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Equivalent life distribution and fatigue failure probability prediction
International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Equivalent life distribution and fatigue failure probability prediction Liyang Xie 1 Northeastern University, Shenyang 110006, People’s Republic of China
Abstract Equivalent fatigue life distributions under two-level cyclic stress are analyzed experimentally, and a model of describing equivalent life distribution is presented. Based on the equivalent life distribution model, an ‘‘equivalent damage–equivalent life distribution’’ method is developed to predict fatigue failure probability under variable amplitude loading. The underlying principles are fatigue damage accumulation rule and ‘‘load cycles–fatigue life’’ interference theorem. In the equivalent damage–equivalent life distribution method, two parameters, i.e. mean and standard deviation of the (equivalent) life distribution are used to describe cumulative fatigue damage effect. A basic feature of the equivalent damage–equivalent life distribution method is its capability of reflecting the loading history dependent change of the equivalent life distribution. Tests under multi-level stress show a good agreement between predicted fatigue failure probability and the test results. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Equivalent life distribution; Fatigue failure probability
1. Introduction Stress–strength interference analysis has been widely applied to fatigue probability analysis [1–5]. Nevertheless, its direct use is limited to the simplest situation, i.e., constant amplitude cyclic loading. Alternatively, the ‘‘conditional reliability – equivalent life method’’ [6] has been applied directly or indirectly to a broad range of fatigue failure probability predictions. This conditional reliability – equivalent life method is also a simple method and therefore, is of rather a limited applicability. More sophisticated studies on fatigue failure probability have been published in recent years [7-18]. In these literature, residual fatigue life distributions under constant amplitude cyclic loading and two-stage loading conditions [13,14], impact of load changing on lifetime distributions [15,16], and computer simulation of fatigue reliability analysis [17,18] are investigated. These studies have provided plenty of insights into fatigue failure probability under variable amplitude loading. In the present paper, an equivalent fatigue life distribution model is presented, and a fatigue failure probability prediction method is proposed based on the equivalent life distribution model, cumulative fatigue damage rule, and load cycles – fatigue life interference theorem.
1 Present address: Otto-von-Guericke-University, IAUT, Department of Plant Design and Safety, Postfach 4120, 39016 Magdeburg, Germany. Tel.: 0049-391-6718113; Fax: 0049-391-6711128. E-mail address: [email protected] (L. Xie)
2. Equivalent life distribution under variable amplitude loading Fatigue life distribution is generally load spectrum dependent. Therefore, a thorough investigation on the relationship between life distribution and the respective loading history is necessary to develop fatigue failure probability model. In this regard, lots of tests have been done in a rotated-bending fatigue test machine, using smooth specimens made of normalized 0.45% carbon steel or hot rolled alloy 16Mn steel. The material compositions, mechanical properties and specimen geometry are the same as presented in [19]. For the carbon steel, tests are carried out under both constant amplitude loading (three different stress levels are used) and two-level cyclic loading (some combinations of the constant amplitude loads). The amplitudes of the three stress are 366 , 331 and 309 MPa, respectively. All of the three stress amplitudes are less than the material yield limit 380 MPa. Fatigue tests under the three constant amplitude cyclic stresses are carried out first, so as to obtain the means and standard deviations of the fatigue lives. After that, two-level stress tests under low ! high (331 ! 366 MPa) and high ! low (331 ! 309 MPa) loading sequences are done in order to investigate fatigue life distribution under variable amplitude loading. Test procedure and data processing for the two-level stress test are as the following: The first-level stress operates only a given cycle number, after that the second-level stress begins to operate and lasts to specimen failure. The cycle ratios (life fractions) of the first-level stress are set as 0.27, 0.53 and
0308-0161/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(98)00117-3
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L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Table 1 Fatigue test results and the life distribution parameters of the 0.45% carbon steel Stress level and/or loading sequence
Sample size
366 Mpa
15
331 Mpa
18
309 Mpa
16
331 ! 366
14
40300
14
80600
15
120900
13
40300
13
80600
12
120900
331 ! 309
Cycles of 1st-level stress n1
Fatigue life a (Ni) or residual life at 1 ) (100 Cycles) 2nd-level stress (N2r
444, 397, 533, 368, 487, 433, 305, 665, 395, 403, 449, 638, 344, 431, 462 2063, 1197, 1168, 1354, 1282, 1564, 1508, 1324, 1159, 1724, 1053, 1364, 2620, 2556, 799, 906, 1975, 1743 6021, 2910, 7099, 9355, 6429, 8790, 6752, 7236, 9042, 9618, 5893, 5519, 7047, 7089, 3274, 3531 381, 250, 303, 271, 469, 444, 183, 315, 402, 223, 429, 421, 356, 325 183, 301, 285, 168, 114, 463, 551, 372, 181, 24, 283, 160, 526, ⫺10 394, 96, ⫺9.9, ⫺58.8, 269, ⫺93.3, ⫺29.1, 206, 337, ⫺18, 252, 168, ⫺84, 71, 146 5186, 1470, 1817, 4248, 1990, 2884, 1899, 2010, 2211,3332, 2508, 4566, 2526 2601, 1225, 762, 993, 2751, 1237, 144, 1277, 558, 1063, 1950, 382, 2107 1435, 1133, 59, 421, 944, 628, ⫺631, ⫺1022, 269, ⫺209, 1284, ⫺1583
Mean life N i or equivalent life (N 12 )
Std. deviation of Life si or equivalent life s12
45027
9917
151422
52286
658944
209045
46069
8846
49688
17389
47442
15830
457825
117361
475304
81816
549032
95151
1 1 1 a Note: Equivalent life N21 N 2 n1 =N 1 ⫹ N2r ; N2r , N2r is test record of the residual life at s 2 after n1 cycles of s 1. In the case of failure occurs at the first-level 1 stress, residual life is negative and is calculated by N2r N1 ⫺ n1 N 2 =N 1 . Here n1 is the given cycle number of the first level stress in the two-level stress loading, and N1 is the cycle number of first-level stress at which failure occurs.
