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526 Appendix D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0 0.5 1.5 2 Step size (σ) Correction of mean standard deviation (σ) 8 specimens 10 specimens 12 specimens 15 specimens 20 specimens 30 specimens 50 specimens 1 Figure D.6. Proposed correction applied to mean of fatigue strength standard deviations. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 0.5 1 1.5 2 Step size (σ) Correction of mean standard deviation (σ) Proposed correction Svensson–Lorén correction Dixon–Mood mean Figure D.7. Comparison of Dixon–Mood, Svensson–Lorén correction, and proposed correction for N = 20 specimens. unbiased. Of course, there will be estimates for standard deviation which are both higher and lower than the mean, and these estimates will not be corrected to the true value using this correction. Compared to both Dixon–Mood and Svensson–Lorén, the proposed correction tends to have more scatter (though less than Braam-van der Zwaag’s correction [8]). Thus, in an average sense, bias is reduced, but for any individual test the results may Appendix D 527 not be improved compared to either Dixon–Mood or Svensson–Lorén. The next section will address the means of reducing this scatter. THE USE OF BOOTSTRAPPING Obviously, one of the biggest problems inherent in small-sample testing is that statistical scatter can greatly impact results. Use of both the Svensson–Lorén correction and the proposed correction increases the already significant scatter inherent in the Dixon–Mood standard deviation estimate. The bootstrapping method was investigated as a possible means to reduce this variance. The bootstrap is a data-based simulation which utilizes multiple random draws from real test data to improve statistical inferences about the underlying population [11]. Essentially, the bootstrap method can be summed up by the following conjecture: Assuming the test data collected accurately represents the true distribution, what other results could have been obtained if the test were repeated? The bootstrap algorithm for the staircase application is based on the associated proba- bilities of failure computed using the number of survivals and failures at each stress level (i.e., P–S data). Using these data, simulated staircase tests can be generated numerically using the same starting stress, step size, and number of specimens. For the first specimen, a random number is drawn and compared to the probability of failure associated with the initial stress level. The stress level is increased or decreased depending on the result of this draw (if the random number is less than the probability of failure, the result is considered a failure, otherwise it is a survival). Another random number is then drawn for the second specimen and compared to the probability of failure for its associated stress level, with the stress level increased or decreased based on this comparison, and the process is repeated until all specimens are used. Note that test data must be bounded by both P(failure) = 0 and P(failure) = 1 stress levels or the staircase may walk to a stress level where no data exists. Through this algorithm, a set of “virtual” staircase tests is generated for the one “real” staircase test. For each virtual staircase, the Dixon–Mood method can be applied to provide a standard deviation estimate. The result is a distribu- tion of standard deviation estimates. Lastly, the revised point estimate can be taken as the mean of this distribution, or another statistic such as the median or other percentile point. The staircase simulation was modified to accommodate this bootstrapping algorithm by simulating a “real” staircase using an assumed underlying fatigue strength distribution, and then using the P–S data from this simulated staircase to bootstrap additional staircases from which a distribution of standard deviation estimates can be analyzed. Figure D.8 shows a comparison of mean bootstrap results to Dixon–Mood statistics for the case of a 12-specimen test conducted at a step size of 1. From Figure D.3, the expected value for standard deviation is approximately 3.7. Fifty staircases were conducted with 528 Appendix D 0.0 0.5 1.0 1.5 2.0 2.5 Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Standard deviation estimate (σ) DM Statistics Alone With Bootstrapping Expected Value Figure D.8. Comparison of bootstrapped results to Dixon–Mood statistics for 12-specimen test with step size 1. 5000 bootstrap replications each, with the results sorted in groups as the staircase test produces discretized results (in this case, there were 10 different values for standard deviation based on the 50 “real” staircase tests). One can see how the standard deviation estimate is closer to the expected value using bootstrapping. Figure D.9 presents this same data through a different means, showing how the distribution of calculated standard deviations is narrowed using bootstrapping. In fact, for this case the standard deviation of the estimates was reduced from 035 to 014 using the bootstrap. The bootstrap provides a measure of protection against large errors associated with small-sample tests due to statistical variance. The bootstrap thus provides the first real tool to reduce the scatter inherent in the Dixon–Mood standard deviation method. The algorithm adjusts the estimate for standard deviation to be closer to the expected value given the test parameters. It has