International Journal of Fatigue 27 (2005) 966–973 www.elsevier.com/locate/ijfatigue
Fatigue life prediction based on variable amplitude tests—specific applications Thomas Svenssona, Pa¨r Johannessona,*, Jacques de Mare´a,b a
Fraunhofer Chalmers Research Centre for Industrial Mathematics, Chalmers Science Park, SE 412 88 Go¨teborg, Sweden b Mathematical Statistics, Chalmers University of Technology, SE 412 96 Go¨teborg, Sweden Received 21 July 2004; received in revised form 31 August 2004; accepted 18 November 2004
Abstract Three engineering components have been tested with both constant amplitude loading and with different load spectra and the results are analysed by means of a new evaluation method. The method relies on the Palmgren-Miner hypothesis, but offers the opportunity to approve the hypothesis validity by narrowing the domain of its application in accordance with a specific situation. In the first case automotive spot weld components are tested with two different synthetic spectra and the result is extrapolated to new service spectra. In the second case, the fatigue properties of a rock drill component are analysed both by constant amplitude tests and by spectrum tests and the two reference test sets are compared. In the third case, butt welded mild steel is analysed with respect to different load level crossing properties and different irregularity factors. q 2005 Elsevier Ltd. All rights reserved. Keywords: Cumulative fatigue; Confidence limits; Life prediction; Model errors
1. Introduction
the Palmgren-Miner rule
Fatigue life assessment in industrial applications involve a lot of sources of uncertainty such as material strength, notch geometries, defect contents and residual stresses. Often, these uncertainties are substantial enough to prevent detailed analysis of the fatigue phenomenon as such, and thereby, force the designer to base the assessment on the global behaviour of the component in question, i.e. from an experimental Wo¨hler curve. When components are subjected to variable amplitude service loads, additional uncertainties arise; how is the loading in laboratory tests related to the loads that could be expected to appear in service? Traditionally this problem is solved by using the simplifying assumption of damage accumulation, and constant amplitude tests in laboratory are transformed to variable amplitude severity by
DZ
* Corresponding author. Tel.: C31 772 4295; fax: C31 772 4260. E-mail address:
[email protected] (P. Johannesson).
0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2004.11.010
m X ni N iZ1 i
which says that a load cycle with amplitude Si adds to the cumulative damage D, a quantity (1/Ni). Here, Ni denotes the fatigue life under constant amplitude loading with amplitude Si and ni is the number of load cycles at this amplitude. The lack of validity of this accumulation rule has been demonstrated in many applications and in consequence its usage will introduce uncertainties which must be compensated for by safety factors, see for instance [1–4]. One possible way to diminish the deviations from the damage accumulation rule is to perform the laboratory experiments closer to the service behaviour with respect to the loads. A method for establishing a Wo¨hler curve based on variable amplitude loads has recently been developed and is presented in a parallel paper [5]. The use of this method should be customised to each specific application by performing laboratory tests with load spectra covering different service requirements. One idea is that service measurements are used to establish a few reference load spectra for use in laboratory tests. Based on the resulting variable amplitude Wo¨hler curve, fatigue
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life can be predicted for load spectra similar to the reference types. Three different components, each with a specific set of load spectra, were investigated. (1) Spot welded components from an automotive application were subjected to spectra of different types, two synthetic spectra and one from an automotive proving ground. (2) Suspension arms from rock drilling machines were subjected to a large number of load spectra with varying irregularity and mean values, all created from field measurements. (3) Welded specimens of mild steel were subjected to four different load spectra, created in the purpose of studying the effect of irregularity and spectrum type. All three applications were investigated with respect to the new method for establishing a variable amplitude Wo¨hler curve. Fatigue life tests were performed and the new method was compared to damage accumulation assessment based on constant amplitude tests. It was found that the prediction errors diminish when constant amplitude reference curves are replaced by curves based on reference spectra. In addition, the influences of irregularity, mean values, and spectrum type are discussed.