0.80, respectively. The cycle number of the first-level stress is converted into an equivalent cycle number of the secondlevel stress according to damage equivalence (here Miner’s rule is used), so that the equivalent life at the second-level stress can be obtained by summing the equivalent cycle number and the residual life under the second-level stress. Test results of the carbon steel are shown in Table 1. Similar tests are carried out for the alloy 16Mn steel. Three chosen stress levels (amplitudes) are 394 , 373 and 344 MPa, respectively, all of them are less than the material yield limit 410 MPa. The cycle ratios of the first-level stress are 0.32, 0.48 and 0.74 in one situation and 0.23, 0.39 and 0.66 in the other, respectively. Test results of the alloy steel are shown in Table 2. From these data one can find that the equivalent fatigue life is not exactly equal to the fatigue life under the pertinent constant amplitude loading. This is owing to the error from Miner’s rule. The same tendencies exist for the two materials, i.e., for the low ! high loading sequences, Miner’s rule works well, for the high ! low loading sequences, Miner’s rule lead to significantly non-conservative errors. It is no doubt that more realistic cumulative fatigue damage models, if available, may yield more realistic equivalent lives. About fatigue life distribution patterns, the log-normal distribution has been widely applied. Other distributions, such as the Weibull distribution, the normal distribution
and the inverse normal distribution, are also applied sometimes [20,21]. Especially, according to [21], the fatigue lives under cyclic loading follow the Weibull distribution, while the fatigue lives under complex loading follow the normal distribution. In the present paper, the normal distribution, instead of the log-normal distribution or the Weibull distribution, is applied to describe distribution characteristics of the fatigue lives, because the test data provided no strong indication that the normal distribution is incorrect. The test results under constant amplitude load show the general tendency – the higher the stress level, the less the standard deviation of the fatigue life, and vice versa. The test results under two-level stress show the following tendency: If lower stress operates first, the standard deviation of the equivalent life under the following higher stress is greater than the standard deviation of the fatigue life under the higher constant stress. If higher stress operates first, the standard deviation of the equivalent life under the following lower stress is less than the standard deviation of the fatigue life under the pertinent lower constant stress. The tendency shows also that the greater the difference between the amplitudes of the higher stress and the lower stress, the greater the difference between the standard deviation of the equivalent life and that of the life under the pertinent constant stress. In conclusion, the preceding stress cycles affect the standard deviation of the equivalent life under the following stress, the degree of the effect depends
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
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Table 2 Fatigue test results and the life distribution parameters of the alloy 16Mn steel Stress level and/or loading sequence
Sample size
394 MPa
15
373 MPa
15
344 MPa
15
373 ! 394
10
62500
10
95200
10
146000
10
26000
10
44000
10
75000
394 ! 373
Cycles of 1st-level stress n1
Fatigue life a (Ni) or residual life at 1 ) (100 Cycles) 2nd-level stress (N2r
Mean life (N i ) or equivalent life (N 12 )
915, 1382, 1066, 1444, 712, 1120, 1422, 916, 1532, 903, 1310, 1350, 919, 1149, 953 2087, 1817, 2262, 1788, 1929, 2113, 1685, 1744, 1646, 2632, 1764, 1833, 2312, 1903, 2011 6944, 6818, 8631, 7631, 5725, 5941, 9868, 6825, 8919, 5305, 7597, 8101, 6933, 6911, 6178 461, 384, 702, 516, 544, 653, 922, 788, 757, 885 1029, 698, 192, 552, 588, 781, 1019, 654, 686, 632 506, 130, 776, 100, 210, 534, 96, 115, 252, 488 1177, 1460, 1103, 1207, 1708, 1157, 1511, 842, 1304, 1541 611, 466, 691, 835, 485, 524, 656, 841, 1117, 898 314, 797, 392, 67, 438, 310, 745, 138, 591, 190
113893
25130
196720
27322
722200
125800
102305
18161
123427
23878
107590
26805
175008
25549
147128
20881
169362
24804
Std. deviation of Life (si ) or equivalent life s12
1 1 1 Note: Equivalent life N21 N 2 n1 =N 1 ⫹ N2r ; N2r , N2r is test record of the residual life at s 2 after n1 cycles of s 1. In the case of failure occurs at the first-level 1 stress, residual life is negative and is calculated by N2r N1 ⫺ n1 N 2 =N 1 . Here n1 is the given cycle number of the first level stress in the two-level stress loading, and N1 is the cycle number of first-level stress at which failure occurs. a
on the relative stress level of the preceding stress as well as its cycle number. Based on regression analysis to the test data, a model is developed next to predict the equivalent life distribution parameters under two-level stress, and it will be extended further to other variable amplitude loading situations. Let N i and si be respectively the mean and standard deviation of fatigue life under ith stress level (i 1, 2,…,), accordingly, N 1 and s1 be respectively the mean and standard deviation of fatigue life under first stress level s 1. Let N 12 and s12 be respectively the mean and standard deviation of the equivalent life under the second stress level s 2 in loading sequence (s 1 ! s 2), with n1 cycles of s 1 applied first. According to the test results, it seems reasonable to predict the equivalent life distribution parameters (N 12 , s12 ) by the following model: N 12 N 2
1
n s12 s2 ⫹ s1 ⫺ s2 1 N1
2
Therefore, the p.d.f. of the equivalent life is f21 N ⬃ N N 12 ; s12 . Obviously, Eq. (1) is self-satisfactory according to the equivalent life definition. Eq. (2) is proposed in the present paper, taken as a primary approximation of the relationship between the standard deviation of the equivalent life and the
preceding loading parameters (all the items such as s1 , n1 and N 1 are connected with the preceding loading parameters), which is mainly based on the test results presented in Tables 1 and 2. Here, two parameters are used to describe the damage effect produced by preceding load history, taking regard to the fact that one parameter is not sufficient to describe cumulative fatigue damage [19]. If there is a third stress, as in the condition of three-level stress or other complex loading histories, the equivalent life 12 distribution parameters N 12 3 ; s3 at the third stress s 3, after n1 cycles of s 1 and n2 cycles of s 2, can be predicted by: N 12 3 N3
3
1 1 1 n2 s12 3 s3 ⫹ s2 ⫺ s3 1 N2
n n n s3 ⫹ s1 ⫺ s3 1 ⫹ s12 ⫺ s3 ⫺ s1 ⫺ s3 1 21 N1 N1 N2 n1 n1 n2 s3 ⫹ s1 ⫺ s3 ⫹ s2 ⫺ s3 1 ⫺ N1 N 1 N 2
4
Similarly, for the ith stress level in condition of multilevel stress or more general, in a variable amplitude loading history, the equivalent life distribution parameters N i12… i⫺1 ; si12… i⫺1 can be predicted by: N i12… i⫺1 N i
5
270
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Fig. 1. Test result and equivalent life standard deviation model of the normalized carbon steel under two level stress (a) standard deviation of equivalent life for the high stress under low-high loading (b) standard deviation of equivalent life for the low stress under high-low loading. 12… i⫺2 si12… i⫺1 si12… i⫺2 ⫹ s i⫺1 ⫺ si12… i⫺2
si ⫹
iX ⫺1 j1
" sj ⫺ si 1 ⫺
jX ⫺1 k1
nk Nk
!
