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A method for fatigue based reliability when the loading of a component is unknown
International Journal of Fatigue 21 (1999) 603–610 www.elsevier.com/locate/ijfatigue
A method for fatigue based reliability when the loading of a component is unknown Xiaobin Le, M.L. Peterson
*
Department of Mechanical Engineering, Colorado State University, Fort Collins, CO 80523, USA Received 27 March 1998; received in revised form 30 November 1998; accepted 8 February 1999
Abstract When the loading of a component is unknown, it is not possible to perform fatigue evaluation without some additional knowledge of the applied loading. A model is presented which considers problems of this type. In this situation, the components will have experienced some degree of fatigue damage during operation. However, the loading during use is unknown, and thus direct calculation of the remaining service life is not possible without additional data. The use of components removed from service to conduct fatigue tests can provide the required test data. The test data contains additional information associated with the loading which can be used to determine the residual fatigue damage resistance of the components. Comparing the residual fatigue damage resistance of the tested components with the fatigue damage resistance of new components can make it possible to, indirectly, describe the fatigue damage accumulation which has occurred in the components during the time in service. Thus, the fatigue damage accumulated in the components can be determined by additional fatigue tests, in spite of the absence of direct knowledge of applied service loading. The fatigue evaluation of untested components can be performed using the probabilistic fatigue model which is presented in this paper. The model is demonstrated by performing an evaluation of compressor blades for which the applied loading is unknown. The calculated results are shown to be close to the statistical results obtained in-service for the blades. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue model; Reliability; Unknown loading; Fatigue test
1. Introduction Before fatigue evaluation of a component can be carried out, in general, three pieces of information must be known: (1) fatigue strength data with appropriate description, (2) loading which will be applied to the component and the loading model and (3) the limit state function or relationship equation. The limit state function, or relationship equation, brings the fatigue strength, cyclic loading, fatigue damage, fatigue life and the reliability index together [1,2]. In a number of engineering applications, the loading of components may not be known or it may not be possible to describe the loading with sufficient accuracy even when the applied loads are known. Incomplete knowledge of the loading is common
* Corresponding author. Tel.: +1-970-491-2813; fax: +1-970-4911055. E-mail address: [email protected] (M.L. Peterson)
not only when the utilization is unknown, but also when uncertainty exists in the operating conditions. For example, unknown loading due to utilization is common in infrastructure when the quantity and weight of the traffic are not known. In automobile or truck applications, the road surface and vehicle utilization are typically not well defined, which results in uncertainty. For aerospace applications, the effect of turbulence and other operating conditions may create significant uncertainty in the applied load. In addition, the loads on a component may be difficult to measure because the component is irregular in shape. In other cases, the load may also be applied to the component in a complicated manner. Actual measurements of the loads encountered in service may not even be possible because of difficulties associated with physical access to the part for measurements. For example, operational access to the compressor blades of an airplane engine is restricted in addition to any measurements being difficult because of the motion of the blades. In situations such as a compressor blade
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in an aircraft engine, it is difficult to describe the operational loading with reasonable accuracy. In order to evaluate the degree of fatigue damage in the blades, blades which have operated under normal service conditions for a certain number of hours can be used to conduct fatigue tests. The fatigue test may be for a simplified loading of the blade, rather than completely replicating the operational loading. This approach is in general also applicable to accelerated testing where the loading may be more severe than in-service loads. In some applications, the fatigue damage status of some of the components may be investigated more economically by conducting operational fatigue tests, rather than trying to measure and then replicate the operational loading. A method is proposed which will address these problems which are associated with fatigue evaluation of a component with unknown loading. The main objective of the model is to present an approach which can utilize fatigue test data performed on used components which have been removed from service prior to failure. The model is used to provide information which will compensate for the limited information about the in-service loading. The model also makes it possible to represent the fatigue damage which has accumulated in the component due to the unknown loading. The fatigue evaluation of compressor blades in a jet engine will be used to show the utility of the approach.
2.1. Model of the fatigue strength data in the test Fatigue failure of components occurs because of accumulated fatigue damage which occurs during cyclic loading. When fatigue damage accumulates to a critical value, fatigue failure will occur. Because of the amount of scatter in test data, it is impossible, at a given stress level, to predict the exact number of cycles at which a fatigue failure will occur [3]. Additionally, it is also not possible to know at what stress level a failure will occur for a given fatigue life. The random variables, which reflect the degree of damage associated with a particular stress level and number of stress cycles, can describe the random event of a fatigue failure. Fatigue damage of a component is a conceptual abstraction which can be indirectly described by a random variable. Miner defines the fatigue damage associated with one stress cycle, d, as [4]: 1 NT
冘冉 冊 nTi NTi
⌬⫽
(2)
where NTi is the number of cycles to failure at the ith stress cycle sTi and nTi is the number of stress cycles at the stress level sTi. The fatigue damage index is then ⌬, which indirectly expresses the fatigue damage to the component which has undergone nTi cycles at a stress level sTi. It is assumed that when the fatigue damage accumulated in the component is greater than ⌬failure then a fatigue failure will occur. In traditional theory, when ⌬ is greater than 1, failure will occur, however this is not necessarily true in a probabilistic sense. Additionally, for the present model, ergodicity of loading is also assumed. In other words, the average characteristics of the loading may be determined from a piece of loading history which is taken over a sufficiently long period of time. Ricles and Leger use a slightly different description of the random variable which represents the fatigue damage due to one stress cycle [5]. The damage from a single cycle can be restated from Eq. (1) as: (sT) d˜ ⫽ K
m
(3)
where m and K are empirical parameters. When the linear cumulative damage hypothesis is applied using this description of damage, the random variable which describes the random event of a fatigue failure is [5]:
2. Fatigue strength and loading
d⫽
sT. Using the linear cumulative hypothesis, the random variable which describes a random event of fatigue failure is Miner’s rule which is:
(1)
where NT is the number of cycles to failure at a set stress level sT from fatigue tests. Therefore, d is the fatigue damage due to one stress cycle at a cyclic stress level
冘
1 ⌬⫽ K
(nTismTi)
(4)
i
The random variables defined by Eqs. (2) and (4) can be used to describe the random event of a fatigue failure if the test data is available from fatigue tests; tests which are normally preformed at constant amplitude. However, the distribution parameters must be obtained from a combination of constant amplitude fatigue tests and block loading fatigue tests [4,6–8]. It is not possible to obtain all of the required statistical information from constant amplitude fatigue tests, which is a significant shortcoming in nearly all of the available test data. In order to obtain the required distribution parameters, it is typically assumed that the same distribution parameters are suitable for all materials [5,9–13]. It is more realistic to assume that different materials will have a different range of ⌬ because of differences in the scatter of the strength data which will in turn result in different distribution parameters [6–8,14]. It is, therefore, not reasonable to assume that the distribution parameters of ⌬ are the same for all materials. Previous research has suggested that the uncertainty in ⌬ is due primarily to the linear cumulative damage hypothesis [5,9,11,13–16].
