Efficient method for estimating sensitivities of the reliability of fatigued structures with respect to pod curve parameters
Fatigue life predictions are obtained by using probabilistic fracture mechanics methods to account for variability in input variables such as initial crack size, crack growth rates, loading, material properties, inspection methods, and inspection times. These life predictions are also known as the system response. Life predictions are used to help quantify the structural reliability of a component due to fatigue by determining: (1) the probability that the component will fail before an observed life, known as probability-of-failure (POF), and (2) the mean and standard deviation of the response. In addition, a probabilistic sensitivity analysis of the POF and response moments is often conducted to help determine how parameters of random variables affect failure. Furthermore, POF calculations that account for inspections using probability of detection (POD) curves have been developed and incorporated into the analysis to assess the benefits of inspection and to optimize the inspection schedule. Dye penetrant, ultrasonics, radiography, and eddy currents are among the common inspection techniques used today in industry. Each inspection technique has a POD curve describing the probability of detecting a crack versus a certain size (length or area). Although calculations of the POF and system response for fatigue analysis and their sensitivities with respect to parameters of random variables are found in common practice, sensitivities of the POF and system response with respect to the parameters of a probability of detection (POD) curve are not. As a result, a methodology is derived here that provides a convenient means to estimate the sensitivities of the probability of detecting and not detecting, POF, mean and standard deviation of the system response with respect to parameters of a POD curve. The sensitivities provide a convenient and low cost method to assess the potential changes in the inspection method. An attractive feature of this methodology is that the sensitivities are computed as a post-processing by-product of the probabilistic analysis. As a result, the computational cost is negligible. The methodology is general and not specific to any particular form of the POD curve as long as it can be represented parametrically. The methodology scales to any number of inspections and can be integrated with existing sampling methods, such as Monte Carlo and Latin hypercube, as a post-processing option. Thus, it can be integrated easily with commercial software such as NASA's crack growth code (NASGRO) and Air Force Growth (AFGROW). Several numerical examples are presented and show excellent agreement with finite difference estimates with significant computational savings.