nj Nj
#
n1 i⫺1 12… i⫺2 N i⫺1
3. Equivalent damage – equivalent life distribution method for fatigue reliability prediction 6
In all the equations, developed previously, deterministic cycle number is used. In the case where the cycles of the applied stress is a random variable, the mean of the random variable can be used to predict the equivalent life distribution parameters for the sake of simplification, because the error caused by replacing the random variable with its mean is secondary in comparison with the change of the life distribution parameters caused by the preceding cyclic loading. The test results and the theoretical model of the normalized carbon steel and the alloy 16Mn steel are shown in Figs. 1 and 2 respectively, in which the abscissas stand for the applied cycle ratio of the first-level stress and the ordinates stand for standard deviation of the equivalent life at the second-level stress. Despite the scatter of the test data, a general trend is shown. What is important is that the data and the model highlighted the variation of the equivalent life distribution parameters under variable amplitude load and provided a basis for fatigue reliability calculation.
Let f N represent the p.d.f. of fatigue life at a given stress level and f n the p.d.f. of the applied stress cycles. Obviously, failure occurs when stress cycles n is greater than life N, therefore fatigue reliability is defined as: R P n ⬍ N
7
According to load cycles – fatigue life interference analysis, reliability function can be derived: Z⫹ ∞ Z⫹ ∞ f n f NdN dn 8 R t 0
n
Based on Eq. (8), fatigue reliability under program loading can be dealt with conveniently. The ‘‘conditional reliability – equivalent life method’’ [4] was proposed for this kind of loading pattern, in which the contribution of every load block to fatigue damage is measured by the failure probability induced by it. That is to say, the equivalent load cycles between different stresses are evaluated according to failure probability equivalence. This method can be stated as following (see Fig. 3). Suppose there is a high-low two-level loading spectrum,
Fig. 2. Test result and equivalent life standard deviation model of the alloy steel (a) standard deviation of equivalent life for the high stress under low-high loading (b) standard deviation of equivalent life for the low stress under high-low loading.
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Fig. 3. Illustration of the conditional reliability – equivalent life method and comparison between failure probability equivalence and damage equivalence.
life p.d.f.s are f1 N and f2 N respectively for the high stress s 1 and the low stress s 2. After n1 cycles of s 1, failure probability equals to P1: Zn1 f1 NdN 9 P1 0
By the conditional reliability – equivalent life method, when stress level converts from s 1 to s 2, the effect of n1 cycles of s 1 is evaluated by the failure probability P1, and its equivalent cycle number n2p under s 2 is calculated by equating the failure probability induced by n2p cycles of s 2 to P1, i.e. Zn2p Zn1 f1 NdN f2 NdN 10 0
0
This is a simple treatment to the equivalence between the cycle numbers of stresses s 1 and s 2 (it can be called ‘‘equivalent probability conversion’’), and it does not concern much on whether the fatigue damage state produced by n1 cycles of s 1 is really equivalent to that produced by n2p cycles of s 2 or not. However, fatigue lives always have a non-zero minimum. In condition of the cycles of the first level stress is quite small, its equivalent cycle numbers to other stress cannot be calculated because the failure probability induced by the small cycle number of the first level stress equals to zero. This implies that the ‘‘conditional probability – equivalent life method’’ can never be applied to variable amplitude loading in which the cycle number at a given stress level is normally equal to 1. It is well known that fatigue is a damage accumulation process. According to the viewpoint of damage equivalence, in the loading sequence (s 1 ! s 2), the equivalent cycle number of stress s 2 to n1 cycles of s 1 should be determined by cumulative fatigue damage model. Take Miner’s rule for example, it can be calculated as: n2d
n1 N 2 N 1
271
Fig. 4. Illustration of the equivalent damage – equivalent life distribution method.
