Optimization of inspection schedule for a surface-breaking crack subject to fatigue loading
Introduction
In recent years, the damage-tolerance design approach to fatigue has supplanted more traditional approaches like infinite-life design and safe-life design (see e.g. Grandt [1]). The damage-tolerance design assumes that the component has some initial damage and deals with the ability of the component to resist a specific amount of damage for a given period of service. In a generalized sense, the successful implementation of the damage-tolerance approach depends on the following factors: (i) predicting macrocrack initiation, (ii) modeling of the macrocrack growth, and (iii) component inspection. In current applications, the predictions of macrocrack initiation are generally left out of consideration. However, a detailed discussion of measuring and quantifying pre-crack fatigue damage and predicting the time for macrocrack initiation has been given by Kulkarni et al. [2]. Modeling of the crack growth is based on the determination of the stress-intensity factor and on establishing a suitable crack growth law. Component inspection deals with nondestructive techniques to detect cracks and with scheduling of inspections. A probabilistic approach to quantifying imperfect inspections, which is relevant to the current paper, has been discussed by Kulkarni and Achenbach [3]. In scheduling inspections, one needs to account for the fact that early in the life cycle the crack growth rates and the crack sizes are small, and hence the chances of cracks being detected by the inspection technique are small. To derive maximum benefit from the damage-tolerance approach, a sufficient but not an excessive number of inspections must be scheduled based on their POD (probability of detection) and the expected crack growth characteristics. This problem of scheduling inspections in the context of implementing the damage-tolerance approach has attracted the attention of researchers, and a brief literature review is presented next.
Yang and Trapp [4] appear to have first formulated the problem of the determination of the optimum inspection frequency as a constrained minimization problem. They indicated that various variables including inspection frequency and inspection quality can be adjusted in such a way so as to minimize a pre-defined cost function. Mizutani and Fujimoto [5] presented a sequential minimization method which aims to find an optimal inspection strategy so that the total cost expected in the period between the present inspection and the next is a minimum. Recently, Tanaka and Toyoda-Makino [6] have theoretically investigated the optimal scheduling of a single inspection based on minimizing a cost function. Random crack growth is accounted for by the diffusive crack model. This work was later extended by Toyoda-Makino [7] to the case of scheduling multiple inspections. All the studies mentioned here assume that the component is replaced if a crack, however small, is detected during the inspection. Naturally, repair of the component is the other option. Lifetime optimization methodology for planning the inspection and repair of structures that deteriorate over time due to factors other than fatigue has also been developed; see Frangopol et al. [8], who applied it to the case of T-girders of a highway bridge that are deteriorating due to corrosion. In this context we should also mention the development of the PROF software by Berens et al. at the University of Dayton Research Institute [9]. In principle this software has the capability to perform similar calculations, with a different algorithm, and without optimization calculations, but with the capability to account for the repair of detected cracks.
For safety-critical components the present practice is to replace a component once a crack has been detected. This paper presents, however, a framework for scheduling inspections which assumes an understanding of crack growth, and accounts for the case that a component may be allowed to continue in service when a crack is detected. Replacement or repair is implemented only if the detected crack exceeds a certain pre-defined size. We also allow for the fact that the crack depth density in a replaced component need not be the same as the crack depth density in a virgin component. In Section 2 we begin our discussion by first describing the problem under consideration along with the growth law used to model the crack growth. In the next section we briefly discuss the model used to describe the inherent uncertainty of an inspection process. The main contribution of the paper is presented in Section 4, where we discuss a framework to optimize the inspection schedule of a component or structure. Expressions for the efficient implementation of the proposed strategy are presented. Numerical results obtained for a sample problem are shown and discussed in Section 5. Finally, conclusions based on the numerical results and general remarks regarding the framework presented are given in the last section.
Section snippets
Modeling of crack growth
To proceed, we need to assume a suitable model for the crack growth which accounts for the variabilities observed in fatigue crack growth. Various such models are available in the literature (see e.g. Sobczyk and Spencer [10]). In the present case, we have represented the crack growth by a nonlinear differential equation with fixed parameters and have introduced the variabilities only through the random size of the macrocrack which appears at the end of the initiation stage.
We restrict our
Characterization of the nondestructive inspection technique
A common approach to model the inherent uncertainty in a nondestructive inspection technique to monitor crack size is to specify the probability of detection (POD) as a function of crack size. In the present paper the POD curves are assumed to be represented by the Log-Odds-Log scale model (see Barnes and Hovey [12]) where is the crack depth and and are regression parameters. Typical POD curves for three different inspection techniques labeled ‘A’, ‘B’ and ‘C’ are shown in
Optimization of the inspection schedule
We now present the framework for finding the optimum inspection schedule for a component subjected to fatigue loading. In the present case, the optimization is carried out by minimizing a predefined cost function which accounts for the cost due to failure, the cost of replacement and also the cost associated with each individual inspection. To mathematically formulate the cost function we make the following assumptions:
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The expected lifetime, , of the component has been estimated.
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The number
Numerical examples
We will now consider the example of a surface-breaking crack whose crack growth is modeled by the Paris law given by Eq. (2). The model parameters are taken as and (see Moran et al. [14]). We apply the methodology presented in the paper to this problem, to obtain an optimum inspection schedule. We consider an expected lifetime of the component, , of 200 000 cycles. The initial crack depth, , distribution is represented by a lognormal distribution with
Concluding remarks
An analytical method has been presented to optimize the inspection schedule for monitoring the growth of fatigue cracks. Explicit results are given for a two-dimensional surface-breaking crack. The point of departure of the analysis is a crack defined by a probability density of crack depths whose growth is governed by the Paris law, and an inspection technique with a known probability of detection. Two critical crack lengths are defined, a crack length at which a component will be replaced,
Acknowledgement
The work presented in this paper was carried out in the course of research funded by the Federal Aviation Administration, monitored by Jim White.
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