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516 Appendix C envelope or within a larger population of engines. Frequency margin can be provided above or below the steady-state speed. Engine components should not fail due to HCF or a combination of HCF and LCF when subject to the maximum attainable combined steady-state and vibratory stresses at a rate above that stated in A.4.13.3. Low-order crossings and all known drivers within two stages either upstream or down- stream of the subject component should have probabilistic design margins for final design that account for the variation in speeds for different operating conditions. A deterministic 10% frequency margin on component modes is acceptable for preliminary design or when there is insufficient confidence in probabilistic methods. In the event that insufficient margin exists to meet the probability of failure requirement, the next level of probabilistic design analysis is required. Such probabilistic design analysis includes physical models of forced response, damping, and mistuning, along with the appropriate probability models for each random variable and correlations between those variables, as appropriate. Analyses on blades, vanes, disks, and integrally bladed rotors (IBRs or blisks) should include the effects of response to unsteady aerodynamic pressures, damping, and mistuning. Probabilistic methods should be applied to predict the probability distribution of the dynamic responses. Without these analyses, HCF response of the component may be seriously underestimated. Appendix D ∗ Evaluation of the Staircase Test Method using Numerical Simulation Major Randall Pollak INTRODUCTION Numerical simulation was used to model the staircase test method for determining the fatigue strength at a given number of cycles. The simulation allows the user to specify the true fatigue strength distribution (modeled as normally distributed in all cases) by specifying the mean and standard deviation of the fatigue strength at the number of cycles of interest. Individual simulated test sequences for the staircase test procedure, described in Chapter 3, are chosen randomly consistent with the assumed normal distribution function and values of the parameters chosen for that function. Numerical simulation was used to model the staircase test in order to determine the effects of starting stress, step size, and number of specimens on parameter estimates using Dixon–Mood statistics as presented in Chapter 3. The objective of these simulations was to evaluate the feasibility of using the staircase test to estimate a material’s fatigue strength distribution using small sample sizes. There were two primary considerations when evaluating the staircase methodology. The first was to assess how accurate the test is, on average. In this sense, the goal was to demonstrate that the test method estimates the mean of fatigue strength distribution parameters as closely to true values as possible—i.e., a test with little bias in results. The second consideration involved the ability to estimate the scatter in results. Because a designer or researcher must make conclusions based on a limited set of data, it is important that a test should not be subject to a wide dispersion in results due to statistical scatter alone. In other words, one would prefer a test method with results as tightly grouped as possible around the true parameter values. These two considerations led to recommending the use of two different techniques to improve the fatigue strength analysis using a small-sample staircase test strategy. The first technique involved the use of a non-linear correction term to the Dixon–Mood standard deviation estimate in order to mitigate the bias inherent in this equation with small sample ∗ This document was prepared by Major Randall Pollak from the work that is part of his PhD dissertation at the Department of Aeronautics and Astronautics, Air Force Institute of Technology. Much of the work in this Appendix was presented at the 10th National Turbine Engine High Cycle Fatigue Conference in New Orleans, LA, 8–11 March 2005. 517 518 Appendix D sizes. The second technique was the incorporation of a simple bootstrapping algorithm to help mitigate scatter in results. Through a combination of these techniques, a modified staircase strategy emerged which is generally more accurate (less bias) and more precise (less scatter) than traditional Dixon–Mood analysis alone. THE STAIRCASE METHOD The staircase method, first analyzed by Dixon and Mood [1] for explosives testing, and later popularized by Little [2–3] for fatigue strength testing, utilizes a very simple protocol in which a specimen is tested at a given starting stress for a specified number of cycles or until failure, whichever occurs first. If the specimen survives, the stress level is increased for the next specimen; likewise the stress is decreased if the specimen fails. This protocol is continued for a batch of specimens with statistical equations applied to provide an estimate for the mean fatigue strength and its standard deviation at the specified number of cycles. Unfortunately, it is difficult in practice to provide accurate estimates of the standard deviation using this method for small-sample test programs typical of high-cycle or ultra high-cycle fatigue testing. Typically, the step size between adjacent