probability conversion. The equivalent probability conversion does not accord with damage equivalence rule. However, conventional accumulative damage model does not accord with failure probability equivalence. Through load cycles – fatigue life interference analysis, taking into account damage equivalence as well as the variation of equivalent life distribution parameters, the following ‘‘equivalent damage – equivalent life distribution’’ method can be developed for calculating fatigue reliability under variable amplitude loading histories. Let us begin with a two-level stress loading spectrum. After n1 cycles of the first level stress s 1, by which a failure probability P1 is produced, the stress changes to s 2. Now the life distribution parameters under stress s 2 are no longer (N 2 ; s2 ) but (N 12 ; s12 ) – N 12 is the same as N 2 , and the standard deviation s12 can be predicted by Eq. (2) or Eq. (6). With the increasing cycles of s 2, failure probability increases continuously. From a life distribution curve, it is easy to know that at different stages in a fatigue process, the failure probability increments DP caused by the same load cycle increment Dn are different. It is believed in the present paper that ‘‘damage equivalence’’ should be the basis of calculating the equivalent cycles of different stresses. From this viewpoint, n1 cycles of s 1 is equivalent to n2d cycles of s 2, and the failure probability increment DP2 should be calculated from the point n2d of the changed p.d.f. curve f21 N when cyclic stress s 2 starts operating (see Fig. 4), i.e., DP2
Zn2d ⫹ n2 n2d
f21 NdN
12
11
This equivalent calculation will be called as ‘‘equivalent damage conversion’’. Generally, n2p 苷 n2d, i.e., the equivalent damage conversion is different from the equivalent
Fig. 5. Schematic diagram of multi-level cyclic loading history.
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L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Table 3 Experiment results under multi-level spectrum load Specimen code
1
2
3
4
5
6
7
8
9
10
Mean
Std.
Total cycles to failure (100)
2440
2066
2197
2340
2134
2089
2171
2265
2436
2433
2257
146.9
With respect to the fact that a failure probability P1 has been induced by n1 cycles of s 1, the total failure probability P12 is then equal to: P12 P1 ⫹ DP2
13
What needs explaining is that the failure probability P2p (shown in Fig. 4), which is the probability of the virgin material failure under n2d cycles of s 2, may not equals to P1. This is the result of the fact that the cumulative damage model does not account for the uncertainty of fatigue life, therefore damage equivalence cannot warrant failure probability equivalence. In order to improve the fatigue reliability model, a better cumulative damage model is needed. However, fatigue damage accumulation is a rather complicated phenomenon, when the equivalent life reaches its median under complex loading, the failure probability may not be equal to 50%. The main reason is the nonuniformity of fatigue life deviations at different stress levels. Other reasons include strengthening or weakening effect occurred during fatigue process, differences in fatigue mechanisms for HCF and LCF, etc. If there is a third-level stress, the total failure probability after the third stress equals to P123: P123 P1 ⫹ DP2 ⫹ DP3
14
where, DP3
Zn3d ⫹ n3
f312 NdN
15
n3d N 3 n1 =N 1 ⫹ n2 =N 2
16
n3d
For a variable amplitude loading history, the similar step can be successively applied to any required point and a pertinent failure probability can be calculated. As to the equivalent damage – equivalent life distribution method, although Miner’s rule is applied to calculate equivalent cycle number, it is not mean that the method is equivalent to predict fatigue life or failure probability directly by Miner’s rule. The main characteristics of the present method is its capability of reflecting the change of (equivalent) life distribution. Besides, this method is a cycle by cycle damage analysis approach, its input about loading is the detailed history rather than distribution parameters of the loading spectrum. Its advantage is that all the information about complex loading history is retained. And its disadvantage of time consuming can be overcome by using computer code.
4. Experimental verification In order to verify the equivalent damage – equivalent life distribution method, fatigue tests were done under multilevel cyclic loading for the hot-rolled alloy 16Mn steel. The spectrum parameters of the multi-level cyclic loading used in the tests is shown in Fig. 5. Each of the six preceding blocks contains 10 000 load cycles, while the last block lasts until to the specimen failure. The life distribution parameters under constant amplitude cyclic stress is obtained by testing fifteen specimens at each of the stress level (see Table 2). Another ten specimens were tested under the multi-level cyclic loading spectrum (results are shown in Table 3). From Fig. 6 one can see that the equivalent life distribution estimated by the equivalent damage – equivalent life distribution method is quite comparable with the test result, though, owing to the small sample number, the tested lives do not follow the normal distribution or others commonly used for fatigue lives. The model gives a conservative estimation of the equivalent fatigue life distribution.
5. Conclusion The equivalent life distribution under variable amplitude loading is studied, and a equivalent damage – equivalent life distribution method for predicting fatigue failure probability is presented. Besides the well known tendency (under constant amplitude load) that the higher the stress level, the less the standard deviation of the fatigue life, and vice versa, the present paper shows also that under two level stress, if lower stress operates first, the standard deviation of the equivalent life under the following higher stress will increase, i.e. the standard deviation of the equivalent life is greater than that of the fatigue life under the constant high stress. If higher stress operates first, the standard deviation of the equivalent life under the following lower
Fig. 6. Test results and the estimated equivalent life distribution.
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
stress will decrease. The tendency is that the greater the difference between the amplitudes of the high stress and the low stress, the greater the change of the standard deviation of the equivalent life. In one word, the preceding stress cycles will affect the standard deviation of the equivalent lives under the following stress, and the effect depends on the relative stress level of the preceding stress and its cycle number as well. The equivalent damage – equivalent life distribution method developed in the present paper can be used to predict fatigue failure probability under variable amplitude loading history. This method differs from the ‘‘conditional reliability – equivalent life method’’ owing to its calculating equivalent cycle number by damage equivalence, and differs from general cumulative damage model owing to its reflecting the change of equivalent life distribution.
Acknowledgements This research was supported by the Fok Yin Tung Education Foundation and the National Education Commission of People’s Republic of China. Appendix A F-statistical significance tests were carried out to examine the significance of the variations in standard deviation of the equivalent lives. For the test results of the carbon steel, six cases 32 ⫹ 32 are tested in total. The outcome is that four out of six cases reject the hypothesis that no significant differences between the standard deviations with 10% significance level, or two out of six cases reject the hypothesis that no significant differences between the standard deviations with 5% significance level. When evaluate the outcomes, one should realize that the differences between the life standard deviations at the two constant stresses and the cycle ratios of the first level stress have evident effect on the significance level examined. For the situation of the alloy 16Mn steel, the two standard deviations under the two constant stresses are nearly the same (25130 vs. 27322), so it is not surprising to see that no significant variations exist between the standard deviations of the equivalent lives.
