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The steady state nodal temperatures are used as a loading condition in elastic stress analysis. The principal stress distribution is shown in Figure 5 for the mean values of input shown in Table 1. Figure 4. Steady state temperature profile resulting from convective thermal loads Figure 5. Principal stress distribution in the component. Uncertainties in gas temperature and film transfer coefficient as boundary conditions in thermal analysis (input 2), elastic modulus and coefficient of thermal expansion as material properties in stress analysis (input 3) are included for demonstration of the methodology. Uncertainties in other input may also exist, but for demonstration purposes, only few are considered. However, the methodology will be able to address them all during a realistic design stage. Only a single value is shown in Table 1 for the parameter for which uncertainty is not considered. The uncertainties are represented as a normal distribution, values for mean, and -3σ and +3σ of the distribution are shown in Table 1 for the variables for which uncertainties are considered (where σ is the standard deviation). Though the ranges given in the uncertainty distribution are assumed here for demonstration purposes, such levels of uncertainty are not unreasonable for gas turbine boundary conditions and material properties. The Weibull modulus has been taken to be 9 (averaged over the temperature range) and characteristic strength as 37 ksi representing volume flaws from uni-axial tension test for SiC [13]. These values are taken as input into CARES (the characteristic strength has to be normalized over the effective volume), and are assumed to be independent of temperature. Though uncertainty in slow crack growth parameters are not considered in this exercise, uncertainty in these parameters can be accounted for in an actual design exercise. Before considering uncertainties, a deterministic stress analysis was done by taking only the mean values and a corresponding probability of failure is determined using CARES. The probability of failure corresponding to mean input values was 0.004. Then the probability of failure for worst case stress state was calculated thru deterministic method. The input values corresponding to the worst case situation corresponding to maximum stress is shown in red in Table 1. The probability of failure for worst case was determined to be 0.49. The probability of failure has been obtained as a distribution when uncertainties in model input are considered. The resulting cumulative distribution of failure probability is shown in Figure 6. It took only 7 iterations of simulations to generate the distribution curve shown in Figure 6. 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 129 Copyright © 2007 by ASME Probability of Failure Figure 6. Predicted probability of failure shown as a cumulative distribution due to uncertainties in modeling input. Since the distribution has been taken as symmetrical everywhere, the deterministic prediction using mean input values and the mean from the distribution obtained from the probabilistic solver should agree. The mean probability of failure from the distribution obtained using the probabilistic solver (corresponding to 50% confidence in Figure 6) is very close to the probability of failure determined deterministically (0.004) taking only mean values for input. This verifies the accuracy of the probabilistic solver used to obtain the Cumulative DistributionPDF Image | ADVANCED MICROTURBINE SYSTEMS Final Report for Tasks 1 Through 4 and Task 6
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