X. Le, M.L. Peterson / International Journal of Fatigue 21 (1999) 603–610
However, this assertion is suspect since ⌬ is a random variable not only because of the linear damage cumulative hypothesis, but also because failure is a random event. In fact, ⌬ is primarily a random variable because of the inherently high level of scatter associated with fatigue strength data in fatigue tests. Consider, instead, an alternative way to describe the random event of fatigue failure. Let the random variable defined by Eq. (5) represent the fatigue damage due to one completely reversed stress cycle of amplitude sT. d⫽(sT)m
(5)
Using the linear cumulative damage hypothesis, the random variable which describes the scatter associated with the fatigue strength data in constant amplitude fatigue tests is:
冘 NTi
Ki⫽
(sTi)m⫽NTi(sTi)m
(6)
i⫽1
where sTi is the completely reversed constant amplitude stress in constant amplitude fatigue tests, NTi is the number of cycles to failure at the stress level sTi, and Ki is a sample value of the random variable K. The power to which the stress amplitude is raised, m, is an empirical parameter. The fatigue tests are conducted at a range of mean stress levels. This type of data is most useful when converted to completely reversed stress [17]. Among the most common methods to account for the effect of mean stress are attributed to Goodman, Gerber and Soderberg [17–20]. In the discussion which follows, Goodman’s method will be used to correct for the effect of a nonzero mean stress [21]. The completely reversed stress amplitude is then: sT ei⫽
sT aisu su−sT mi
(7)
where sT ai and sT mi are the stress range and the mean stress respectively, su is tensile strength and sT ei is the equivalent completely reversed stress range. Based on the use of this conversion, the data which was obtained using a non-zero mean stress will be assumed to be converted to an equivalent completely reversed stress range. Consider a group of constant amplitude completely reversed stress fatigue tests in which data pairs (sT ij,NT ij where i =1,2%I and j=1,2%J) are collected. Notationally, the subscript T indicates a fatigue test, the subscript i is the ith stress level in the fatigue tests sequence and j is the jth fatigue strength test data point at the ith stress level. The data sets are paired because the fatigue mechanism does not change for high cycle fatigue tests although the stress level is different. The variable Kij is then defined as: Kij ⫽NTij (sTij )m where i⫽1,2%I and j⫽1,2%J
(8)
605
where Kij belongs to the same population. Due to the inherent uncertainty of fatigue data both Kij and m are random variables. Because data variances are associated with the same test data, these are dependent random variables. Both variables reflect, to a certain degree, the same source of uncertainty, so just one variable is required to describe the scatter of the fatigue strength test data [22]. In this paper, the variable K is chosen as a random variable in order to account for the scatter of fatigue strength data and m is treated as a deterministic parameter. This approach significantly reduces the complexity associated with the use of K in reliability fatigue theory while still including consideration of the uncertainty of the fatigue tests data. The value of m will be determined empirically from the fatigue strength test data and K will be treated as a lognormally distributed random variable [5,9–11,23–25]. When random variables are described by Eqs. (2) and (4), then the scatter of the fatigue tests are assumed to be the same for all materials. Therefore, K is assumed to have the same distribution parameters for all materials [5,9–11,23–25]. However, realistically, materials will be expected to have different amounts of scatter as a result of variability in the mechanisms of failure which will result in different distribution parameters [5,15,16,26–28]. As a result the distribution parameters for K will be obtained from fatigue test data for the material of interest. The distribution parameters will be obtained assuming that K is a random variable with a lognormal distribution. Using a single subscript for the indices i and j, Eq. (8) may be rewritten as: Kl⫽NTl(sTl)m where l⫽1,2,3,%I·J
(9)
where the subscript l represents a single fatigue stress data point at a single stress level. In order to determine the statistical distribution parameters of K, a value of m must first be calculated. Eq. (9) can be rewritten as: ln(Kl)⫽ln(NTl)⫹mln(sTl)
(10)
Using a least squares fit to the fatigue strength data at all stress levels, m is found as [22]:
冋冘 L
− m⫽
l⫽1
(lnsTllnNTl)−
冘
冉冘 冊冉冘 冊册
1 L
L
L
L
lnsTl
l⫽1
冘 L
lnNTl
l⫽1
(11)
1 (lnsTl) − (lnNTl)2 L l⫽1 l⫽1 2
After the value of m is found then sample values for the random variable K can be calculated using the fatigue test data. The distribution of K is assumed to be lognormal. The lognormal distribution is convenient for K since the logarithm of K, ln K, will be a normally distributed random variable. The choice of distribution in this case should not significantly impact the result and is rather a matter of convenience. The mean and standard deviation
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of the logarithm of K are then mlnK and slnK which may be obtained from:
冘 L
mlnK⫽
冘 L
1 m (lnNTl)⫹ (lnsTl) Ll⫽1 L i⫽1
冪
slnK⫽
冘
(12)
L
(lnNTl=mlnsTl−mlnK)2
l⫽1
(13)
L
It is evident that the scatter of the fatigue strength test data is represented by mlnK and slnK for a particular value of m. The value of K is indirectly used to include the scatter of the fatigue strength tests data at all of the stress levels considered. From definitions in Eqs. (5) and (6), K can be clearly seen to be the critical value of the fatigue damage accumulation, or the point at which failure is likely to occur. Since K represents the ability of a material to resist fatigue failure, it will be referred to as the fatigue damage resistance of the material. 2.2. Model of loading Using the definition from Eq. (5), a model of the loading can be used to indirectly reflect the fatigue damage accumulation in the component. The fatigue damage accumulated due to the cyclic loading applied is:
冘 j⫽J
DC ⫽
Based on the definitions in Eqs. (5) and (13), DC can indirectly reflect the fatigue damage status of the components during the design fatigue life based on either design or in-service loading conditions. For example, DC indirectly represents the fatigue damage which has accumulated due to the cyclic loading in components during the fatigue life T. In the follow discussion, D will be referred to as the fatigue damage accumulated of components during the fatigue life T.