2. Estimating material fatigue properties from variable amplitude tests The traditional Palmgren-Miner method is adjusted here for more accurate life predictions. The theory is evaluated in a parallel paper [5] and is summarized here: The variable amplitude load is described as a spectrum of load amplitudes, calculated by for instance Rain Flow Count and discretized into m load amplitudes Si with corresponding relative frequency of occurrence, ni: fðni ; Si Þ; i Z 1; 2; .; mg:
(1)
The Wo¨hler curve is described in the form N ¼ a,SKb eq where a and b are material parameters and Seq is the damage equivalent load amplitude for the actual load spectrum defined as " #1=b m X b n i Si : Seq Z iZ1
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The primary aim of the laboratory test is to estimate the material parameters a and b, which is straightforward in the case of constant amplitude tests. However, model (2) is nonlinear in the parameter b since the equivalent load Seq,j depends on b. This complication is solved in [5] by using the maximum-likelihood method. From each experiment, we can then find the estimated parameters, denoted a^ and b^ from any set of spectrum tests. This method thereby estimates parameters for a certain set of spectrum test results, and the choice of reference spectra and the amount of extrapolation in a specific application can be chosen with respect to engineering judgements.
3. Statistical evaluation of test results The results from the new estimation procedure can be evaluated in detail using standard statistical tools. Except the parameter estimates, one also obtains an estimate of the ^ Z s2 , and variance of the random property 3; Var½3 uncertainties in parameters, estimated lives, and predicted lives can be evaluated. In particular, prediction intervals can be calculated for a single spectrum prediction or for a group of predictions giving the possibility to judge if there are systematic deviations from the damage accumulation rule or if the deviation may be a result of randomness. Here, we will study mainly two questions about the cumulative damage results. The first is an extension of the usual question: Can fatigue life be predicted for a spectrum result if the material properties are estimated from constant amplitude tests? Here, this question can be extended thus: Can fatigue life be predicted for a certain spectrum result if the material properties are estimated from other spectra, and how close should the reference spectra be to the prediction situation? A second question is sometimes raised in view of spectrum fatigue test results: is the exponent b different for different types of spectra or can it be regarded as a pure component property? The first question can be studied with the following property Nrel Z
Nf ; Npred
(3)
The corresponding statistical model for the fatigue test is ln Nj Z ln a K b ln Seq;j C 3j ;
j Z 1; 2; .; n
(2)
where n specimens have been tested, each at the equivalent load amplitude Seq,j resulting in the fatigue life Nj. The randomness due to the specimen strength is described by the independent additive random variables 3j, each assumed to follow the Gaussian distribution 3j wNð0; s2 Þ; j Z 1; 2; .; n:
i.e. the relative life described as the experimental life Nf compared to the predicted life, Npred. By the methodology developed in [5], one can calculate prediction limits for this property, either for a single prediction or for the geometric mean of several predictions. If such prediction limits cover unity, the Palmgren-Miner rule cannot be rejected, but the mean deviation may be explained by random behaviour. Throughout this paper we will use 95% confidence and prediction limits, which means that if the test procedure
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was repeated a large number of times, 95% of the intervals would cover the true value. The second question can be evaluated by studying the differences between exponents. Assume that the parameters {a1, b1} and {a2, b2} have been estimated from two different sets of fatigue test results and that one wants to check if the exponents b1 and b2 differ significantly. The variance of the difference is
s212
1 2 1 ^ ^ ^ ^ Z Var½b1 K b2 Z Var½b1 C Var½b2 Z s C q1 q2 ðn K 2Þs21 C ðn2 K 2Þs22 1 1 C z 1 ð4Þ q1 q2 n1 C n2 K 4
where we have assumed that the two data sets are independent and that the variance of the random variable 3 is the same in both experiments; s2. The approximation indicates that we use the estimated standard deviations s1 and s2 from each set of test result. The number of specimens in each test are n1 and n2 respectively, and the quantities q1 and q2 depend on which spectra that have been used in the different tests. Their expressions are quite complex and are not given here, but is defined in the parallel paper [5].
4. Three applications The method has been applied to three different engineering problems, namely one related to automotive applications, testing spot welds, one dealing with service loads on suspension arms for a rock drill rig, and one studying welded mild steel. All three applications focus on engineering components expected to contain defects large enough to support an assumption that crack growth is the dominating damage mechanism. Consequently, the linear damage accumulation concept is quite plausible.
Fig. 1. Load spectra used for the spot welded specimens.