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Equivalent life distribution and fatigue failure probability prediction Liyang Xie 1 Northeastern University, Shenyang 110006, People’s Republic of China
Abstract Equivalent fatigue life distributions under two-level cyclic stress are analyzed experimentally, and a model of describing equivalent life distribution is presented. Based on the equivalent life distribution model, an ‘‘equivalent damage–equivalent life distribution’’ method is developed to predict fatigue failure probability under variable amplitude loading. The underlying principles are fatigue damage accumulation rule and ‘‘load cycles–fatigue life’’ interference theorem. In the equivalent damage–equivalent life distribution method, two parameters, i.e. mean and standard deviation of the (equivalent) life distribution are used to describe cumulative fatigue damage effect. A basic feature of the equivalent damage–equivalent life distribution method is its capability of reflecting the loading history dependent change of the equivalent life distribution. Tests under multi-level stress show a good agreement between predicted fatigue failure probability and the test results. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Equivalent life distribution; Fatigue failure probability
1. Introduction Stress–strength interference analysis has been widely applied to fatigue probability analysis [1–5]. Nevertheless, its direct use is limited to the simplest situation, i.e., constant amplitude cyclic loading. Alternatively, the ‘‘conditional reliability – equivalent life method’’ [6] has been applied directly or indirectly to a broad range of fatigue failure probability predictions. This conditional reliability – equivalent life method is also a simple method and therefore, is of rather a limited applicability. More sophisticated studies on fatigue failure probability have been published in recent years [7-18]. In these literature, residual fatigue life distributions under constant amplitude cyclic loading and two-stage loading conditions [13,14], impact of load changing on lifetime distributions [15,16], and computer simulation of fatigue reliability analysis [17,18] are investigated. These studies have provided plenty of insights into fatigue failure probability under variable amplitude loading. In the present paper, an equivalent fatigue life distribution model is presented, and a fatigue failure probability prediction method is proposed based on the equivalent life distribution model, cumulative fatigue damage rule, and load cycles – fatigue life interference theorem.
1 Present address: Otto-von-Guericke-University, IAUT, Department of Plant Design and Safety, Postfach 4120, 39016 Magdeburg, Germany. Tel.: 0049-391-6718113; Fax: 0049-391-6711128. E-mail address: [email protected] (L. Xie)
2. Equivalent life distribution under variable amplitude loading Fatigue life distribution is generally load spectrum dependent. Therefore, a thorough investigation on the relationship between life distribution and the respective loading history is necessary to develop fatigue failure probability model. In this regard, lots of tests have been done in a rotated-bending fatigue test machine, using smooth specimens made of normalized 0.45% carbon steel or hot rolled alloy 16Mn steel. The material compositions, mechanical properties and specimen geometry are the same as presented in [19]. For the carbon steel, tests are carried out under both constant amplitude loading (three different stress levels are used) and two-level cyclic loading (some combinations of the constant amplitude loads). The amplitudes of the three stress are 366 , 331 and 309 MPa, respectively. All of the three stress amplitudes are less than the material yield limit 380 MPa. Fatigue tests under the three constant amplitude cyclic stresses are carried out first, so as to obtain the means and standard deviations of the fatigue lives. After that, two-level stress tests under low ! high (331 ! 366 MPa) and high ! low (331 ! 309 MPa) loading sequences are done in order to investigate fatigue life distribution under variable amplitude loading. Test procedure and data processing for the two-level stress test are as the following: The first-level stress operates only a given cycle number, after that the second-level stress begins to operate and lasts to specimen failure. The cycle ratios (life fractions) of the first-level stress are set as 0.27, 0.53 and
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Table 1 Fatigue test results and the life distribution parameters of the 0.45% carbon steel Stress level and/or loading sequence
Sample size
366 Mpa
15
331 Mpa
18
309 Mpa
16
331 ! 366
14
40300
14
80600
15
120900
13
40300
13
80600
12
120900
331 ! 309
Cycles of 1st-level stress n1
Fatigue life a (Ni) or residual life at 1 ) (100 Cycles) 2nd-level stress (N2r
444, 397, 533, 368, 487, 433, 305, 665, 395, 403, 449, 638, 344, 431, 462 2063, 1197, 1168, 1354, 1282, 1564, 1508, 1324, 1159, 1724, 1053, 1364, 2620, 2556, 799, 906, 1975, 1743 6021, 2910, 7099, 9355, 6429, 8790, 6752, 7236, 9042, 9618, 5893, 5519, 7047, 7089, 3274, 3531 381, 250, 303, 271, 469, 444, 183, 315, 402, 223, 429, 421, 356, 325 183, 301, 285, 168, 114, 463, 551, 372, 181, 24, 283, 160, 526, ⫺10 394, 96, ⫺9.9, ⫺58.8, 269, ⫺93.3, ⫺29.1, 206, 337, ⫺18, 252, 168, ⫺84, 71, 146 5186, 1470, 1817, 4248, 1990, 2884, 1899, 2010, 2211,3332, 2508, 4566, 2526 2601, 1225, 762, 993, 2751, 1237, 144, 1277, 558, 1063, 1950, 382, 2107 1435, 1133, 59, 421, 944, 628, ⫺631, ⫺1022, 269, ⫺209, 1284, ⫺1583
Mean life N i or equivalent life (N 12 )
Std. deviation of Life si or equivalent life s12
45027
9917
151422
52286
658944
209045
46069
8846
49688
17389
47442
15830
457825
117361
475304
81816
549032
95151
1 1 1 a Note: Equivalent life N21 N 2 n1 =N 1 ⫹ N2r ; N2r , N2r is test record of the residual life at s 2 after n1 cycles of s 1. In the case of failure occurs at the first-level 1 stress, residual life is negative and is calculated by N2r N1 ⫺ n1 N 2 =N 1 . Here n1 is the given cycle number of the first level stress in the two-level stress loading, and N1 is the cycle number of first-level stress at which failure occurs.