[Ncj (scj )m]⫽T ⫻ E[(sc)m]
(14)
j
where the subscript C means components or structures, sCj is jth cyclic stress level, and NCj is the number of cyclic stress sCj. The value of m is found from the definition in Eq. (11). T is the design fatigue life of the component or structure being considered. The expected value of the mth moment of the cyclic loading stress sC is E((sC)m). The fatigue damage accumulated in components or structures due to the cyclic loading is DC. For simple loading, the cyclic loading applied on components or structures can be treated as a series of constant amplitude stress levels. For a narrow band random loading process, the cyclic loading sC can be described by a probability density function [1]. In the wide band random loading process, the rainflow counting algorithms should be used to obtain the cyclic stresses (sCj,NCj) [29,30]. When the cyclic stress is nonzero-mean cyclic stress, it is converted as described above using Goodman’s method. After using rainflow counting the data can be treated as a series of constant cyclic stress levels or described by a probability density function. Regardless of the loading, simple, narrow band, or wide band, when the description of cyclic loading is obtained, Eq. (13) can be used to calculate DC which is completely determined by the loading on the components or structures.
3. Probabilistic fatigue model After the concepts of two random variables K and D are developed, a limit state function can be constructed. However before K can be used to compare with DC, the differences between the test specimens and the actual components should be taken into account. This means that K must first be modified before it can be used in fatigue evaluation of components. The main differences between the components and the test specimen are the surface roughness and the dimensions of the part versus the test specimen. Additionally, in the actual part, stress concentrations may also exist. Following general design practice, correction coefficients can be used to accommodate the differences. For example, three random variables e, b and Ks can be used to describe the differences which exist, the size coefficient, the surface roughness coefficient and the stress concentration coefficient respectively [2,18,31,32]. Usually, all of the variables are treated as normal random variables [8,32]. After differences which exist between the test specimen and the components are identified, the corrected value of K can be described as:
冉 冊 冉冊
KC⫽NT
m eb eb m sT ⫽ ⫻K Ks Ks
(15)
Because differences between the specimens and the components are included in KC, it is reasonable to assume that KC indirectly reflects the inherent capability of components or structures to resist the fatigue damage. Thus, KC may be referred to as the fatigue damage resistance of a component. When the distribution parameters of e, b and Ks are known, the distribution parameters of the random variables KC can be calculated from Eq. (15). Therefore, based on the definitions of KC and DC which have been developed, the limit state function is Z⫽KC⫺DC
(16)
Based on the limit state function, the reliability of a component or structure is: PR⫽P(Z⬎0)⫽P(KC⬎DC)
(17)
In general, an exact solution for Eq. (16) is not practical. Instead, an iterative approximate method is employed
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[1]. In the iterative solution technique proposed by Rackwitz [33], a convergent value denoted as B, which is referred to as a reliability index, is obtained. The reliability of a component is then determined by: PR⫽⌽(B)
(18)
where ⌽ is the standard normal cumulative density function and B is the reliability index. Eqs. (16)–(18) bring the fatigue strength, fatigue damage, loading, fatigue life and reliability index together. The fatigue evaluation and fatigue design can be carried out based on Eqs. (16) and (17) if component loading is known. However, the objective is to consider the fatigue evaluation of components with unknown loading. In order to use the model to perform the fatigue evaluation of a component, it is necessary to express DC in terms of additional information which is associated with the applied loading. Fatigue test data of used components is used to obtain the information required to understand the applied loading and to calculate DC. 3.1. Residual fatigue damage resistance If the component loading is unknown additional information is required to perform the fatigue evaluation. As noted previously, the additional information on the inservice applied loading may be obtained from used components. If the accumulated fatigue damage from a particular length of time in-service can be measured, fatigue evaluation of the component can still be performed without explicit knowledge of the applied loading. As components suffer fatigue damage, the components loose their capacity for resisting fatigue damage. In the following discussion, a used component refers to a component which has been in operation for some period of time, but which has not failed. Used components with the same in-service time should be used to conduct the fatigue tests. The distribution of the test data describes the residual fatigue damage resistance and the applied loading of the used components. The fatigue damage resistance of the used components can then be compared to new parts to describe the fatigue damage which has accumulated in the parts during the time in service. In order to explain the use of fatigue test data of used components to describe the loading of the components, first, temporarily suppose that fatigue damage resistance K is a deterministic constant. Suppose that the number of cycles to failure of the component in a normal fatigue test is N1 when the stress range is s1. Then, according to the definition of the fatigue damage resistance of the component, KC is: KC⫽N1(s1)m
(19)
After the component has been in service for some time, the component is removed to conduct fatigue test. The
607
number of cycles to failure of the used component at a stress range of s1 is N2. The residual fatigue damage resistance of the used component is then: KR⫽N2sm1
(20)
Because the fatigue damage resistance of the component, KC, is assumed to be a constant, the fatigue damage accumulated in the component due to cyclic constant amplitude loading during the time in service is: DC⫽KC⫺KR⫽(N1⫺N2)(s1)m
(21)
It is clear that although the loading conditions is unknown, information on the fatigue damage accumulated during the time in service can be obtained from the fatigue test data of the used component. The change in the remaining fatigue life of the component as time-inservice increases can also provide additional information on the applied loading which can be used to calculate DC . Because of high degree of scatter in fatigue strength, the test data on used components, like new components, will also have a lot of scatter. The residual fatigue damage resistance of the used components, KR, will be a random variable that indirectly reflects the remaining capability of the used component to resist fatigue damage relative to a new component. The distribution parameters of KR are determined from the fatigue test data of the used components. Suppose used components with the same time in service, t, are used to conduct the constant amplitude fatigue tests. A group of test pairs (sRi,NRi, i=1,2,3,...,nR) are collected. Subscript R indicates fatigue tests of used components, NRi is ith number of cycles to failure at the test stress range sRi, and nR is the sample size for the fatigue tests. The nR data pairs can then be used to calculate the nR sample values of the residual fatigue resistance, KR: KRi⫽NRi(sRi)m where (i⫽1,2,3,...,nR)
(22)
Using nR sample values of the residual fatigue resistance, KR, the distribution parameters of KR can be determined statistically. It is also assumed that KR observes a lognormal distribution. Although a new value of m can be calculated by the group of test pairs (sRi,NRi, i=1,2,3,...,nR) using the equation similar to Eq. (11), the value of m in Eq. (22) must be equal to the value defined by Eq. (11), which is calculated from test data of new components. Two reasons exist for the use of the same m. It is assumed that the fatigue damage in the new component is zero because it has not experienced cyclic loading so the fatigue damage accumulated in the used component is compared with the new component. The value of m calculated from the test data on the used components contains uncertainty associated with loading. Based on Eqs. (5) and (6), the value of m should be associated only with the uncertainty of the fatigue strength.