stress levels is held constant (approximately equal to the standard deviation of fatigue strength), in which case the statistics of Dixon and Mood may be applied directly to estimate mean and standard deviation of the fatigue strength at a given number of cycles. Even though the true standard deviation in fatigue strength is one of the unknowns, Dixon notes that it is not too important if the interval is actually incorrect with respect to the true standard deviation by as much as 50% [4]. However, Little notes that arbitrary spacing between stress levels may be used when accompanied by a probit-type analysis [2]. In fact, tests conducted with non-uniform spacing may be more statistically efficient than uniform spacing; however, the analysis becomes much more tedious and the equations and tables derived for uniformly spaced tests are no longer useful [5]. The equations for the mean, , and standard deviation, , from the staircase method (denoted by subscript “sc”) are (see Chapter 3): SC =S i=0 +s ⎛ ⎜ ⎜ ⎝ i max i=0 im i i max i=0 m i ±05 ⎞ ⎟ ⎟ ⎠ (D.1) SC =162s ⎛ ⎜ ⎜ ⎜ ⎝ i max i=0 m i i max i=0 i 2 m i − i max i=0 im i 2 i max i=0 m i +0029 ⎞ ⎟ ⎟ ⎟ ⎠ (D.2) Appendix D 519 In these equations, s denotes the step size and the parameter “i” is an integer denoting the stress level, with i max corresponding to the highest stress level in the staircase. If the majority of specimens failed, then the lowest stress level at which a survival occurs (i.e., run-out) corresponds to the i = 0 level and m i corresponds to the number of specimens which survived each stress level. The next highest stress level would be the i =1 level, and the stress level one above that would be i = 2, etc. If the majority of specimens survived the given number of cycles, then the lowest stress level at which a failure was observed is denoted as the i =0 level and m i corresponds to the number of specimens which failed at each stress level. The standard deviation equation above, Equation (D.2), should only be used if the quotient term in this equation is greater than or equal to 0.3, otherwise the standard deviation is estimated as 053s. The primary condition of the analysis on which these equations is based is that the variate under consideration must be normally distributed. In the case of fatigue testing, this would imply that the fatigue strength distribution must be normally distributed. This condition would seem to be very restrictive and rule out staircase testing for many materials; however, a transformation of the stress values may be applied. For example, the fatigue strength distribution of some materials may be reasonably represented using a lognormal distribution, in which case the logarithm of stress values would be normally distributed. The staircase test would then be conducted using steps of equal intervals with respect to the natural logarithm of stress. Depending on the skewness of the distribution, other transformations may be applied to make the distribution reasonably normal, although there is no guarantee that every distribution can be transformed to approach normality. In addition to logarithmic transformations, power transformations are often applied for such a purpose. Squared and cubic transformations (as well as other powers greater than one) are used to reduce negative skewness, while logarithmic and powers less than 1 (such as a square-root transformation) as well as negative reciprocals (−1/x) are often used to reduce positive skewness [6]. Thus, the normality condition is not as restrictive as it first appears. However, it is often likely to be the case where there are not enough data to actually estimate the distribution shape prior to conducting a staircase test. Thus, one must either make a guess as to the shape based on data from a similar material and apply an appropriate transformation if non-normal, or simply accept the initial assumption of normality and conduct the test without a stress transformation and then use the data from the experiment itself to estimate the distribution. STAIRCASE SIMULATION A computer simulation was designed to analyze the effects of staircase parameter settings. The simulation allows the user to specify the true fatigue strength distribution (mean and standard deviation). The underlying distribution was modeled as Gaussian in all cases. 520 Appendix D The staircase test parameters (starting stress, step size, and number of specimens) were then specified. For each specimen, the simulation calculated a random fatigue strength based on the specified underlying distribution and compared this value to the current stress level to determine if the specimen failed or survived. The stress level was then increased or decreased for the next specimen according to the staircase protocol. This procedure was repeated 1000 times in order to provide a distribution of calculated means and standard deviations according to the Dixon–Mood equations. Comparison of runs made with 1000 replications versus 10,000 replications showed that 1000 replications provided more than adequate statistical significance. The simulation was later expanded to allow the use of bootstrapping and various iterative test schemes. STAIRCASE PARAMETER ANALYSIS The first set of simulations focused on the effects of starting stress on parameter estimates using the Dixon–Mood equations. Figure D.1a shows