0.80, respectively. The cycle number of the first-level stress is converted into an equivalent cycle number of the secondlevel stress according to damage equivalence (here Miner’s rule is used), so that the equivalent life at the second-level stress can be obtained by summing the equivalent cycle number and the residual life under the second-level stress. Test results of the carbon steel are shown in Table 1. Similar tests are carried out for the alloy 16Mn steel. Three chosen stress levels (amplitudes) are 394 , 373 and 344 MPa, respectively, all of them are less than the material yield limit 410 MPa. The cycle ratios of the first-level stress are 0.32, 0.48 and 0.74 in one situation and 0.23, 0.39 and 0.66 in the other, respectively. Test results of the alloy steel are shown in Table 2. From these data one can find that the equivalent fatigue life is not exactly equal to the fatigue life under the pertinent constant amplitude loading. This is owing to the error from Miner’s rule. The same tendencies exist for the two materials, i.e., for the low ! high loading sequences, Miner’s rule works well, for the high ! low loading sequences, Miner’s rule lead to significantly non-conservative errors. It is no doubt that more realistic cumulative fatigue damage models, if available, may yield more realistic equivalent lives. About fatigue life distribution patterns, the log-normal distribution has been widely applied. Other distributions, such as the Weibull distribution, the normal distribution
and the inverse normal distribution, are also applied sometimes [20,21]. Especially, according to [21], the fatigue lives under cyclic loading follow the Weibull distribution, while the fatigue lives under complex loading follow the normal distribution. In the present paper, the normal distribution, instead of the log-normal distribution or the Weibull distribution, is applied to describe distribution characteristics of the fatigue lives, because the test data provided no strong indication that the normal distribution is incorrect. The test results under constant amplitude load show the general tendency – the higher the stress level, the less the standard deviation of the fatigue life, and vice versa. The test results under two-level stress show the following tendency: If lower stress operates first, the standard deviation of the equivalent life under the following higher stress is greater than the standard deviation of the fatigue life under the higher constant stress. If higher stress operates first, the standard deviation of the equivalent life under the following lower stress is less than the standard deviation of the fatigue life under the pertinent lower constant stress. The tendency shows also that the greater the difference between the amplitudes of the higher stress and the lower stress, the greater the difference between the standard deviation of the equivalent life and that of the life under the pertinent constant stress. In conclusion, the preceding stress cycles affect the standard deviation of the equivalent life under the following stress, the degree of the effect depends
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Table 2 Fatigue test results and the life distribution parameters of the alloy 16Mn steel Stress level and/or loading sequence
Sample size
394 MPa
15
373 MPa
15
344 MPa
15
373 ! 394
10
62500
10
95200
10
146000
10
26000
10
44000
10
75000
394 ! 373
Cycles of 1st-level stress n1
Fatigue life a (Ni) or residual life at 1 ) (100 Cycles) 2nd-level stress (N2r
Mean life (N i ) or equivalent life (N 12 )
915, 1382, 1066, 1444, 712, 1120, 1422, 916, 1532, 903, 1310, 1350, 919, 1149, 953 2087, 1817, 2262, 1788, 1929, 2113, 1685, 1744, 1646, 2632, 1764, 1833, 2312, 1903, 2011 6944, 6818, 8631, 7631, 5725, 5941, 9868, 6825, 8919, 5305, 7597, 8101, 6933, 6911, 6178 461, 384, 702, 516, 544, 653, 922, 788, 757, 885 1029, 698, 192, 552, 588, 781, 1019, 654, 686, 632 506, 130, 776, 100, 210, 534, 96, 115, 252, 488 1177, 1460, 1103, 1207, 1708, 1157, 1511, 842, 1304, 1541 611, 466, 691, 835, 485, 524, 656, 841, 1117, 898 314, 797, 392, 67, 438, 310, 745, 138, 591, 190
113893
25130
196720
27322
722200
125800
102305
18161
123427
23878
107590
26805
175008
25549
147128
20881
169362
24804
Std. deviation of Life (si ) or equivalent life s12
1 1 1 Note: Equivalent life N21 N 2 n1 =N 1 ⫹ N2r ; N2r , N2r is test record of the residual life at s 2 after n1 cycles of s 1. In the case of failure occurs at the first-level 1 stress, residual life is negative and is calculated by N2r N1 ⫺ n1 N 2 =N 1 . Here n1 is the given cycle number of the first level stress in the two-level stress loading, and N1 is the cycle number of first-level stress at which failure occurs. a
on the relative stress level of the preceding stress as well as its cycle number. Based on regression analysis to the test data, a model is developed next to predict the equivalent life distribution parameters under two-level stress, and it will be extended further to other variable amplitude loading situations. Let N i and si be respectively the mean and standard deviation of fatigue life under ith stress level (i 1, 2,…,), accordingly, N 1 and s1 be respectively the mean and standard deviation of fatigue life under first stress level s 1. Let N 12 and s12 be respectively the mean and standard deviation of the equivalent life under the second stress level s 2 in loading sequence (s 1 ! s 2), with n1 cycles of s 1 applied first. According to the test results, it seems reasonable to predict the equivalent life distribution parameters (N 12 , s12 ) by the following model: N 12 N 2
1
n s12 s2 ⫹ s1 ⫺ s2 1 N1
2
Therefore, the p.d.f. of the equivalent life is f21 N ⬃ N N 12 ; s12 . Obviously, Eq. (1) is self-satisfactory according to the equivalent life definition. Eq. (2) is proposed in the present paper, taken as a primary approximation of the relationship between the standard deviation of the equivalent life and the
preceding loading parameters (all the items such as s1 , n1 and N 1 are connected with the preceding loading parameters), which is mainly based on the test results presented in Tables 1 and 2. Here, two parameters are used to describe the damage effect produced by preceding load history, taking regard to the fact that one parameter is not sufficient to describe cumulative fatigue damage [19]. If there is a third stress, as in the condition of three-level stress or other complex loading histories, the equivalent life 12 distribution parameters N 12 3 ; s3 at the third stress s 3, after n1 cycles of s 1 and n2 cycles of s 2, can be predicted by: N 12 3 N3
3
1 1 1 n2 s12 3 s3 ⫹ s2 ⫺ s3 1 N2
n n n s3 ⫹ s1 ⫺ s3 1 ⫹ s12 ⫺ s3 ⫺ s1 ⫺ s3 1 21 N1 N1 N2 n1 n1 n2 s3 ⫹ s1 ⫺ s3 ⫹ s2 ⫺ s3 1 ⫺ N1 N 1 N 2
4
Similarly, for the ith stress level in condition of multilevel stress or more general, in a variable amplitude loading history, the equivalent life distribution parameters N i12… i⫺1 ; si12… i⫺1 can be predicted by: N i12… i⫺1 N i
5
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Fig. 1. Test result and equivalent life standard deviation model of the normalized carbon steel under two level stress (a) standard deviation of equivalent life for the high stress under low-high loading (b) standard deviation of equivalent life for the low stress under high-low loading. 12… i⫺2 si12… i⫺1 si12… i⫺2 ⫹ s i⫺1 ⫺ si12… i⫺2
si ⫹
iX ⫺1 j1
" sj ⫺ si 1 ⫺
jX ⫺1 k1
nk Nk
!
