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distribution of probability of failure due to uncertain input values in the thermal stress model. The FORM algorithm used also captures sensitivity of statistical parameters contributing to failure prediction. The relative contribution or rank ordering of uncertainties in parameters is shown in Figure 7. In this example, uncertainty in thermal boundary conditions and material coefficient of thermal expansion appears to have more influence on uncertainty in reliability prediction. confidence is 0.2, it suggests that the compromise in reliability can be as high as 196 more parts failing per 1000. Table 2. Probability of Failure determined through different approaches Treatment of Probability Model Input of failure Deterministic, Mean value 0.004 Deterministic, extreme value 0.49 Probabilistic, at 95% confidence 0.2 CTE 23% E 4% H2 11% Thot 20% Tcold -20% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% 0.0 Components unnecessarily rejected at proof test Reliability Compromised H1 Figure 7. Sensitivity ranking of uncertainties in model input Though only few input parameters are chosen as random variables in this example for demonstration, the method is certainly applicable to actual design where more input parameters will have to be treated as random variables. This slightly enhances the complexity in coding or scripting and increases the number of iterations, but both are achievable within reasonable limits. DISCUSSION Uncertainties governing the probability of failure have implications on design and selection of proof stress level. Since the predicted probability of failure itself is a distribution when uncertainties in model input are considered, the Y axis in the cumulative distribution shown in Figure 6 can be interpreted as confidence of prediction due to uncertainties in model input. Figure 8 is reproduced from Figure 6 where the X axis is linear instead of logarithmic. Table 2 lists probability of failure obtained from deterministic approach for mean input values and extreme condition input values. Also shown in Table 2 is the predicted probability of failure with 95% confidence when considering uncertainties in model inputs thru the probabilistic approach. As shown in this example, one would expect the probability of failure to be 0.004 when mean input values are taken for prediction. If the actual boundary conditions or material properties are unfavorably different than the mean value considered for prediction, then the resulting failure probability will be higher than 0.004; that is more parts are likely to fail in service than predicted. In this case, the reliability is compromised. Since the predicted reliability with 95% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 130 Copyright © 2007 by ASME Probability of Failure Figure 8. Cumulative distribution of probability of failure due to uncertainties in model input. A conservative approach will be to predict probability of failure using only extreme conditions. Based on this approach, the predicted probability of failure is 0.49 (from Table 2). Since this level of probability of failure can not be accepted in service, the parts would have to be proof screened prior to shipment. Since the parts are likely to be proof screened at or near use stress level (corresponding to worst case stress prediction), nearly 49% of parts are expected to fail at proof screening. This may be too conservative. The likelihood of all factors considered at the extreme condition occurring simultaneously in reality is going to be very small. Reliability with 95% confidence may be adequate and provides enough conservatism since uncertainties are already taken into account. In this case, the proof stress level can be reduced appropriately and nearly 30% good components can be saved. The sensitivity analysis helps the design engineer to focus their effort on better quantification of the most sensitive uncertain input parameters to the model. The design engineer also can pay attention to the design such that the range of uncertainty in critical parameters can be reduced to enable robust design. ConfidencePDF Image | ADVANCED MICROTURBINE SYSTEMS Final Report for Tasks 1 Through 4 and Task 6
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