the effect of starting stress on estimates of mean fatigue strength. Figure D.1b shows the starting stress impact on standard deviation estimates. As one would expect, starting higher than the true mean leads to an estimate for mean fatigue strength that is slightly too high on average, and starting lower leads to lower estimates. As the number of specimens increases, the importance of starting stress on calculations of mean fatigue strength is diminished. This analysis reaffirms a fact that is generally already accepted—namely, that one should start the staircase test as closely to the true mean as possible. However, there are techniques already developed in the literature to account for staircase results which begin with a string of failures (when starting at stresses higher than the mean) or survivals (when starting too low). Brownlee et al. first investigated this problem in 1953 and proposed a modified equation for the mean [7]. Dixon later proposed a set of lookup tables for the mean in 1965 [4], and Little expanded these tables in 1972 [5]. In short, there are adequate methods to estimate the mean fatigue strength using small-sample data quite accurately. For standard deviation estimation, Figure D.1b shows that even for small samples, the effect of starting stress is rather small. However, use of offset starting stresses does have some beneficial effect for small sample sizes by alleviating some of the standard deviation bias. This result is due to the fact that when one starts above or below the mean, it is statistically more likely that a failure will be observed below the mean (if starting below) or a survival will be encountered above the mean (if starting above), simply because one is conducting more testing above or below the mean, on average. In a small-sample test, just one of these outcomes can lead to a higher estimate of standard deviation, and therefore, on average, there is a slight reduction of the standard deviation underestimating bias for small-sample tests. This effect is not overwhelming enough to Appendix D 521 398.5 399.0 399.5 400.0 400.5 401.0 401.5 0 50 100 150 200 0 50 100 150 200 Number of specimens Point estimate of FS mean Start at mean Start 2 steps below Start 2 steps above 2 3 4 5 6 Number of specimens Point estimate of FS standard deviation Start at mean Start 2 steps below Start 2 steps above (a) (b) Figure D.1. Effect of starting stress on (a) mean and (b) standard deviation estimates for true underlying distribution Normal ( =400 =5) and step size 1. eliminate the bias altogether, but does provide some means of softening it. For the data in Figure D.1b, the mean standard deviation for N =10 samples is 3.68 (26.3% error) when starting at the mean, but 4.06 (18.9% error) when starting two steps below the mean. The traditional approach of starting the staircase at the mean in order to maximize accuracy of the estimate for mean fatigue strength may be modified in order to improve standard 522 Appendix D deviation estimation, especially since adequate means of handling offset starting stresses exist for mean fatigue strength estimation. Next, the effect of step size was analyzed. The choice of step size had little effect on the calculated fatigue strength mean in an average sense, although Figure D.2 shows that as step size increases, there is more scatter in fatigue strength mean estimates. This result is rather intuitive since one would expect a tighter grouping around the mean for small steps. Thus, a single staircase test is more likely to have error in calculated mean fatigue strength for larger steps versus smaller steps. Figure D.3 shows the calculated standard deviation as a function of step size and number of specimens. This figure represents the expected value of standard deviation for any combination of step size and number of specimens. When normalized by the true standard deviation, these expected values are independent of the true standard deviation. Note that in the limit (simulated using 1000 specimens), the calculated standard deviation is close to the true standard deviation for all step sizes ranging from 01 to 17. However, for small-sample tests, the standard deviation is significantly underestimated. Braam and van der Zwaag [8], along with Svensson and de Maré [9], also noted this underestimating bias for small-sample tests. Note that existing guidance stemming from Dixon and Mood’s original work suggests a step size in the 2/3to3/2 range. To remove standard deviation bias, however, Figure D.3 shows that step sizes must be larger, on the order of 16 to 175. Beyond this region, the standard deviation estimates converge to the 053s line. It is clear from Figure D.3 that standard deviation bias is magnified as either step size or sample size are reduced. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1 1.5 2 Step size (σ) Standard deviation of FS mean 8 specimens 10 specimens 12 specimens 15 specimens 20 specimens 30 specimens 50 specimens 100 specimens 1000 specimens Figure D.2. Effect of step size on scatter of fatigue strength mean estimates for true underlying distribution Normal ( =400 =5). Appendix D 523 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0 0.5 1.5 2 Step size (σ) Point estimate of standard deviation (σ) 8 specimens 10 specimens 12 specimens 15 specimens 20 specimens 30 specimens 50 specimens 100 specimens 1000 specimens 1 Figure D.3. Effect of step size on estimated fatigue strength standard deviation. In addition to being more accurate on average, use of a higher step size reduces the variance in standard deviation estimates compared to the estimates obtained using a step of 2/3to3/2, as illustrated by Figure D.4. This effect is due in part to the larger steps producing more results using the 053s standard deviation estimate, which is constant for a given step size. In general, it appears that use of steps in the 16 to 175 range will result in both less bias and less scatter than smaller step sizes. 0.0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 Step size (σ) Standard deviation of D–M (σ) 8 specimens 10 specimens 12 specimens 15 specimens 20 specimens 30 specimens 50 specimens Figure D.4. Effect of step size on scatter of fatigue strength standard deviation estimates. 524 Appendix D Thus, there is a bit of a tradeoff here. Larger step sizes on the order of 17 provide generally more accurate estimates of standard deviation with less scatter, but result in slightly more scatter in fatigue strength mean. Remedies to this tradeoff will be presented in the next section. CORRECTION FOR STANDARD DEVIATION Due to the bias in standard deviation estimates for small sample sizes, a correction term was investigated to allow the use of Dixon–Mood equations at smaller steps to avoid the increased scatter in fatigue strength estimates inherent in the use of higher step sizes. A starting point for such a correction was the linear correction factor presented by Svensson et al. [10]. This work addressed the problem of standard deviation bias, noting that no theoretical solution to this bias was apparent. A correction factor for the standard deviation was proposed based on the number of tested specimens, given as: corr = DM N N −3 (D.3) where corr is the modified (corrected) standard deviation, DM is the standard deviation calculated using the Dixon–Mood equation, and N is the number of specimens. This linear correction (which will be called the “Svensson–Lorén correction” herein) was applied to the standard deviation estimates from the simulation results from Figure D.3, with corrected standard deviations shown in Figure D.5. Notice that the step 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 100 specimens 1000 specimens 1 Figure D.5. Svensson–Lorén correction for fatigue strength standard deviation estimates. Appendix D 525 size for unbiased standard deviation estimates has shifted from 17 to approximately 1. This result is significant since it allows the use of smaller step sizes without standard deviation bias. However, since the true standard deviation is unknown, it is not possible to plan a test at precisely this unbiased step size. If one tests at too low a step size, standard deviation is still underestimated (on average). Likewise, the standard deviation is overestimated at higher step sizes. One would prefer a correction which allowed a greater range of unbiased step sizes. A more elaborate correction factor was developed based on analysis of simulation results and is proposed for small-sample staircase tests. This correction factor accounts for both sample size and step size, and has the form: corr =A DM N N −3 B DM s m (D.4) In this equation, A B, and m are constants based on the number of specimens, with other variables as previously defined. Table D.1 shows the values for these constants based on simulation results for selected numbers of specimens. Since standard deviation bias is independent of both the true mean and true standard deviation of the underlying fatigue strength distribution, these constants are not specific to a particular simulation case. These constants were determined in order to calibrate the mean standard deviation estimate from Dixon–Mood analysis to be as unbiased as possible. Figure D.6 shows the corrected mean estimates for standard deviation using this non-linear correction, with Figure D.7 comparing the standard deviation estimates for Dixon–Mood, the Svensson– Lorén correction, and this non-linear correction for N = 20 specimens. These figures suggest that the proposed correction allows a greater range of unbiased (or only slightly biased) step sizes. Thus, when using this correction, one is less reliant on the initial guess of true standard deviation in order to get meaningful results. However, it should be noted that it is only the mean standard deviation from Dixon–Mood that is Table D.1. Non-linear correction constants based on number of specimens Number of specimens AB m 8 1.30 1.2 1.72 10 1.08 1.2 1.10 12 1.04 1.2 0.78 15 0.97 1.2 0.55 20 1.00 1.2 0.45 30 1.00 1.2 0.22 50 1.00 1.2 0.15 >50 Use Svensson–Lorén. . conducting a staircase test. Thus, one must either make a guess as to the shape based on data from a similar material and apply an appropriate transformation if non-normal, or simply accept the initial. document was prepared by Major Randall Pollak from the work that is part of his PhD dissertation at the Department of Aeronautics and Astronautics, Air Force Institute of Technology. Much of the. many materials; however, a transformation of the stress values may be applied. For example, the fatigue strength distribution of some materials may be reasonably represented using a lognormal
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