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Since the components have been in service for time t the components have fatigue damage prior to testing. The fatigue damage accumulated in the components while in service can be determined from: DR⫽KC⫺KR
(23)
where DR is also a random variable, whose distribution parameters can be determined by using the distribution parameters of KC and KR. The loading conditions on the components have changed and can not be predicted. However, when the function of the components is determined, even when the loading conditions have changed, the loading on the components in a sufficient long statistical unit can be assumed to be an ergodic random process. The length of load time will vary with the application. For example, when considering components in airplane engine, one statistical unit may be 100 hours. In contrast, for components installed in machinery one statistical unit may be 1 years. Based on this assumption, the distribution law and parameters of the loading on the parts in each statistical unit are the same. Thus, the fatigue damage accumulated in the used components with the same time in service, t, in one statistical unit will be: 1 Duni t⫽ (KC⫺KR) t
(24)
Then, the fatigue damaged accumulated in the components during a fatigue life of T will be: T DC⫽T ⫻ Duni t⫽ (KC⫺KR) t
(25)
After DC is determined, Eqs. (16) and (17) can be used to perform the fatigue evaluation of components with unknown loading.
4. Fatigue evaluation of compressor blades In order to demonstrate the approach described, an example is presented. Compressor blades in an airplane engine are well suited to this sort of approach because of the large number of components which can be tested, and because the risks of failure are high. In addition, the loading of the blades is not well characterized. The primary reason for fatigue failure of the compressor blades in an airplane engine is stress caused by high frequency vibration of the blades [34]. Stresses due to the vibrations are a random variable. When 100 hours of service is taken as one statistical unit, vibration stress can be assumed to be an ergodic random process. The distribution law and parameters of the stresses are then assumed to be the same for each statistical unit. Two groups of blades were used to conduct the fatigue tests [34]. One group is new blades with sample size of 76
blades. Another group, with sample size of 102 blades, are the used blades which have been in service for 300 hours. Using available test data [34] and the model developed above, the lognormal distribution parameters of the fatigue data for the new blade and used blades can be calculated. The distribution parameters calculated for the new blades are: m⫽16.3322 mln K⫽118.6632 (Mpa) sln K⫽0.1995 (Mpa) The distribution parameters calculated for the used blades are: m⫽16.3322 mln K⫽118.3982 (Mpa) sln K⫽0.1168 (Mpa) Using 100 hours as one statistical unit, the fatigue damage accumulated in the blades during service time T is: T DC⫽T ⫻ Duni t⫽ (KC⫺KR) 3
(26)
Using the limit state function Eq. (16) and the iterative solution technique [33], the fatigue evaluation of the blades are shown in Tables 1 and 2. In Tables 1 and 2, the reliability index is defined by Eq. (18). If the reliability level is defined by technical requirements, the corresponding fatigue life can be calculated at the given reliability. Table 1 shows the calculation results for this application. For example, in Table 1, if the reliability requirement is 0.999 (0.0001 probability of failure is allowed), the blades which have operated for 221 hours should be replaced. It may also be required that the reliability level of the blades at a particular length of service time be known. Table 2 shows the calculation result in this situation. For example, in Table 2, after blades have been in service for 200 hours, calculations indicate that 4 out of 10 000 blades will have failed during the 200 hours in service (evaluated reliability level is 0.9996 and evaluated failure probability is 0.0004). The calculation results shown in Table 2 are reasonably close to the statistical results from in-situ testing of the blades [34].
