpose, the present work hinges on the Cross-Entropy method (CE method) introduced by Rubinstein in [7] for the estimation of probabilities of rare event and in [8] for combinatorial optimization. In the probability estimation setting, the CE method is a form of adaptive importance sampling and therefore belongs to the broad class of variance reduction techniques. The CE method was successfully applied to Markovian reliability systems for which failure is defined by the state of a Markov chain, as described in [9] for specific applications involving the failure of highly reliable repairable systems. The application of the CE method to failure of damage tolerant structures in the present work emanates from such successful applications to Markovian reliability systems. It is however worth mentioning that discrete-time Markov processes are not used here to describe the state of the system itself but an input quantity of the model used for simulating its behavior, namely the random fatigue load. The CE method is firstly applied to random loads modeled by FMC. In References [4], [5], HMM were found to be more accurate than FMC for modeling random fatigue loads in damage tolerance applications due to their ability to capture extreme load values via the Generalized Pareto Distribution (GPD). The CE method is therefore further extended to HMM with continuous state space which constitutes a novel approach from the authors' viewpoint.
The paper is organized as follows. The setup of the general formulation of the damage tolerance reliability problem is detailed in Section 2. This section also presents the simplified problem which is addressed here by considering the sole scatter in loads modeled by discrete-time Markov processes. The proposed methodology based on the CE method is then detailed in Section 3 as a solution to assess the failure probability of cracked structures subjected to random Markovian loads, either modeled by FMC or HMM. This methodology is subsequently applied to a structural component considered as representative of real engineering problems in Section 4. Results are compared to those obtained by crude Monte Carlo simulations and the strengths and limits of the proposed method are finally discussed.
Section snippets
Failure criteria
We assume here that failure is triggered by the unstability of the crack growth and we restrict our approach to the opening mode of the crack (mode I). Static failure is therefore defined when the crack driving force expressed in terms of Stress Intensity Factor (SIF) K exceeds the fracture toughness :
The tension load σ in Eq. (1) should be viewed here either as the peak stress of the nth fatigue loading cycle or a prescribed load such as the so-called limit load defined by aviation
Importance Sampling
Importance Sampling (IS) represents an alternative technique to crude Monte Carlo Simulation (MCS) aiming at reducing the variance of the failure probability estimator for a given computational budget (number of samples). The key idea consists in generating samples which preferably populate the failure domain and which therefore contribute to the estimation of the failure probability Pf. The original pdf pX is replaced by an instrumental pdf qX which must dominate :
Reliability analysis of cracked structures under Markovian loads
In this section, the CE method is applied to the simplified reliability problem with random fatigue loading described in Section 2.4.2. The accuracy, efficiency and robustness of the CE method are assessed over simple crack growth applications supposed representative of real cracked aircraft components. The results are compared to those of MC simulations considered as references.
Conclusion and perspectives
This paper addresses the reliability analysis of structural aircraft components under random Markov fatigue loads. A first part of the paper presents a formulation of the problem with aleatory uncertainties taken in a general context, including initial crack size, material properties and sustained fatigue load uncertainties. It is shown that the reliability problem is time-dependent in its most general formulation if we consider the exceedance of the residual strength at each cycle of the
Acknowledgments
The work presented in this paper was financially supported by DGA which is gratefully acknowledged. They also express their gratitude to the editor and the two anonymous reviewers for their suggestions and comments.
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