9IDM- Bayesian Technique or Reducing Uncertainty in Fatigue Failure Model Sriram Pattabhiraman and Nam H. Kim University o Florida, Gainesville, FL, 36 Copyright 8 SAE International ABSTRACT In this paper, Bayesian statistics is utilized to update uncertainty associated with the atigue lie relation. The distribution or atigue strain at a constant load cycle is determined using the tial uncertainty rom analytical prediction and likelihood unctions associated with data. The Bayesian technique is a good method to reduce uncertainty and at the same time provides a conservative estimate, given the distribution o analytical prediction errors and variability o data. First, the distribution o analytical atigue model error is estimated using Monte Carlo simulation with uniormly distributed parameters. Then the error distribution is progressively updated by using the variability as a likelihood unction, which is obtained rom ield data. The sensitivity o estimated distribution with respect to the tial error distribution and the selected likelihood unction is studied. The proposed method is applied to estimate the atigue lie o turbine blade. It is ound that the proposed Bayesian technique reduces the scatteredness o atigue lie by almost %, while maintang the conservative lie estimate at a given atigue strain. In addition, a good conservative estimate o atigue lie prediction has been proposed using a knockdown actor that is obtained rom the distribution o lowest data. INTRODUCTION In general, there are two dierent lie prediction models in atigue analysis: stress-lie and strain-lie models. The ormer is oten used or high-cycle atigue analysis in which the stress-strain relation is in the linear region. The latter is requently used or low- and medium-cycle atigue in which plastic deormation contributes to the atigue lie. Although the basic concept in the proposed Bayesian approach is the same, the strain-lie model will be investigated in this paper. In strain-lie atigue analysis, the total atigue strain ( t ) is decomposed by elastic strain ( e ) and plastic. For this analysis, the strain-lie curve is de- strain ( p ) ined [3] as: t s e p E b N e N c () where e is the coeicient o atigue ductility, s the coeicient o atigue strength, b the eponent o atigue strength, c the eponent o atigue ductility, and E the Young s modulus. The coeicients in the strain-lie curve are obtained using curve-itting o data. However, due to variability in and material, the results are oten scattered. In the material handbook, or eample, the standard values o the coeicients are available. However, a particular batch o material may have dierent properties. In addition, a particular machine may have dierent atigue properties due to manuacturing process used and possibility o residual stresses. Thus, an important question is how to ind more accurate lie estimate or a speciic machine when several data are available. Traditionally, saety actors have been adopted as a measure to counter variability. But, presently, there is growing interest in replacing saety actor-based determstic design with reliability-based design (e.g., Wirsching [7], SAE Aerospace Inormation Report 8 [] ). In addition, the goal o is oten to ind conservative estimate o the predicted lie. Obviously taking the lowest data can be a choice, but its variability will be high and in many cases, the lowest data will not provide enough conservativeness []. Thus, another important question is how to ind the best way o predicting conservative atigue lie o a machine. The objective o this paper is to investigate the possibility o using the Bayesian statistics in order to reduce scatteredness o the atigue lie distribution when additional data are available. In addition, a good way o conservatively estimating atigue lie is proposed using a knockdown actor a term introduced or correcting analytical predictions based on results [8] ) that is obtained rom the distribution o lowest data. The concept o Graduate student, Department o Mechanical and Aerospace Engineering, psriram8@ul.edu. Associate Proessor, Department o Mechanical and Aerospace Engineering, nkim@ul.edu.