5. Summary and discussions When the loading of the components is unknown, it is not possible to perform fatigue evaluation of the components without additional information. A method Table 1 The evaluated fatigue life at given reliability index Reliability requirement 0.9999 Failure probability 0.00001 requirement Reliability index 3.72 Evaluated fatigue life 172.5 (hours)
0.999
0.99
0.9
0.0001
0.001
0.1
3.09
2.33
1.282
221.4
306.6
511.6
X. Le, M.L. Peterson / International Journal of Fatigue 21 (1999) 603–610
609
Table 2 The evaluated reliability at given fatigue life Service time (hours) Evaluated reliability level Evaluated failure probability Reliability index
100 0.9999997 0.0000003 5.0
has been described which shows how to obtain information about the loading from the performance of used components in fatigue tests. A probabilistic fatigue model for this problem is developed and can be used to perform the fatigue evaluation of the component with unknown loading. After a component experiences cyclic loading, some amount of fatigue damage will accumulate in the component. Because fatigue damage of the component is permanent damage, it will gradually accumulate as the time in service increases. This means, that the components have lost some of their capability for resisting fatigue damage. According to the linear cumulative damage assumption, a certain fatigue damage accumulated in the component will be associated with the number of cycles (time in service) and applied loading. So the accumulated fatigue damage in the component will provide the additional information which is needed regarding the applied loading. Using components which have been removed from service after the same number of service hours, fatigue tests can be conducted which will provide the required test data. The test data can then be used to determine the residual fatigue damage resistance of the used components. Comparing the residual fatigue damage resistance of the used components with the fatigue damage resistance of the new components can indirectly express the fatigue damage accumulated in the used components during time in service. As a result, fatigue evaluation of components can be performed based on the model developed in this paper. In order to obtain the required accuracy, the used components must be carefully selected for an appropriate time in service. The used components must have been in service for at least one statistical unit in order for KR to be able to express the fatigue damage with sufficient accuracy. For example, the typical time in service must be more than 100 hours for components in an airplane engine, or one year for components of an industrial machine. If time in service for the components is too short, KR will not contain sufficient information with respect to actual loading conditions. For example, some loading conditions will not occur during the service interval if the time is too short. However, the service time cannot be so long that fatigue failures of the components have occurred. The suitable time in service for conducting fatigue tests on the used components will depend on the design and the working conditions. The required
200 0.9996 0.0004 3.945
300 0.9912 0.0088 2.375
400 0.9606 0.0396 1.752
500 0.9072 0.0828 1.325
time ion-service corresponds to a reliability level of 0.99, for the compressor blades example. The suitable time in service of the blade prior to conducting fatigue tests will be in the range of 100 to 306 hours based on the calculation results in Table 2. The potential of this approach is significant, however, important restrictions exist in the manner in which it may be applied as well as the type of application for which it is suitable. This model applies to situations where the loading can be assumed to be ergodic. In addition, a sufficient number of components must exist which can be removed from service after a sufficient service interval. If these criteria are met however, a new approach is possible for determining the fatigue resistance of an important type of service application.
Acknowledgements This work was supported in part by the RockwellAnderson Chair in the Department of Mechanical Engineering at Colorado State University. Additional support was provided by Storage Technology Corporation and Eastman Kodak Company.
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babilistic Mechanics and Structural Reliability. Worchester (MA, USA), 1996, August. Zhao Z, Haldar A, Breen FL. Fatigue-reliability evaluation of steel bridges. Journal of Structural Engineering 1994;120(5):1608–23. Wirsching PH. Fatigue reliability for off-shore structures. Journal of Structural Engineering 1984;110(10):2340–56. Fatigue reliability instruction. Journal of Structural Engineering 1982;108(ST1):3–23. Wirsching PH, Light MC. Probability based fatigue design criteria for off-shore structures. Final Report, American Petroleum Institute PRAC Project 79-15, 1979. Fatigue reliability, variable amplitude loading. Journal of Structural Engineering 1982;108(ST1):47–69. Byers WG, Marley MJ. Fatigue reliability reassessment procedures: state-of-the-art paper. Journal of Structural Engineering 1997;123(3):271–6. Byers WG, Marley MJ. Fatigue reliability reassessment procedures: state-of-the-art paper. Journal of Structural Engineering 1997;123(3):277–85. ASM Handbook Committee, Metals Handbook, Failure analysis and prevention, Vol. 11., 9th ed. 1978 Edwin HG, Charles NG. Structural Engineering Handbook. 4th ed McGraw-Hill, 1997. Properties and Selection, Irons, Steel and High Performance Alloys, Vol. 1. In: Metals Handbook., 10th ed, 1990. Osgood CC. Fatigue Design. Oxford, New York: Pergamon Press, 1982. Goodman J. Mechanics Applied to Engineering. London: Longman, Green and Company, 1899. Kececioglu D. Reliability Engineering Handbook. Englewood Cliffs (NJ): Prentice Hall, 1991. Wirsching PH. Linear models in probabilistic fatigue design.
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Journal of the Engineering Mechanics Division, ASCE 1980;106(EM6):1265–77. Wirsching PH, Ortiz K, Chen YN. Fracture mechanics fatigue model in a reliability format. In: Proc. 6th International Symposium on OMAE, Houston (TX), 1987. Fatigue reliability, development of criteria for design. Journal of Structural Engineering 1982;108(ST1):71–8. Hanna SY, Karsan D. Fatigue data for reliability-based platform inspection and repair. J Structural Engineering ASCE 1991;117(10):3168–85. ESC. Steel in Marine Structures. In: Proc Third Int European Coal and Steel Community (ESC) Offshore Conf Delft (The Netherlands), 1987. Department of Energy. New fatigue design guidance for steel welded joints in offshore structures. Recommendations, Her Majesty’s Stationary Office (HMSO) London, 1982. Laman JA. Probabilistic Fatigue Models for Bridge Evaluation. In: Proceedings of the Seventh Specialty Conference, Probabilistic Mechanics and Structural Reliability, 1996 August 7–9. Worcester (Massachusetts, USA). Downing SD, Socie DF. Simple rainflow counting algorithms. Int J Fatigue 1981;3(1):31–40. Pilkey WD. Formulas for stress, strain, and structural materials. John Wiley and Sons, 1994. Hu Z, Le X. Probabilistic design of machinery and structure. Shanghai (China): Shanghai Jiaotong University Press, 1995. Rackwitz R. Practical probabilistic approach to design. Bull. 112, Comite European du Beton, Paris (France), 1976. Aviation Industry Ministry of P.R. China. The research report on the fatigue tests and life evaluation of WP6 airplane engine. Chapter 4: The test research of fatigue life on compressor blades I. Aviation Industry Ministry of P.R. China, 1988.