nj Nj
#
n1 i⫺1 12… i⫺2 N i⫺1
3. Equivalent damage – equivalent life distribution method for fatigue reliability prediction 6
In all the equations, developed previously, deterministic cycle number is used. In the case where the cycles of the applied stress is a random variable, the mean of the random variable can be used to predict the equivalent life distribution parameters for the sake of simplification, because the error caused by replacing the random variable with its mean is secondary in comparison with the change of the life distribution parameters caused by the preceding cyclic loading. The test results and the theoretical model of the normalized carbon steel and the alloy 16Mn steel are shown in Figs. 1 and 2 respectively, in which the abscissas stand for the applied cycle ratio of the first-level stress and the ordinates stand for standard deviation of the equivalent life at the second-level stress. Despite the scatter of the test data, a general trend is shown. What is important is that the data and the model highlighted the variation of the equivalent life distribution parameters under variable amplitude load and provided a basis for fatigue reliability calculation.
Let f N represent the p.d.f. of fatigue life at a given stress level and f n the p.d.f. of the applied stress cycles. Obviously, failure occurs when stress cycles n is greater than life N, therefore fatigue reliability is defined as: R P n ⬍ N
7
According to load cycles – fatigue life interference analysis, reliability function can be derived: Z⫹ ∞ Z⫹ ∞ f n f NdN dn 8 R t 0
n
Based on Eq. (8), fatigue reliability under program loading can be dealt with conveniently. The ‘‘conditional reliability – equivalent life method’’ [4] was proposed for this kind of loading pattern, in which the contribution of every load block to fatigue damage is measured by the failure probability induced by it. That is to say, the equivalent load cycles between different stresses are evaluated according to failure probability equivalence. This method can be stated as following (see Fig. 3). Suppose there is a high-low two-level loading spectrum,
Fig. 2. Test result and equivalent life standard deviation model of the alloy steel (a) standard deviation of equivalent life for the high stress under low-high loading (b) standard deviation of equivalent life for the low stress under high-low loading.
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
Fig. 3. Illustration of the conditional reliability – equivalent life method and comparison between failure probability equivalence and damage equivalence.
life p.d.f.s are f1 N and f2 N respectively for the high stress s 1 and the low stress s 2. After n1 cycles of s 1, failure probability equals to P1: Zn1 f1 NdN 9 P1 0
By the conditional reliability – equivalent life method, when stress level converts from s 1 to s 2, the effect of n1 cycles of s 1 is evaluated by the failure probability P1, and its equivalent cycle number n2p under s 2 is calculated by equating the failure probability induced by n2p cycles of s 2 to P1, i.e. Zn2p Zn1 f1 NdN f2 NdN 10 0
0
This is a simple treatment to the equivalence between the cycle numbers of stresses s 1 and s 2 (it can be called ‘‘equivalent probability conversion’’), and it does not concern much on whether the fatigue damage state produced by n1 cycles of s 1 is really equivalent to that produced by n2p cycles of s 2 or not. However, fatigue lives always have a non-zero minimum. In condition of the cycles of the first level stress is quite small, its equivalent cycle numbers to other stress cannot be calculated because the failure probability induced by the small cycle number of the first level stress equals to zero. This implies that the ‘‘conditional probability – equivalent life method’’ can never be applied to variable amplitude loading in which the cycle number at a given stress level is normally equal to 1. It is well known that fatigue is a damage accumulation process. According to the viewpoint of damage equivalence, in the loading sequence (s 1 ! s 2), the equivalent cycle number of stress s 2 to n1 cycles of s 1 should be determined by cumulative fatigue damage model. Take Miner’s rule for example, it can be calculated as: n2d
n1 N 2 N 1
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Fig. 4. Illustration of the equivalent damage – equivalent life distribution method.
probability conversion. The equivalent probability conversion does not accord with damage equivalence rule. However, conventional accumulative damage model does not accord with failure probability equivalence. Through load cycles – fatigue life interference analysis, taking into account damage equivalence as well as the variation of equivalent life distribution parameters, the following ‘‘equivalent damage – equivalent life distribution’’ method can be developed for calculating fatigue reliability under variable amplitude loading histories. Let us begin with a two-level stress loading spectrum. After n1 cycles of the first level stress s 1, by which a failure probability P1 is produced, the stress changes to s 2. Now the life distribution parameters under stress s 2 are no longer (N 2 ; s2 ) but (N 12 ; s12 ) – N 12 is the same as N 2 , and the standard deviation s12 can be predicted by Eq. (2) or Eq. (6). With the increasing cycles of s 2, failure probability increases continuously. From a life distribution curve, it is easy to know that at different stages in a fatigue process, the failure probability increments DP caused by the same load cycle increment Dn are different. It is believed in the present paper that ‘‘damage equivalence’’ should be the basis of calculating the equivalent cycles of different stresses. From this viewpoint, n1 cycles of s 1 is equivalent to n2d cycles of s 2, and the failure probability increment DP2 should be calculated from the point n2d of the changed p.d.f. curve f21 N when cyclic stress s 2 starts operating (see Fig. 4), i.e., DP2
Zn2d ⫹ n2 n2d
f21 NdN
12
11
This equivalent calculation will be called as ‘‘equivalent damage conversion’’. Generally, n2p 苷 n2d, i.e., the equivalent damage conversion is different from the equivalent
Fig. 5. Schematic diagram of multi-level cyclic loading history.