knockdown actor has been adopted in aerospace structures. randomly chosen within the variability limits with the analytical value as the mean. INPUTS AND ASSUMPTIONS It is important to understand the assumptions that are used in this paper. Some o them are or convenience, and others represent the lack o knowledge. I additional inormation is available, the latter can be improved. It is irst assumed that the coeicients in Eq. () are randomly distributed and their statistical distribution parameters are known. These distributions represent the prior knowledge or analytical prediction. This inormation can be obtained by studying strain-lie data and by estimating upper- and lower-bounds o data rom the mean curve. When this inormation is not available, it is possible to assume that these parameters ( s, e, b, and c) are distributed uniormly with given bounds rom their nominal values. This will serve as a prior knowledge o the atigue ailure distribution. Even i input parameters are uniormly distributed, the atigue ailure strain will not be uniormly distributed. The conidence interval is a measure o our conidence in the analytically predicted value. It would be the lower and upper bounds o the strain-lie data. However, since only the distribution o each parameter is considered known, the bounds or lie cycle value are determined rom the distribution o all parameters. Considering cases, where only one o the parameters varies at a time, while the rest o them are at their mean values, the sensitivity o lie cycle value to each parameter has been determined, and ound to be positive or all parameters. Hence, the upper and lower bounds or lie cycle would be when all parameters are simultaneously at their algebraic maimum and mmum values, respectively. The bounds o the conidence interval are usually epressed as a percentage o analytical value. I the bounds are asymmetrically distributed about the analytical value, the maimum variation is considered. The distribution o error is the distribution o lie cycle due to the distributions o the parameters. This distribution o error is generated using the Monte Carlo Simulation (MCS). In this paper, the MCS is perormed by generating, values or each parameter, governed by its variability. These randomly generated numbers are used to calculate, values o strain lie, rom which the distribution o error can be estimated. Test variability is the scatteredness o the data. Distribution o variability is the histogram o strain-lie data at the known constant ailure atigue strain ( t ) value. However, or the want o data, the distribution o variability is assumed to be normal with same parameters as that o the distribution o error. Even i there is no limitation on the number o data, three data are assumed available through the o a speciic component. These three data are BAYESIAN UPDATE FOR FATIGUE FAILURE STRAIN NORMALIZATION: Although the raw data and tial distribution can be used or Bayesian update, it is oten more convenient to normalize all data and distributions. All the actors aecting the Bayesian update, such as the conidence interval, error distribution, variability, and results are normalized with respect to the tial value o atigue lie, N. Since the conidence interval is epressed as a percentage o analytical value, it is not aected by normalization. The parameters o variability are epressed as a raction o mean value. In the normalized distribution, the standard deviation becomes identical to the coeicient o variance (COV). LIKELIHOOD FUNCTION: The likelihood unction o a result is the probability o obtang that result, given the value o actual atigue lie and the variability. It is actually the ordinate value o probability distribution unction (PDF) o variability with the actual atigue lie as its mean, when the abscissa is equal to the result. The likelihood would be a single value, i the actual atigue lie is known. But, since only the bounds or the actual atigue lie are known, the likelihood unction varies within that conidence interval. The likelihood o the given result can be ound by considering each point within the error bounds as the actual atigue lie. The likelihood unction or a given result would be the variation o these likelihood values with the actual atigue lie values. The likelihood unction or each o the three results can be determined in a similar ashion. BAYESIAN UPDATE: The Bayesian update is based on the theory o conditional probability, which states p true p true p true () p The epression or Bayesian update is very similar, i.e. upd N N N N N de Here, N is the likelihood unction or the given result. It could also be seen as probability o obtang the result given the true value o atigue lie. N is the tial distribution o N. For updating with the irst result, this distribution is taken as the error distribution. This updated distribution is used as the tial t (3)