A method for fatigue based reliability when the loading of a component is unknown Xiaobin Le, M.L. Peterson
*
Department of Mechanical Engineering, Colorado State University, Fort Collins, CO 80523, USA Received 27 March 1998; received in revised form 30 November 1998; accepted 8 February 1999
Abstract When the loading of a component is unknown, it is not possible to perform fatigue evaluation without some additional knowledge of the applied loading. A model is presented which considers problems of this type. In this situation, the components will have experienced some degree of fatigue damage during operation. However, the loading during use is unknown, and thus direct calculation of the remaining service life is not possible without additional data. The use of components removed from service to conduct fatigue tests can provide the required test data. The test data contains additional information associated with the loading which can be used to determine the residual fatigue damage resistance of the components. Comparing the residual fatigue damage resistance of the tested components with the fatigue damage resistance of new components can make it possible to, indirectly, describe the fatigue damage accumulation which has occurred in the components during the time in service. Thus, the fatigue damage accumulated in the components can be determined by additional fatigue tests, in spite of the absence of direct knowledge of applied service loading. The fatigue evaluation of untested components can be performed using the probabilistic fatigue model which is presented in this paper. The model is demonstrated by performing an evaluation of compressor blades for which the applied loading is unknown. The calculated results are shown to be close to the statistical results obtained in-service for the blades. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue model; Reliability; Unknown loading; Fatigue test
1. Introduction Before fatigue evaluation of a component can be carried out, in general, three pieces of information must be known: (1) fatigue strength data with appropriate description, (2) loading which will be applied to the component and the loading model and (3) the limit state function or relationship equation. The limit state function, or relationship equation, brings the fatigue strength, cyclic loading, fatigue damage, fatigue life and the reliability index together [1,2]. In a number of engineering applications, the loading of components may not be known or it may not be possible to describe the loading with sufficient accuracy even when the applied loads are known. Incomplete knowledge of the loading is common
* Corresponding author. Tel.: +1-970-491-2813; fax: +1-970-4911055. E-mail address: [email protected] (M.L. Peterson)
not only when the utilization is unknown, but also when uncertainty exists in the operating conditions. For example, unknown loading due to utilization is common in infrastructure when the quantity and weight of the traffic are not known. In automobile or truck applications, the road surface and vehicle utilization are typically not well defined, which results in uncertainty. For aerospace applications, the effect of turbulence and other operating conditions may create significant uncertainty in the applied load. In addition, the loads on a component may be difficult to measure because the component is irregular in shape. In other cases, the load may also be applied to the component in a complicated manner. Actual measurements of the loads encountered in service may not even be possible because of difficulties associated with physical access to the part for measurements. For example, operational access to the compressor blades of an airplane engine is restricted in addition to any measurements being difficult because of the motion of the blades. In situations such as a compressor blade
0142-1123/99/$ - see front matter. 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 9 ) 0 0 0 1 6 - X
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in an aircraft engine, it is difficult to describe the operational loading with reasonable accuracy. In order to evaluate the degree of fatigue damage in the blades, blades which have operated under normal service conditions for a certain number of hours can be used to conduct fatigue tests. The fatigue test may be for a simplified loading of the blade, rather than completely replicating the operational loading. This approach is in general also applicable to accelerated testing where the loading may be more severe than in-service loads. In some applications, the fatigue damage status of some of the components may be investigated more economically by conducting operational fatigue tests, rather than trying to measure and then replicate the operational loading. A method is proposed which will address these problems which are associated with fatigue evaluation of a component with unknown loading. The main objective of the model is to present an approach which can utilize fatigue test data performed on used components which have been removed from service prior to failure. The model is used to provide information which will compensate for the limited information about the in-service loading. The model also makes it possible to represent the fatigue damage which has accumulated in the component due to the unknown loading. The fatigue evaluation of compressor blades in a jet engine will be used to show the utility of the approach.
2.1. Model of the fatigue strength data in the test Fatigue failure of components occurs because of accumulated fatigue damage which occurs during cyclic loading. When fatigue damage accumulates to a critical value, fatigue failure will occur. Because of the amount of scatter in test data, it is impossible, at a given stress level, to predict the exact number of cycles at which a fatigue failure will occur [3]. Additionally, it is also not possible to know at what stress level a failure will occur for a given fatigue life. The random variables, which reflect the degree of damage associated with a particular stress level and number of stress cycles, can describe the random event of a fatigue failure. Fatigue damage of a component is a conceptual abstraction which can be indirectly described by a random variable. Miner defines the fatigue damage associated with one stress cycle, d, as [4]: 1 NT
冘冉 冊 nTi NTi
⌬⫽
(2)
where NTi is the number of cycles to failure at the ith stress cycle sTi and nTi is the number of stress cycles at the stress level sTi. The fatigue damage index is then ⌬, which indirectly expresses the fatigue damage to the component which has undergone nTi cycles at a stress level sTi. It is assumed that when the fatigue damage accumulated in the component is greater than ⌬failure then a fatigue failure will occur. In traditional theory, when ⌬ is greater than 1, failure will occur, however this is not necessarily true in a probabilistic sense. Additionally, for the present model, ergodicity of loading is also assumed. In other words, the average characteristics of the loading may be determined from a piece of loading history which is taken over a sufficiently long period of time. Ricles and Leger use a slightly different description of the random variable which represents the fatigue damage due to one stress cycle [5]. The damage from a single cycle can be restated from Eq. (1) as: (sT) d˜ ⫽ K
m
(3)
where m and K are empirical parameters. When the linear cumulative damage hypothesis is applied using this description of damage, the random variable which describes the random event of a fatigue failure is [5]:
2. Fatigue strength and loading
d⫽
sT. Using the linear cumulative hypothesis, the random variable which describes a random event of fatigue failure is Miner’s rule which is:
(1)
where NT is the number of cycles to failure at a set stress level sT from fatigue tests. Therefore, d is the fatigue damage due to one stress cycle at a cyclic stress level
冘
1 ⌬⫽ K
(nTismTi)
(4)
i
The random variables defined by Eqs. (2) and (4) can be used to describe the random event of a fatigue failure if the test data is available from fatigue tests; tests which are normally preformed at constant amplitude. However, the distribution parameters must be obtained from a combination of constant amplitude fatigue tests and block loading fatigue tests [4,6–8]. It is not possible to obtain all of the required statistical information from constant amplitude fatigue tests, which is a significant shortcoming in nearly all of the available test data. In order to obtain the required distribution parameters, it is typically assumed that the same distribution parameters are suitable for all materials [5,9–13]. It is more realistic to assume that different materials will have a different range of ⌬ because of differences in the scatter of the strength data which will in turn result in different distribution parameters [6–8,14]. It is, therefore, not reasonable to assume that the distribution parameters of ⌬ are the same for all materials. Previous research has suggested that the uncertainty in ⌬ is due primarily to the linear cumulative damage hypothesis [5,9,11,13–16].