4.2. Load spectra In all applications, tests have been performed both at constant amplitude (CA) and at variable amplitude (VA) loading. In the variable amplitude case, blocks of load cycles have been applied, and each block has been repeated until failure. The life is measured as the number of individual load cycles until failure. The load spectra are described by means of their Rain Flow Count spectra (1) and their block length. In Fig. 11, parts of the sample paths are shown and in Figs. 1–3 the spectra are illustrated. In each diagram the x-axis shows how large a proportion of the load ranges exceeds the corresponding y-axis range value. For additional characterisation of the load spectra, we also define the irregularity factor I and the load ratio R
IZ
#fup K crossings of the mean levelg S and R Z min #flocal maximag Smax
4.1. Laboratory test All three application tests were performed in servohydraulic test machines in laboratory environment. The load signals were monitored during the tests and their deviations from the target signal were found negligible. The specimen for the spot weld application consisted of two 1-mm thick steel sheets connected with one spot weld with nominal diameter 5.6 mm, see [6] for further details. The welded specimen consisted of two steel plates, 10-mm thick and 30-mm wide, welded together by a transversal butt weld. The suspension arms were cast steel components, hardened and tempered. They were subjected to a combination of tensile and bending stresses with the highest stresses at the quite rough surface.
Fig. 2. Load spectra used for the suspension arm.
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Table 2 Suspension arm spectra
Fig. 3. Load spectra used for the butt welded specimens.
where the # symbol denotes the number of events, and Smin and Smin are the global load minimum and maximum respectively. The spot weld application uses three different variable amplitude load signals, two synthetic spectra denoted ‘linear’ and ‘Gaussian’, referring to their level crossing properties. The third signal is condensed from a test track at an automotive proving ground. All three spectra are scaled to give different equivalent load amplitudes. They are illustrated in Fig. 1 and their properties are summarized in Table 1. Constant amplitude tests have been performed at three different load ratios, 10 specimens at RZK1, 7 at RZ0.1, and 12 at RZ0.5. For the suspension arm application, 11 different load spectra have been extracted from service measurements. All measured time signals are filtered by rain flow count filtering in order to be able to perform laboratory tests in reasonable time. They have also been scaled in such a way that the maximum load in each spectrum reflects the capacity of the actual testing machine. The resulting spectra have block lengths ranging from 122 to 4271 cycles. Their irregularity factors range from 0.35 to 1 and their load ratio from K1 to K0.6 (Table 2). In Fig. 2, the cycle contents of all 11 spectra are illustrated. Constant amplitude tests were performed with eight specimens at the load ratio RZK1. For the last application, the butt welded joints, four different synthetic spectra are used. They represent two different level crossing distributions, called ‘concave’ and ‘convex’, each applied at two different irregularity factors, IZ0.5 and IZ0.99.
Name
Block length
R
I
Number of spec.
Spect 1 Spect 7 Spect 8 Spect 9 Spect 10 Spect 11 Spect 12 Spect 13 Spect 14 Spect 15 Spect 16
166 122 232 293 957 835 4271 337 124 962 146
K0.7 K1 K1 K0.7 K0.6 K0.6 K0.6 K1 K1 K1 K1
1.00 0.35 0.76 0.97 0.88 0.56 0.58 1.00 0.85 0.99 0.99
1 1 1 1 1 1 4 1 2 1 4
Each type of spectrum is scaled and translated in different ways giving different R-ratios. Illustrations are in Fig. 3 and properties in Table 3. Constant amplitude tests have been performed with 10 specimens at RZ0. 4.3. Results The results from the analysis are given in different ways. The main idea behind the developed method is to estimate the Wo¨hler curve using a pre-chosen set of experimental results, and then use this result for predictions including prediction intervals for 95% probability of coverage. In our three applications, this is demonstrated by using a certain set of the experimental results for estimation, the estimate set, and another set for check of predictions, the validation set. By comparing the results from the validation set with the prediction limits one can judge if the damage accumulation theory is valid, or if systematic differences can be detected. This is done in two ways: (i) by including both sets in figures, and (ii) by checking if the prediction intervals for the geometric mean of the relative life, N^ rel , cover the observed value. In each figure the experimental results from the estimation set are illustrated as triangles and the results from the validation experiments as dots. The estimated Wo¨hler curve is a solid line, the 95% confidence band for this curve is dash–dot lines and the 95% prediction band is given as dashed lines. 4.3.1. Spot welds For the automotive application with spot welds the two synthetic spectra can be considered as reference spectra for determining the fatigue properties, replacing the ordinary Table 3 Butt weld spectra
Table 1 Spot weld spectra Name
Block length
R
I
Number of spec.