distribution or updating with second result and so on. The denominator is simply the integration o numerator. This is interpreted as the normalization o PDF such that the area under the distribution becomes one. Dierent techniques like Trapezoidal rule, Simpson s Rule, etc, can be used or this purpose. Trapezoidal rule has been used in this paper or numerical integration. Ater the tial distribution is updated using the irst upd data, N is replaced by N, and the above procedure is repeated or the net results. The mean o the distribution obtained ater updating with inal result, is called the Bayesian Lie. KNOCKDOWN FACTOR: It is common in practice to take the mmum value o the results as the actual atigue lie. Such consideration implicitly applies a knockdown actor on average atigue lie value. In this paper, an eplicit knockdown actor is calculated rom statistics and multiplied to the mean value o the updated Bayesian lie to calculate a conservative estimate o the atigue lie. It is known that the lowest o the results ollows an etreme value distribution [6]. The mean o this etreme value distribution is used or the knockdown actor. Knock down actor calculation depends on material variability only. I is the cumulative distribution unction (CDF) o material variability ater normalization, the etreme value distribution is given by, 3 3 F ( ) (4) The mean o this etreme value distribution ( F ) is the knockdown actor. [] The conservative estimate o atigue lie is the product o the Bayesian Lie and knockdown actor. NUMERICAL EXAMPLE CONSTANT CYCLE TEST As a case, the Bayesian technique is applied to steel 434 material, whose strain-lie atigue parameters are shown in Table [4]. The strain-lie curve or steel 434 material is shown as Mean curve in Figure. The uncertainty in the parameters leads to uncertainty in strain amplitude ( t ). First the sensitivity o strain amplitude with respect to each o the parameters has been determined, and ound to be positive or all parameters. The eect o the uncertainty in strain amplitude has been plotted through the Maimum and Mmum curves in Figure. These curves have been plotted considering each o the parameter at its algebraic maimum and mmum respectively, governed by its variability. 3 In the irst eample, it is assumed that the material is ailed at N =, reversals and three data are available at that reversal value. Since the crossover reversal or this material is in the order or,, the atigue strain is in the elastic region. For the material parameters in Table, the value o analytical atigue strain is.3. t Table : Strain-lie atigue parameters or steel 434 Parameter Value Elastic stiness (E) 8,9 MPa Fatigue ductility coeicient (e ).83 Fatigue ductility eponent (c) -.6 Fatigue strength coeicient (s ),73 MPa Fatigue strength eponent (b) -.9 e t..... 3 4 6 7 N Figure : Strain-lie curve or steel 434 9 8 7 6 4 3 Mean Mmum Maimum. 3 3. 4 4. -3 Figure : Histogram o atigue strain at N =, In order to ind the tial error bounds o the atigue model, all parameters aecting the atigue strain are assumed to vary uniormly % o their nominal value. When all the parameters are varying simultaneously, the value o t varies between.4 and.4. Then, the error bounds are calculated considering the maimum deviation rom the mean, i.e.,.4.3 =.3, which is about 39% o the mean. Hence, the error bounds are 39% o the mean.
() In addition to the error bounds, detailed distribution o t can be plotted using MCS. First,, samples o material parameters are randomly generated according to uniorm distribution. Then, the histogram o t is plotted by applying these samples to Eq. (). Figure shows the histogram o atigue strain at N =, reversals. The solid curve connects the midpoint o each bin in the histogram. The PDF o atigue strain is obtained by scaling down the curve, such that the area under the curve is unity. It turns out that the atigue strain distribution has ollowing parameters: Mean.33 SD.33 In Figure 3, the normal distribution with the same mean and standard deviation is plotted. It is clear that the histogram looks close to a normal distribution with same parameters. Although variability associated with atigue should be obtained rom more rigorous method, it is assumed that the variability is normally distributed with mean.33 and COV %. With error bounds and variability, now it is possible to perorm Bayesian update. Let us consider that three atigue s are perormed. Ater normalizing by analytical atigue strain, the three data strains are.8,., and.. These three normalized data correspond to strain values o.8,.346, and.379, respectively. 4 8 6 4 Normal distribution pd o total strain. 3 3. 4 4. -3 Figure 3: Fitting the distribution o atigue strain using normal distribution Figure 4 shows distribution o estimated atigue strain at each stage o Bayesian update. The dotted lines show the likelihood unction or each result, while the solid lines show the updated distributions. The inal updated distribution o the atigue strain has the ollowing parameters: Mean.37 SD.7 () (6) Note that the mean does not change signiicantly, while the standard deviation o the updated Bayesian distribution is about % o that o the original distribution. Thus, the three data eectively reduce the uncertainty in the atigue ailure strain. Table tabulates the variation o the parameters o the distribution o atigue ailure strain with each stage o Bayesian update. Since the atigue strain is distributed, it is better to provide a conservative estimate o the atigue strain using a knockdown actor. When the variability is normalized, the mean is shited to., while retang the COV. Hence, the knockdown actor or the normalized variability, governed by N(.,. ) is calculated to be.97, using the mean value o etreme distribution in Eq. (4). Hence, the conservative estimate o atigue ailure strain at, reversals becomes t.97. 4 6 4 Initial Distribution Initial.7.8.9...3.4 with irst.7.8.9...3.4 with second 3 3.7.8.9...3.4 Final Bayesian stress with third.7.8.9...3.4 Figure 4: Bayesian update history o ailure atigue strain Table : Variation o parameters with each stage o Bayesian update Mean SD Initial..4 Ater irst.9.74 Ater second.9667.9 Ater third..