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However, this assertion is suspect since ⌬ is a random variable not only because of the linear damage cumulative hypothesis, but also because failure is a random event. In fact, ⌬ is primarily a random variable because of the inherently high level of scatter associated with fatigue strength data in fatigue tests. Consider, instead, an alternative way to describe the random event of fatigue failure. Let the random variable defined by Eq. (5) represent the fatigue damage due to one completely reversed stress cycle of amplitude sT. d⫽(sT)m
(5)
Using the linear cumulative damage hypothesis, the random variable which describes the scatter associated with the fatigue strength data in constant amplitude fatigue tests is:
冘 NTi
Ki⫽
(sTi)m⫽NTi(sTi)m
(6)
i⫽1
where sTi is the completely reversed constant amplitude stress in constant amplitude fatigue tests, NTi is the number of cycles to failure at the stress level sTi, and Ki is a sample value of the random variable K. The power to which the stress amplitude is raised, m, is an empirical parameter. The fatigue tests are conducted at a range of mean stress levels. This type of data is most useful when converted to completely reversed stress [17]. Among the most common methods to account for the effect of mean stress are attributed to Goodman, Gerber and Soderberg [17–20]. In the discussion which follows, Goodman’s method will be used to correct for the effect of a nonzero mean stress [21]. The completely reversed stress amplitude is then: sT ei⫽
sT aisu su−sT mi
(7)
where sT ai and sT mi are the stress range and the mean stress respectively, su is tensile strength and sT ei is the equivalent completely reversed stress range. Based on the use of this conversion, the data which was obtained using a non-zero mean stress will be assumed to be converted to an equivalent completely reversed stress range. Consider a group of constant amplitude completely reversed stress fatigue tests in which data pairs (sT ij,NT ij where i =1,2%I and j=1,2%J) are collected. Notationally, the subscript T indicates a fatigue test, the subscript i is the ith stress level in the fatigue tests sequence and j is the jth fatigue strength test data point at the ith stress level. The data sets are paired because the fatigue mechanism does not change for high cycle fatigue tests although the stress level is different. The variable Kij is then defined as: Kij ⫽NTij (sTij )m where i⫽1,2%I and j⫽1,2%J
(8)
605
where Kij belongs to the same population. Due to the inherent uncertainty of fatigue data both Kij and m are random variables. Because data variances are associated with the same test data, these are dependent random variables. Both variables reflect, to a certain degree, the same source of uncertainty, so just one variable is required to describe the scatter of the fatigue strength test data [22]. In this paper, the variable K is chosen as a random variable in order to account for the scatter of fatigue strength data and m is treated as a deterministic parameter. This approach significantly reduces the complexity associated with the use of K in reliability fatigue theory while still including consideration of the uncertainty of the fatigue tests data. The value of m will be determined empirically from the fatigue strength test data and K will be treated as a lognormally distributed random variable [5,9–11,23–25]. When random variables are described by Eqs. (2) and (4), then the scatter of the fatigue tests are assumed to be the same for all materials. Therefore, K is assumed to have the same distribution parameters for all materials [5,9–11,23–25]. However, realistically, materials will be expected to have different amounts of scatter as a result of variability in the mechanisms of failure which will result in different distribution parameters [5,15,16,26–28]. As a result the distribution parameters for K will be obtained from fatigue test data for the material of interest. The distribution parameters will be obtained assuming that K is a random variable with a lognormal distribution. Using a single subscript for the indices i and j, Eq. (8) may be rewritten as: Kl⫽NTl(sTl)m where l⫽1,2,3,%I·J
(9)
where the subscript l represents a single fatigue stress data point at a single stress level. In order to determine the statistical distribution parameters of K, a value of m must first be calculated. Eq. (9) can be rewritten as: ln(Kl)⫽ln(NTl)⫹mln(sTl)
(10)
Using a least squares fit to the fatigue strength data at all stress levels, m is found as [22]:
冋冘 L
− m⫽
l⫽1
(lnsTllnNTl)−
冘
冉冘 冊冉冘 冊册
1 L
L
L
L
lnsTl
l⫽1
冘 L
lnNTl
l⫽1
(11)
1 (lnsTl) − (lnNTl)2 L l⫽1 l⫽1 2
After the value of m is found then sample values for the random variable K can be calculated using the fatigue test data. The distribution of K is assumed to be lognormal. The lognormal distribution is convenient for K since the logarithm of K, ln K, will be a normally distributed random variable. The choice of distribution in this case should not significantly impact the result and is rather a matter of convenience. The mean and standard deviation
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of the logarithm of K are then mlnK and slnK which may be obtained from:
冘 L
mlnK⫽
冘 L
1 m (lnNTl)⫹ (lnsTl) Ll⫽1 L i⫽1
冪
slnK⫽
冘
(12)
L
(lnNTl=mlnsTl−mlnK)2
l⫽1
(13)
L
It is evident that the scatter of the fatigue strength test data is represented by mlnK and slnK for a particular value of m. The value of K is indirectly used to include the scatter of the fatigue strength tests data at all of the stress levels considered. From definitions in Eqs. (5) and (6), K can be clearly seen to be the critical value of the fatigue damage accumulation, or the point at which failure is likely to occur. Since K represents the ability of a material to resist fatigue failure, it will be referred to as the fatigue damage resistance of the material. 2.2. Model of loading Using the definition from Eq. (5), a model of the loading can be used to indirectly reflect the fatigue damage accumulation in the component. The fatigue damage accumulated due to the cyclic loading applied is:
冘 j⫽J
DC ⫽
Based on the definitions in Eqs. (5) and (13), DC can indirectly reflect the fatigue damage status of the components during the design fatigue life based on either design or in-service loading conditions. For example, DC indirectly represents the fatigue damage which has accumulated due to the cyclic loading in components during the fatigue life T. In the follow discussion, D will be referred to as the fatigue damage accumulated of components during the fatigue life T.