Gaussian Linear Test track
66,667 66,667 914
0 0 K1
0.99 0.79 0.83
7 5 7
Name
Block length
I
Number of spec.
Concave Concave Convex Convex
96,000 96,000 96,000 96,000
0.99 0.5 0.99 0.5
9 8 9 8
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T. Svensson et al. / International Journal of Fatigue 27 (2005) 966–973 Table 4 Estimated parameters for the spot weld case Estimation set Constant amplitude, all Constant amplitude, RO0 Linear and Gaussian
a^ 6
3.8!10 4.0!106 9.3!106
b^
s
4.15 4.75 6.09
0.60 0.41 0.27
Table 5 Estimated relative life for spot weld predictions
Fig. 4. Wo¨hler curve for spot welds. The estimation set is linear and Gaussian spectra and the validation set is the proving ground spectrum. Confidence and prediction bands for the curve are included.
constant amplitude reference tests. The results are then used to predict the Test track spectrum tests. The result is illustrated in Fig. 4 and as a comparison the result from using constant amplitude as reference is shown in Fig. 5. It can be seen in the figures that the predictions fall well within the 95% prediction limits and both methods seem to be useful. However, the uncertainty around the Wo¨hler curve based on constant amplitude tests is much larger, and thereby, demands larger safety factors. The parameter estimates in the three cases are given in Table 4 where it is seen that the estimated standard deviation is considerably larger for the constant amplitude cases. We next turn to the question if there is a significant systematic deviation in the predictions. This is not expected for the three actual analyses since the figures suggest that
Estimation set
N^ rel
95% prediction interval
Constant amplitude, all Constant amplitude, RO0 Linear and Gaussian
0.92 1.22 1.19
[0.56; 1.51] [0.84; 1.76] [0.91; 1.56]
the predictions fall well within the 95% limits. However, a more objective answer can be obtained by formally testing the simultaneous prediction results using the relative life property (3). The results from such an analysis is shown in Table 5. The prediction intervals all cover unity and no significant deviations from the Palmgren-Miner rule can be detected. Finally, we ask the question if there is a significant difference in slopes for the Wo¨hler curve estimates. The result using procedure (4) is for all CA-tests compared to the Linear/Gaussian combination bC K bLG Z 1:94G1:66 ð95%Þ: The interval does not cover zero, which means that there is a significant difference in slopes. The same calculation for the CA-tests with positive load ratios compared to the Linear/Gaussian combination gives bC;RO0 K bLG Z 1:34G1:25 ð95%Þ: This interval does not cover zero either, which means that even with the restriction to positive load ratios the significant difference remains.
Fig. 5. Wo¨hler curve for spot welds. The estimation set is constant amplitude with RO0 and the validation set is the proving ground spectrum. Confidence and prediction bands for the curve are included.
4.3.2. Suspension arm For the rock drill application, a large number of spectra have been used and a specific reference set was not defined beforehand. One possible way to compare the constant amplitude reference with a spectrum reference is to use the two spectra with the highest and lowest equivalent loads, respectively. These are the two spectra denoted spect 16 and spect 12 in Table 2, and these spectra also have been replicated. The results are then used to predict the rest of the spectrum tests. The result is illustrated in Fig. 6 and as a comparison the result from using constant amplitude as estimation set is shown in Fig. 7. The parameter estimates in the two cases are given in Table 6. Is there a significant systematic deviation in the predictions? This seems to be the case when the constant amplitude tests are used as reference according to Fig. 7.
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Table 7 Estimated relative life for suspension arm predictions Estimation set
N^ rel
95% prediction interval
Constant amplitude Spectra 12 and 16
0.36 0.76
[0.17; 0.73] [0.35; 1.69]
The difference between the slopes for the two Wo¨hler curve estimates is estimated at bC K bS Z 0:79G1:99 ð95%Þ which clearly covers zero and is far from significant.