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Table 3 Experiment results under multi-level spectrum load Specimen code
1
2
3
4
5
6
7
8
9
10
Mean
Std.
Total cycles to failure (100)
2440
2066
2197
2340
2134
2089
2171
2265
2436
2433
2257
146.9
With respect to the fact that a failure probability P1 has been induced by n1 cycles of s 1, the total failure probability P12 is then equal to: P12 P1 ⫹ DP2
13
What needs explaining is that the failure probability P2p (shown in Fig. 4), which is the probability of the virgin material failure under n2d cycles of s 2, may not equals to P1. This is the result of the fact that the cumulative damage model does not account for the uncertainty of fatigue life, therefore damage equivalence cannot warrant failure probability equivalence. In order to improve the fatigue reliability model, a better cumulative damage model is needed. However, fatigue damage accumulation is a rather complicated phenomenon, when the equivalent life reaches its median under complex loading, the failure probability may not be equal to 50%. The main reason is the nonuniformity of fatigue life deviations at different stress levels. Other reasons include strengthening or weakening effect occurred during fatigue process, differences in fatigue mechanisms for HCF and LCF, etc. If there is a third-level stress, the total failure probability after the third stress equals to P123: P123 P1 ⫹ DP2 ⫹ DP3
14
where, DP3
Zn3d ⫹ n3
f312 NdN
15
n3d N 3 n1 =N 1 ⫹ n2 =N 2
16
n3d
For a variable amplitude loading history, the similar step can be successively applied to any required point and a pertinent failure probability can be calculated. As to the equivalent damage – equivalent life distribution method, although Miner’s rule is applied to calculate equivalent cycle number, it is not mean that the method is equivalent to predict fatigue life or failure probability directly by Miner’s rule. The main characteristics of the present method is its capability of reflecting the change of (equivalent) life distribution. Besides, this method is a cycle by cycle damage analysis approach, its input about loading is the detailed history rather than distribution parameters of the loading spectrum. Its advantage is that all the information about complex loading history is retained. And its disadvantage of time consuming can be overcome by using computer code.
4. Experimental verification In order to verify the equivalent damage – equivalent life distribution method, fatigue tests were done under multilevel cyclic loading for the hot-rolled alloy 16Mn steel. The spectrum parameters of the multi-level cyclic loading used in the tests is shown in Fig. 5. Each of the six preceding blocks contains 10 000 load cycles, while the last block lasts until to the specimen failure. The life distribution parameters under constant amplitude cyclic stress is obtained by testing fifteen specimens at each of the stress level (see Table 2). Another ten specimens were tested under the multi-level cyclic loading spectrum (results are shown in Table 3). From Fig. 6 one can see that the equivalent life distribution estimated by the equivalent damage – equivalent life distribution method is quite comparable with the test result, though, owing to the small sample number, the tested lives do not follow the normal distribution or others commonly used for fatigue lives. The model gives a conservative estimation of the equivalent fatigue life distribution.
5. Conclusion The equivalent life distribution under variable amplitude loading is studied, and a equivalent damage – equivalent life distribution method for predicting fatigue failure probability is presented. Besides the well known tendency (under constant amplitude load) that the higher the stress level, the less the standard deviation of the fatigue life, and vice versa, the present paper shows also that under two level stress, if lower stress operates first, the standard deviation of the equivalent life under the following higher stress will increase, i.e. the standard deviation of the equivalent life is greater than that of the fatigue life under the constant high stress. If higher stress operates first, the standard deviation of the equivalent life under the following lower
Fig. 6. Test results and the estimated equivalent life distribution.
L. Xie / International Journal of Pressure Vessels and Piping 76 (1999) 267–273
stress will decrease. The tendency is that the greater the difference between the amplitudes of the high stress and the low stress, the greater the change of the standard deviation of the equivalent life. In one word, the preceding stress cycles will affect the standard deviation of the equivalent lives under the following stress, and the effect depends on the relative stress level of the preceding stress and its cycle number as well. The equivalent damage – equivalent life distribution method developed in the present paper can be used to predict fatigue failure probability under variable amplitude loading history. This method differs from the ‘‘conditional reliability – equivalent life method’’ owing to its calculating equivalent cycle number by damage equivalence, and differs from general cumulative damage model owing to its reflecting the change of equivalent life distribution.
Acknowledgements This research was supported by the Fok Yin Tung Education Foundation and the National Education Commission of People’s Republic of China. Appendix A F-statistical significance tests were carried out to examine the significance of the variations in standard deviation of the equivalent lives. For the test results of the carbon steel, six cases 32 ⫹ 32 are tested in total. The outcome is that four out of six cases reject the hypothesis that no significant differences between the standard deviations with 10% significance level, or two out of six cases reject the hypothesis that no significant differences between the standard deviations with 5% significance level. When evaluate the outcomes, one should realize that the differences between the life standard deviations at the two constant stresses and the cycle ratios of the first level stress have evident effect on the significance level examined. For the situation of the alloy 16Mn steel, the two standard deviations under the two constant stresses are nearly the same (25130 vs. 27322), so it is not surprising to see that no significant variations exist between the standard deviations of the equivalent lives.
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