Probabilistic damage tolerance analyses with inspections using the first- and second-order reliability analysis methods




Zaman, Shah Md Yasin

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Most reliability problems are characterized by small Probabilities-of-Failure (POF) that typically require a large number of Limit-State Function (LSF) evaluations. In many cases, due to the high cost computational cost of each evaluation of an LSF, it is often desirable to limit the number of evaluations while keeping the accuracy of the POF as high as possible. The trade-off between the cost and accuracy is still a challenge. In this research, the applicability of the First-Order Reliability Method (FORM) and the Second-Order Reliability Method (SORM) was tested in a probabilistic damage tolerance analysis methodology with and without non-destructive inspections. The purposes of the research were to identify limitations and apply improvements to algorithms of these methods in order to make them more robust and efficient when an external crack growth code is to be used in realistic applications. An efficient method, called 'Varying Step Lengths Method', was introduced in this research that can obtain suitable step lengths for gradients of the LSF by the finite difference method. Also, the bisection method was applied to find the next search point in the optimization algorithm of FORM analyses. In addition, the suitability of SORM analyses after FORM analyses was tested in this context. To estimate the POF, FORM assumes a linearized Limit-State Surface (LSS) instead of the actual surface at the design point in the standard normal space. The method computes the shortest distance of the linearized LSS from the origin, and estimates the POF as the probability content outside the hyper-plane using an attractively low number of LSF evaluations. The accuracy of the POF obtained from the FORM analysis can be improved by subsequently applying the SORM analysis that takes the amount of curvature of the actual LSS into account and considers a hyperbolic LSS at the design point. The method estimates the POF as the probability content outside the hyper-paraboloid. In this research, the Augmented FORM algorithm was tested to estimate the POF with and without inspections. It was found that the algorithm can give much quicker estimate of the POF than other costly methods, e.g. Monte Carlo and numerical integration methods. Also, improvements were made to the algorithm for performing Fracture mechanics calculations using the NASA developed well-known Fracture mechanics and fatigue crack growth analysis software NASGRO. Three important random variables: Initial crack size, Load, and Paris crack growth constant C were considered. Subsequently, SORM was applied to improve the FORM results. The results were verified with those obtained from numerical integration and sampling methods used in the FAA-sponsored structural integrity software SMART|DT that also used NASGRO for crack growth calculation. The verification showed that the improved Augmented FORM can give efficient estimates of the POF with and without inspections using NASGRO. It was also revealed that in many cases SORM can improve the POF estimates.


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Damage Tolerance Analysis, First-Order Reliability Method (FORM), NASGRO, NDE Inspections, Second-Order Reliability Method (SORM), Structural Reliability



Mechanical Engineering