Fig. 6. Wo¨hler curve for suspension arm. The estimation set is spectra 12 and 16 and the validation set is the other spectra. Confidence and prediction bands for the curve are included.
4.3.3. Butt welds For the butt weld specimens four spectra have been used which originally were constructed to test the PalmgrenMiner rule [7]. Here it is interesting to see if one type of spectra can be predicted by the other or if the irregularity factor influences the prediction results. The results are illustrated in Figs. 8–10. The parameter estimates in the three cases are given in Table 8. Is there a significant systematic deviation in the predictions? This seem to be the case according to Figs. 8 and 9. The results from formal tests using the relative life property (3) are shown in Table 9. Both intervals are far from unity and significant systematic deviations are detected. The difference between the slopes for the two first Wo¨hler curve estimates is bC K bConcave Z K0:37G1:82 ð95%Þ where no statistical significant difference can be detected.
Fig. 7. Wo¨hler curve for suspension arms. The estimation set is constant amplitude and the validation set is the other spectra. Confidence and prediction bands for the curve are included.
The results from formal tests using the relative life property (3) is shown in Table 7. No significant deviations from the Palmgren-Miner rule can be detected when spectra 12 and 16 are used as reference, but predictions based on the constant amplitude reference show significantly nonconservative results. Table 6 Estimated parameters for the suspension arm case Estimation set Constant amplitude Spectra 12 and 16
a^ 14
25!10 0.15!1014
b^
s
3.93 3.14
0.56 0.67
Fig. 8. Wo¨hler curve for butt welds. The estimation set is the convex spectra and the validation set is the concave spectra. Confidence and prediction bands for the curve are included.
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T. Svensson et al. / International Journal of Fatigue 27 (2005) 966–973 Table 8 Estimated parameters for the butt weld case Estimation set Constant amplitude Convex Concave
a^ 15
1.6!10 8.4!1015 4.9!1015
b^
s
4.19 4.56 4.29
0.35 0.52 0.42
Table 9 Estimated relative life for the butt weld predictions
Fig. 9. Wo¨hler curve for butt welds. The estimation set is constant amplitude and the validation set is the concave spectra. Confidence and prediction bands for the curve are included.
5. Discussion The three different applications show different results in this analysis, which makes it difficult to draw any overall conclusions about the use of the Palmgren-Miner rule. The primary aim of this work is not to make thorough analyses of each specific application, but rather to present some tools for such an analysis. Therefore, we only give some overall comments to the different cases. 5.1. Spot welds In this case, it seems to be an advantage to use the spectrum laboratory tests for determining the material properties. The scatter diminishes compared to the CA reference case and the predictions for the test track
Fig. 10. Wo¨hler curve for butt welds. The estimation set is the spectra with IZ0.99 and the validation set is the spectra with IZ0.5. Confidence and prediction bands for the curve are included.
Estimation set
N^ rel
95% prediction interval
Constant amplitude Convex
1.86 2.66
[1.24; 2.78] [1.82; 3.89]
spectrum show no systematic errors. We here compared the spectrum reference method with CA at three different load ratios, which may be seen unfair to the CA case. However, the prediction set was actually subjected to the global load ratio RZK1, but the reference spectra were applied at zero load ratio. In [6,8] it is concluded that constant amplitude at zero load ratio is a good reference test for spot welds, and as can be seen from our results the scatter is diminished by excluding the results from CA, RZK1. Of course, when many different types of tests are available, one can always find the best combination for a certain result, but in practice, this situation is not often at hand. It would then be good to use reference spectra as close to the application as possible to avoid systematic errors. There is a significant difference in slopes for the spot weld case, which should be further investigated to look for physical explanations. 5.2. Suspension arm In this case, the flexibility of the new method has been utilized by applying several different spectra for the experiments. In our analysis, we have chosen to compare the constant amplitude reference to a two spectra reference set. Here, we see a clear difference in the ability of predicting other spectrum tests, since the CA reference gives significantly nonconservative results, in contrast to the VA reference. This result could be explained by the fact that the CA tests are performed at quite high load levels and that the predictions are mainly extrapolations with respect to the equivalent load. However, the VA spectra also include very high load ranges. The load ratio influence is not expected to be important in this application, since all tests are performed with globally negative load ratios, within the interval R2[K1;K0.55]. However, in the irregular sequences, the mean value influence may be important, since many small cycles then have positive load ratios. Further investigations into this problem are planned.