() () () NUMERICAL EXAMPLE CONSTANT STRAIN TEST For the constant strain, it is assumed that the amplitude o strain applied to the machine is constant at.. For the material properties in Table, the value o analytical atigue lie is, reversals. Since the crossover reversal value or this material is 4,3 reversals, the assumed atigue strain is in the elastic region. Due to computational diiculties, the strain-lie epression is solved or y = log (N), rather than or the strain lie, N. Hence, the analytical value or y would be 4.. In order to ind the tial error bounds o the atigue model, all parameters aecting the atigue strain are assumed to vary uniormly % o their nominal value. The error bounds or y have been determined to be 6% o the mean. In addition to the error bounds, detailed distribution o y can be plotted using MCS. First,, samples o material parameters are randomly generated according to uniorm distribution. Then, the histogram o y is plotted by applying these samples to Eq. (). Figure shows the PDF o y at atigue strain, t.. It turns out that y has the ollowing distribution parameters: Mean 3.8937 SD.99 (7) For the want o results, the variability o the results have been assumed to a normal distribution with mean 3.8937 and COV.% The three additional cases or y have been assumed to at.8,. and. times the analytical value... pd o log(n) Normal Distribution 3 3. 4 4. Figure 6: PDF o log (N) and that o normal distribution with the same mean and standard deviation PDF o log (N).6.4. PDF o product unction Polynomial itted..8.6.4. 3 3. 4 4. Figure : PDF o log (N) at ied strain t. The PDF o log (N) can be modeled as a product o PDF o a normal distribution and a polynomial. The normal distribution that has the same parameters as that o the PDF o log (N) is shown as the curve with dotted lines in Figure 6. Figure 7 plots the product unction determined rom the two curves in Figure 6. An 8th degree polynomial is itted to estimate the product unction. The dashed lines show the polynomial in Figure 7.. 3 3. 4 4. Figure 7: Product unction that models the dierence between PDF o log (N) and that o normal distribution Figure 8 shows distribution o estimated atigue lie at each stage o Bayesian update. The dashed lines show the likelihood unction or each result, while the solid lines show the updated distributions. The inal updated distribution o the y has the ollowing parameters: Mean 4.76 SD.4 Since the atigue lie is distributed, it is better to provide a conservative estimate o the atigue lie using knockdown actor. When the variability is normalized, the mean is shited to., while retang the COV. Hence, the (8)
knockdown actor or the normalized variability, governed by N(,.) is calculated to be.967. Hence, the conservative estimate o atigue ailure lie at strain amplitude o. is N = 3.889 = 769 reversals. Table 3 tabulates the variation o the parameters o the distribution o atigue ailure strain with each stage o Bayesian update. The Bayesian update or the atigue ailure lie has been perormed in Eample above, considering the PDF o lie as tial distribution. The importance o prior or tial distribution could be emphasized by perorming a Bayesian update with prior as uniorm distribution between error bounds and comparing results. The Bayesian update history o ailure atigue lie with a uniorm distributed prior is shown in Figure 9. Table 4 compares the results o these two Bayesian updates. It is noted that when the actual PDF o lie is used as prior, the mean o inal distribution decreases by.3% and the standard deviation o the inal distribution reduces by 3.39%. This suggests that having a better knowledge o the prior distribution results in a better coeicient o variance or the distribution ater update. Initial Distribution Initial Initial Distribution Initial.8.9...3 with irst.8.9...3 with second.8.9...3 with third 3 3 Final Bayesian strain lie.8.9...3.8.9...3 with irst.8.9...3 with second.8.9...3 with third 3 3 Final Bayesian strain lie.8.9...3 Figure 8: Bayesian update history o ailure atigue lie Table 3: Variation o parameters with each stage o Bayesian update Mean SD Initial.. Ater irst.936. Ater second.96.88 Ater third.44.8 EFFECT OF INITIAL DISTRIBUTION Figure 9: Bayesian update history o ailure atigue lie with a uniorm distribution or prior inormation Table 4: Comparison o parameters o distribution ater Bayesian update when the prior distribution is uniorm or the actual PDF Prior Distribution Parameter Uniorm Distribution Actual Distribution Mean o inal.67.44 distribution SD o inal distribution.9.8