been shown to adjust Dixon–Mood estimates towards their expected value. Thus, if one tests in an unbiased region, such as the s =17 region, then the bootstrap should drive Dixon–Mood estimates towards the true standard deviation. However, when combined with a correction for bias, the expected value should be closer to the true standard deviation, and therefore the bootstrap will drive the results towards the true standard deviation value. One important note, however, is that the bootstrap tends to have little influence on staircase tests resulting in just two or three stress levels as these tests result in standard deviation estimates which are based almost solely on the 053s rule. Investigation of the bootstrap in conjunction with both Svensson–Lorén and the proposed bias corrections shows that for test results with at least four stress levels, use of the average of the 60th and 65th percentiles of the bootstrapped standard deviation distribution provides a better estimate than use of the mean of bootstrapped distribution. Appendix D 529 0 5 10 15 20 25 30 2101112 Standard deviation estimate Frequency 456 987 3 (a) (b) 0 5 10 15 20 25 30 Standard deviation estimate Frequency 2101112456 9873 Figure D.9. Histogram of Dixon–Mood ((a) top) and bootstrap ((b) bottom) standard deviation estimates for same conditions as Figure D.8. CONCLUSIONS The results of the analysis presented here are summarized as follows: 1. The Dixon–Mood equation using small-sample data generally underestimates standard deviation using traditional 2/3to3/2 step sizes. 2. Step sizes on the order of 17 reduce both standard deviation bias and scatter, but larger step sizes increase scatter in fatigue strength mean. 530 Appendix D 3. Use of a non-linear correction factor was shown to allow a larger range of unbiased standard deviation estimation (in an average sense) in the 2/3to3/2 range. This correction factor generally increases scatter in results, however. 4. Use of a bootstrapping algorithm was shown to significantly reduce scatter in standard deviation estimates. This simple numerical procedure therefore provides a measure of protection against large errors when conducting small-sample staircase tests. The bootstrap requires test results with at least 4 stress levels to be effective. 5. Combination of bias correction and bootstrapping provides a generally more reliable and robust means of calculating standard deviation using Dixon–Mood staircase analysis by improving expected values and reducing scatter. REFERENCES 1. Dixon, W.J. and Mood, A.M., “A Method for Obtaining and Analyzing Sensitivity Data”, Journal of the American Statistical Association, 43, 1948, pp. 109–126. 2. Little, R.E., Manual on Statistical Planning and Analysis, American Society for Testing and Materials, STP 588, 1975. 3. Little, R.E. and Jebe, E.H., Statistical Design of Fatigue Experiments, John Wiley & Sons, New York, 1975. 4. Dixon, W.J., “The Up-and-Down Method for Small Samples”, Journal of the American Sta- tistical Association, 60, 1965, pp. 967–978. 5. Little, R.E., “Estimating the Median Fatigue Limit for Very Small Up-and-Down Quantal Response Tests and for S–N Data with Runouts”, Probabilistic Aspects of Fatigue, American Society for Testing and Materials, STP 511, 1972. 6. Neter, J., Kutner, M.H., Nachtsheim, C.J., and Wasserman, W., Applied Linear Statistical Models, 4th Ed., Irwin, Chicago, 1996. 7. Brownlee, K.A., Hodges, J.L., Jr., and Rosenblatt, M., “The Up-and-Down Method with Small Samples”, Journal of the American Statistical Association, 48, 1953, pp. 262–277. 8. Braam, J.J. and van der Zwaag, S., “A Statistical Evaluation of the Staircase and the ArcSin √ P Methods for Determining the Fatigue Limit”, Journal of Testing and Evaluation, 26, 1998, pp. 125–131. 9. Svensson, T. and de Maré, J., “Random Features of the Fatigue Limit”, Extremes, 2, 1999, pp. 165–176. 10. Svensson, T., de Maré, J., Wadman, B., and Lorén, S., “Statistical Models of the Fatigue Limit”, Project paper, www.sp.se/mechanics/Engelska/FoU/ModFatigue.htm, 2000. 11. Efron, B. and Tibshirani, R.J., An Introduction to the Bootstrap, Chapman & Hall, New York, 1993. Appendix E ∗ Estimation of HCF Threshold Stress Levels in Notched Components Alan Kallmeyer INTRODUCTION The determination of the allowable threshold stress level, or fatigue limit stress, for notched components is an important consideration in an HCF design methodology. It has long been recognized that the use of the peak notch-root stress, calculated using the theoretical stress concentration factor k t , results in overly conservative life-predictions. Even when localized plasticity at the notch root (resulting in shakedown) is taken into account, the peak stress alone is still insufficient to predict the fatigue life or limiting stress from smooth specimen data. Although there are a number of contributing factors, it is generally acknowledged that the stress gradients in the vicinity of the notch play a significant role. Evidence has shown that the steeper the stress gradient, the more non- conservative the prediction using k t . This phenomenon can be explained by recognizing that, under equivalent peak stress conditions, a steeper stress gradient results in a smaller volume of material subjected to a high stress level. Utilizing a stochastic viewpoint of the fatigue process, cracks would be less likely to initiate from a smaller highly stressed volume of material than in