PSI - Issue 19

A. Halfpenny et al. / Procedia Structural Integrity 19 (2019) 150–167 Author name / Structural Integrity Procedia 00 (2019) 000–000

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3. Reliability simulation

The fatigue results returned by each stochastic fatigue simulation are tabulated under the following column headings: Simulation ID – reference to the parameter configuration used in the current simulation Failure Mode ID – reference to the node, element or group ID of the current failure mode Life – the life to failure in some desired units (cycles, time, distance, etc.) End State – whether the life value pertains to a recorded failure life or a Suspension , this is explained later 3.1. Reliability parameter estimation Reliability parameter estimation is concerned with fitting a probability of failure distribution through the simulated (or measured) life data. A comprehensive introduction to reliability life data analysis is given by [REL 15b]. The following distributions are commonly used: Gaussian-normal distribution

Log-normal distribution Exponential distribution Weibull distribution This paper is concerned mostly with the Weibull distribution. The Weibull distribution is given in Equation (6): ( ) = � � � �� � � � ��� �� �� � � � �

(6)

Where p(x) is the Weibull Probability Density Function (PDF) of life x in the range (x ≥γ≥ 0). η is the characteristic life. β is the shape parameter (or slope). γ is the location parameter (or failure-free life if γ >0). The location parameter is used to shift a distribution in one direction or another. The location parameter defines the location of the origin of a distribution and can be either positive or negative. In terms of lifetime distributions, the location parameter represents a time shift. (This is commonly referred to as a 3-parameter Weibull curve. In the case of a 2 parameter Weibull curve, the location parameter γ=0) The left-hand plot shows the life data plotted on Weibull probability paper. The solid line describes the regression line and the dotted lines the 2-sided 90% confidence bounds. This implies, with a confidence of 90%, that the true regression line will lie somewhere between these two extremities. The right-hand plot shows an alternative view of the Weibull parameters. The contour plot describes the possible range of parameters β and η within the 90% confidence bounds. Knowledge of the confidence bounds are especially important where the number of tests may be small. These include physical tests but also many FEA simulations as well. Mixed-Weibull vs. 3-parameter Weibull vs. 2-parameter Weibull In the case of measured test data, it is often sufficient to use a 2-parameter Weibull distribution (i.e. location parameter γ=0). However, fatigue life simulations are based on a stress-life (SN) or strain-life (EN) fatigue curve. Typically, these are nonlinear when plotted on log-log axes. It therefore follows that these are also nonlinear when plotted on a linearized Weibull plot. An example is shown in the case study in Fig. 12 (red line). For EN-based fatigue simulation, the 3-parameter Weibull curve is often found to give a better representation of the life plot. In this case the regression analysis described above is slightly more complicated and is described in [REL 15b], chapter 4. Similarly, the Weibull curve may follow an S-shaped profile, and this is better suited to the mixed Weibull approach described in [REL 15a], chapter 11.