EFFECT OF TEST CASES The normalized cases considered in the eample above, are.8,. and.. The Bayesian update perormed with an altogether dierent set o normalized results such as.,. and. yields an interesting result. Figure compares the distribution and mean o atigue ailure lie ater Bayesian update or the two cases o s discussed above. Case reers to the result set o [.8,.,.]. Case reers to the result set o [.,.,.]. It is noted that both the distributions are one and the same. 4 8 6 4 Srain lie ater Bayesian update Case Case mean Case Case mean.8.9...3 Figure : Strain lie ater Bayesian update or two dierent set o cases considered or the update It is noted that the average o the two sets o results is same, which is.67.now, Case has two results in the etremes o the error bounds, and hence, the likelihood unction o those results are truncated. Case has all results centered on. and hence their complete likelihood unction is utilized in Bayesian update. Yet, the strain lie distribution ater Bayesian update is one and the same or both cases. Hence, it can be concluded that the distribution o atigue ailure lie ater Bayesian update will depend on the average o the normalized cases considered. The dependence o Bayesian update on the average o the three cases have been urther veriied by considering a set o identical values, [.67,.67,.67] which resulted in an identical distribution as shown in Figure. It has also been veriied that the changing the order in which the cases are considered or the update does not aect the distribution o strain lie ater Bayesian update. CONCLUSIONS Bayesian update has been demonstrated as a good method to reduce uncertainty in number o reversals (N) or constant strain amplitude case and also to reduce uncertainty in strain amplitude or constant lie case, with the knowledge o cases. It is seen that the standard deviation o the distribution obtained ater the Bayesian update, is almost hal o that o the tial error distribution. Since only the average o the three cases is the main parameter, there is no eect o having repetitive values or the cases. Also, changing the order, in which the cases are considered or the Bayesian update, doesn t aect the actual atigue lie. Having a better knowledge or the prior leads to less scatteredness in distribution obtained ater Bayesian update. REFERENCES. Jungeun An, Erdem Acar, Raphael T. Hatka, Nam H. Kim, Peter G. Iju, Theodore F. Johnson, Is the Lowest Test Data Conservative Enough?. E. Acar, J. An, R. Hatka, N. Kim, P. Iju, and T. Johnson, Options or Using Test Data to Update Failure Stress, AIAA-7-97, 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conerence, Honolulu, Hawaii, Apr. 3-6, 7 3. Arthur P. Boresi, Richard J. Schmidt, Advanced Mechanics o Materials Sith Edition. 4. Fracture Control Program Report 3. Society o Automotive Engineers (SAE), Integration o Probabilistic Methods into the Design Process,, Aerospace Inormation Report 8, Warrendale, PA, 997. 6. J.R.M. Hosking, J.R. Wallis, and E.F. Wood, Estimation o the Generalized Etreme-Value Distribution by the Method o Probability-Weighted Moments, Technometrics, Vol.7, No.3, August 98, pp. -6. 7. Wirsching, P.H., Literature Review on Mechanical Reliability and Probabilistic Design, Probabilistic Structural Analysis Methods or Select Space Propulsion System Components (PSAM), NASA Contractor Report 899, Vol. III, Washington, D.C., 99. 8. Acar, E., Kale, A., and Hatka, R.T., "Comparing Eectiveness o Measures that Improve Aircrat Structural Saety," ASCE Journal o Aerospace Engineering, Vol., No.3, July 7, pp. 86-99.