a larger highly stressed volume such as found in smooth, axially loaded specimens. A common design methodology that has been used to account for the observed behavior defines a “fatigue notch factor,” k f , where k f ≤k t . k f is often defined in terms of a “notch sensitivity factor,” q, which is dependent on the material and the notch-root radius (which affects the stress gradients). As the notch radius decreases, q decreases, reflecting a greater deviation between k t and k f . Although this approach has been successfully applied to some common notch geometries under uniaxial loading conditions, the definitions of q and k f are ambiguous for more complex notch/component geometries or under multiaxial loading conditions. More recent approaches for estimating notch threshold stress levels have attempted to account for the complexities associated with the steep stress gradients on crack initiation ∗ This appendix was prepared by Dr. Alan Kallmeyer, Department of Mechanical Engineering, North Dakota State University. Much of the material is extracted from the final report of the National Turbine Engine High Cycle Fatigue program [1]. Very special thanks to Dr. Stephen J. Hudak, Jr. and his collaborative researchers at Southwest Research Institute, San Antonio, TX, for their contributions to the research on very sharp notches, both experimental and analytical, particularly the development of the Worst Case Notch concept. 531 532 Appendix E and propagation in a more systematic manner. Two such models are described below. The Stressed Surface Area approach, relying on a feature-stress (F s) weighting term, is a crack initiation model that incorporates a modification to the peak notch-root stress based on the stress gradients around the notch. The Worst Case Notch (WCN) model is a fracture mechanics approach that assumes that microcracks initiate early in the life of notched components, and considers the unique behavior of small cracks in the crack growth analysis. The two methods are compared to experimental data from fatigue tests on specimens containing small, sharp notches with severe stress gradients, as may be typically observed in turbine blades subjected to foreign object damage (FOD) or near edge-of-contact conditions. STRESSED SURFACE AREA (F s) APPROACH The Stressed Surface Area, or F s approach, accounts for the stress gradient effect through consideration of the stress distribution on the surface of a component in the vicinity of a notch. This approach is based on the “weakest-link” concept, which assumes that fatigue failure will initiate at the weakest grain or defect in a material and, therefore, will be dependent on the volume of material that is exposed to the highest stresses. Since crack initiation normally takes place on the surface of a material, particularly in titanium, the Fs approach was developed recognizing that there is a higher probability of finding a weak grain on the surface of a large notch or smooth component than on the surface of a small notch [2]. Thus, the stressed surface area was evaluated as compared to volume in the Fs approach. As explained by Murthy et al. [2], in using the weakest link approach the probability of survival, R, of a small surface area A i , subjected to a stress of i , is assumed to be R 0 i A i = exp − i A i (E.1) This formulation adopts a two-parameter Weibull distribution for the fatigue strength with as the shape factor (a measure of dispersion) and as the scale parameter (a measure of the mean stress value). The probability of survival of all the stressed areas (a total of n areas) would then be given by lnR = n i=1 ln R 0 i A i (E.2) If two specimens have the same fatigue life, then their probability of survival is the same. By equating the two probabilities, we get max1 Fs 1 = max2 Fs 2 (E.3) Appendix E 533 where the parameter Fs is given by Fs j = n i=1 − ij maxj A ij (E.4) and max is the maximum stress on the surface of the specimen (e.g., the peak notch- root stress in a notched specimen). From Equations (E.3) and (E.4), an equivalent stress corresponding to smooth bar data is obtained as eq1 = Fs 2 Fs 1 1 eq2 (E.5) where eq2 is the equivalent peak stress in the notched component and eq1 is the equivalent stress in a smooth bar specimen that has the same life as that of the notched component. The equivalent stresses in Equation (E.5) can be defined in terms of any equivalent-stress-based fatigue parameter, allowing for the inclusion of mean stress effects or multiaxial loadings. In addition, note that the negative sign in Equation (E.4) can be dropped in the computation since a ratio of Fs terms is computed in Equation (E.5). The Fs approach has been successfully applied to describe the fatigue behavior of specimens containing both large and small blunt notches [1]. To evaluate the capabilities of the Fs method in the analysis of components containing severe stress gradients, such as may occur near FOD impact sites or edge-of-contact conditions, the method was used to estimate the HCF threshold stress levels for Ti-6AL-4V specimens containing very small, sharp notches. The double-edge-notched specimens had a rectangular cross-section with gross-section dimensions of 0235 " ×0125 " 60mm×32mm. The specimens contained small notches produced by EDM, followed by a stress relief and chem mil to leave a smooth surface free of residual stresses [1]. The effective stress concentration factors and stress gradients in the specimens were varied by adjusting the notch depth, b, and notch root radius, , as illustrated in Figure E.1. The specimens were tested in load control at R =05 to experimentally determine the threshold stress levels, defined as the 10 7 cycle notch fatigue strength, using a step-test procedure. b ρ Figure E.1. Small, sharp notch details. 