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From the two industrial applications, one can conclude that it is possible to find spectrum tests that diminish the scatter around the estimated Wo¨hler curve but are still valid for predictions without systematic errors. The result of such a procedure could lower the safety factors due to better precision. Statistical confidence and prediction intervals are very useful here. Combinations of arbitrary spectrum tests can be tested for systematic errors. Thereby, it is possible to rigorously investigate influences from mean values, irregularity, residual stresses, spectrum type, and other factors. The method gives a choice for the engineer: use all possible spectra for estimation and accept large safety factors, or choose a reference set of spectra that can be judged to comply with the specific application, giving more accurate predictions. Acknowledgements Fig. 11. Sample paths from loads of the three different applications.
No significant difference can here be detected between the CA and the VA slopes. 5.3. Butt welds In the butt weld case the spectra were constructed with the intention to test the Palmgren-Miner rule and not for a specific service application. The results also clearly show systematic deviations when predicting the concave spectrum, both when using CA reference and using the convex reference. This result shows that the question that should be asked is not whether constant amplitude or spectrum tests as reference should be used, but rather how to find reference spectra that are close enough to the applications to give reasonable predictions. The large differences between the convex and concave spectra cannot be seen in Fig. 3, since this only describes the cycle contents. The differences are more pronounced in the level crossing properties, reflecting the tendency that small cycles in the convex time histories have large mean levels than the concave ones. These results would be very interesting to analyse further for studying mean level influences on individual cycles. The irregularity factor does not seem not to be very influential in this case as can be seen from Fig. 10, but the large scatter due to the spectrum type influence makes this judgement uncertain.
6. Conclusions The newly developed tool for estimating the Wo¨hler curve from arbitrary sets of spectrum tests has proved useful.
This work was done in close cooperation with Kenneth Weddfeldt at Atlas Copco Rock Drills, Hans-Fredrik Henrysson at Volvo Cars, and Erland Johnson at SP, the Swedish National Testing and Research Institute. The project was also funded by Bombardier Transportation, Sandvik Materials and Technology, and the Swedish foundation for applied mathematics (STM). References [1] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. Int J Fatigue 1998;20(1):9–34. [2] Schu¨tz W. Fatigue life prediction—a review of the state of the art. In: Rossmanith HP, editor. Structural failure, product liability and technical insurance. Amsterdam: Elsevier; 1993. [3] Gurney TR. Comparative fatigue tests on fillet welded joints under various types of variable amplitude loading, vol. III-B. Proceedings of the 12th international offshore mechanics and arctic engineering conference. New York: Materials Engineering, American Society for Mechanical Engineers, USA. p. 537–42. [4] Berger C, Eulitz K-G, Hauler P, Kotte K-L, Maundorf H, Schu¨tz W, et al. Betriebsfestigkeit in Germany—an overview. Int J Fatigue 2002;24:603–25. [5] Johannesson P, Svensson T, de Mare´ J. Fatigue life prediction based on variable amplitude tests-methodology. Presented at the conference: cumulative damage, Seville; 2003. [6] Henrysson H-F. A comparison of variable amplitude fatigue life predictions for spot welds. In: Blom AF, editor. Proceedings of the eighth international fatigue congress, fatigue 2002, Stockholm, Sweden, 2002. p. 397–404. [7] Holmgren M, Svensson T. Investigation and improvement of a new model for fatigue life prediction based on level crossing. In: Schueller GI, Shinozuka M, Yao JTP, editors. Proceedings of ICOSSAR ’93—structural safety and reliability, vol. 2. Rotterdam, Netherlands: Balkema; 1993. p. 1181–4. [8] Sheppard S, Pan N, Bai Z, Sheu Y-C. Refinement and verification of the structural stress method for fatigue life prediction of resistance sport welds under variable amplitude loads. SAE Technical Paper, 2000-012727; 2000.