534 Appendix E In the application of the Fs approach to these specimens, the equivalent stresses in Equation (E.5) were calculated using the Modified Manson-McKnight (MMM) multiaxial fatigue model, which allowed for the multiaxial stress state at the notch root and mean stress effects to be accounted for. Using this model, the equivalent stress parameter is calculated as [3] eqv =05 psu w max 1−w (E.6) where psu is a “pseudo-stress range” defined as psu = 1 √ 2 xx − yy 2 + yy − zz 2 + zz − xx 2 +6 2 xy + 2 yz + 2 zx (E.7) and max is calculated as max = mean +05 psu , where mean = 2 √ 2 xx − yy 2 + yy − zz 2 + zz − xx 2 +6 2 xy + 2 yz + 2 zx (E.8) In these equations, ij and ij define the stress range and summed stress, respectively, for each stress component based on the maximum and minimum points in the fatigue cycle, and the Manson-McKnight coefficient, , is defined as [2] = 1 + 3 1 − 3 (E.9) where, 1 and 3 are the sums of the first and third principal stresses, respectively, at the maximum and minimum stress points in the fatigue cycle. The exponent w is a material- and temperature-dependent constant that collapses variable mean stress data into a single curve. The value of Fs for each notched specimen was calculated using Equation (E.4). An elastic finite element analysis was first performed to obtain the peak notch root stress max and the surface stress distribution across each specimen. The summation in Equation (E.4) was performed over all the elements in the model adjacent to a free surface of the specimen, with i taken as the first principal stress associated with the nodes on the free surface of element i, and A i the free-surface area of that element. Since the value of Fs is calculated from a ratio of stresses, it does not change appreciably with load level, provided notch-root yielding is small; thus, the use of an elastic finite element analysis is appropriate for this calculation. However, in Equation (E.5), the calculation of the equivalent peak stress in the notched component, eq2 , must take into account localized yielding. Therefore, an elastic-plastic finite element analysis was performed to determine Appendix E 535 this value, using Equations (E.6)–(E.9) for cyclic loading. Finally, Equation (E.5) was used to determine the adjusted equivalent stress for the notched component, eq1 , which was then compared to smooth specimen results using the following relationship between the MMM parameter and fatigue life for Ti-6Al-4V (stresses in ksi): 05 psu w max 1−w =35018N −05164 +3674N 000068 (E.10) In these computations, a value of = 35 was used, which was previously obtained by correlating smooth and notched test data for Ti-6Al-4V [1], and F s 1 = 0161 was used as the reference value for a smooth, axial specimen. The notch dimensions( and b) used inthis study were selected to correspond to the aver- age measured notch dimensions of the specimens tested. Due to the difficulty in machining the very small notches, in many cases themeasured notch dimensions deviated substantially from the target values, and there were significant deviations from the average for certain specimens withina group. Consequently,only a limited number ofdirect comparisons could be made between the experimental and predicted threshold stress levels. The notch dimensions and associated elastic stress concentration factors for the samples considered in this study are shown in Table E.1. The stress concentration factors were calculated from the elastic FEA solution. Two values of k t are included in the table for each specimen, one based on the net-section area through the notch (the traditional definition of k t ), and the other based on the cross-section area 0235 " ×01257 " . The net-section k t values range from approximately 3.2 to 5. The comparison of the predicted and experimental threshold stress levels, based on the gross cross-section dimensions, is shown in Table E.2. Although limited in number, these Table E.1. Small notch dimensions and elastic stress concentration factors Notch (in.) b (in.) k t (net section) k t (gross section) 100064 00080 324 348 200062 00230 456 567 300064 00490 499 855 Table E.2. Threshold Stress Estimates for Small Notched Specimens using the Fs Approach Predicted Experimental Notch (in.) b (in.) max (ksi) th (ksi) (in.) b (in.) th (ksi) 100064 00080 432 21.6 00064 00089 163 200062 00230 262 13.1 00060 00236 111 3 0.0064 0.0490 17.5 8.75 00064 00495 60 00064 00484 79 . Small Samples”, Journal of the American Statistical Association, 48, 1953, pp. 262–277. 8. Braam, J.J. and van der Zwaag, S., A Statistical Evaluation of the Staircase and the ArcSin √ P Methods. increase scatter in fatigue strength mean. 530 Appendix D 3. Use of a non-linear correction factor was shown to allow a larger range of unbiased standard deviation estimation (in an average sense). the highest stresses. Since crack initiation normally takes place on the surface of a material, particularly in titanium, the Fs approach was developed recognizing that there is a higher probability
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