[Ncj (scj )m]⫽T ⫻ E[(sc)m]
(14)
j
where the subscript C means components or structures, sCj is jth cyclic stress level, and NCj is the number of cyclic stress sCj. The value of m is found from the definition in Eq. (11). T is the design fatigue life of the component or structure being considered. The expected value of the mth moment of the cyclic loading stress sC is E((sC)m). The fatigue damage accumulated in components or structures due to the cyclic loading is DC. For simple loading, the cyclic loading applied on components or structures can be treated as a series of constant amplitude stress levels. For a narrow band random loading process, the cyclic loading sC can be described by a probability density function [1]. In the wide band random loading process, the rainflow counting algorithms should be used to obtain the cyclic stresses (sCj,NCj) [29,30]. When the cyclic stress is nonzero-mean cyclic stress, it is converted as described above using Goodman’s method. After using rainflow counting the data can be treated as a series of constant cyclic stress levels or described by a probability density function. Regardless of the loading, simple, narrow band, or wide band, when the description of cyclic loading is obtained, Eq. (13) can be used to calculate DC which is completely determined by the loading on the components or structures.
3. Probabilistic fatigue model After the concepts of two random variables K and D are developed, a limit state function can be constructed. However before K can be used to compare with DC, the differences between the test specimens and the actual components should be taken into account. This means that K must first be modified before it can be used in fatigue evaluation of components. The main differences between the components and the test specimen are the surface roughness and the dimensions of the part versus the test specimen. Additionally, in the actual part, stress concentrations may also exist. Following general design practice, correction coefficients can be used to accommodate the differences. For example, three random variables e, b and Ks can be used to describe the differences which exist, the size coefficient, the surface roughness coefficient and the stress concentration coefficient respectively [2,18,31,32]. Usually, all of the variables are treated as normal random variables [8,32]. After differences which exist between the test specimen and the components are identified, the corrected value of K can be described as:
冉 冊 冉冊
KC⫽NT
m eb eb m sT ⫽ ⫻K Ks Ks
(15)
Because differences between the specimens and the components are included in KC, it is reasonable to assume that KC indirectly reflects the inherent capability of components or structures to resist the fatigue damage. Thus, KC may be referred to as the fatigue damage resistance of a component. When the distribution parameters of e, b and Ks are known, the distribution parameters of the random variables KC can be calculated from Eq. (15). Therefore, based on the definitions of KC and DC which have been developed, the limit state function is Z⫽KC⫺DC
(16)
Based on the limit state function, the reliability of a component or structure is: PR⫽P(Z⬎0)⫽P(KC⬎DC)
(17)
In general, an exact solution for Eq. (16) is not practical. Instead, an iterative approximate method is employed
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[1]. In the iterative solution technique proposed by Rackwitz [33], a convergent value denoted as B, which is referred to as a reliability index, is obtained. The reliability of a component is then determined by: PR⫽⌽(B)
(18)
where ⌽ is the standard normal cumulative density function and B is the reliability index. Eqs. (16)–(18) bring the fatigue strength, fatigue damage, loading, fatigue life and reliability index together. The fatigue evaluation and fatigue design can be carried out based on Eqs. (16) and (17) if component loading is known. However, the objective is to consider the fatigue evaluation of components with unknown loading. In order to use the model to perform the fatigue evaluation of a component, it is necessary to express DC in terms of additional information which is associated with the applied loading. Fatigue test data of used components is used to obtain the information required to understand the applied loading and to calculate DC. 3.1. Residual fatigue damage resistance If the component loading is unknown additional information is required to perform the fatigue evaluation. As noted previously, the additional information on the inservice applied loading may be obtained from used components. If the accumulated fatigue damage from a particular length of time in-service can be measured, fatigue evaluation of the component can still be performed without explicit knowledge of the applied loading. As components suffer fatigue damage, the components loose their capacity for resisting fatigue damage. In the following discussion, a used component refers to a component which has been in operation for some period of time, but which has not failed. Used components with the same in-service time should be used to conduct the fatigue tests. The distribution of the test data describes the residual fatigue damage resistance and the applied loading of the used components. The fatigue damage resistance of the used components can then be compared to new parts to describe the fatigue damage which has accumulated in the parts during the time in service. In order to explain the use of fatigue test data of used components to describe the loading of the components, first, temporarily suppose that fatigue damage resistance K is a deterministic constant. Suppose that the number of cycles to failure of the component in a normal fatigue test is N1 when the stress range is s1. Then, according to the definition of the fatigue damage resistance of the component, KC is: KC⫽N1(s1)m
(19)
After the component has been in service for some time, the component is removed to conduct fatigue test. The
607
number of cycles to failure of the used component at a stress range of s1 is N2. The residual fatigue damage resistance of the used component is then: KR⫽N2sm1
(20)
Because the fatigue damage resistance of the component, KC, is assumed to be a constant, the fatigue damage accumulated in the component due to cyclic constant amplitude loading during the time in service is: DC⫽KC⫺KR⫽(N1⫺N2)(s1)m
(21)
It is clear that although the loading conditions is unknown, information on the fatigue damage accumulated during the time in service can be obtained from the fatigue test data of the used component. The change in the remaining fatigue life of the component as time-inservice increases can also provide additional information on the applied loading which can be used to calculate DC . Because of high degree of scatter in fatigue strength, the test data on used components, like new components, will also have a lot of scatter. The residual fatigue damage resistance of the used components, KR, will be a random variable that indirectly reflects the remaining capability of the used component to resist fatigue damage relative to a new component. The distribution parameters of KR are determined from the fatigue test data of the used components. Suppose used components with the same time in service, t, are used to conduct the constant amplitude fatigue tests. A group of test pairs (sRi,NRi, i=1,2,3,...,nR) are collected. Subscript R indicates fatigue tests of used components, NRi is ith number of cycles to failure at the test stress range sRi, and nR is the sample size for the fatigue tests. The nR data pairs can then be used to calculate the nR sample values of the residual fatigue resistance, KR: KRi⫽NRi(sRi)m where (i⫽1,2,3,...,nR)
(22)
Using nR sample values of the residual fatigue resistance, KR, the distribution parameters of KR can be determined statistically. It is also assumed that KR observes a lognormal distribution. Although a new value of m can be calculated by the group of test pairs (sRi,NRi, i=1,2,3,...,nR) using the equation similar to Eq. (11), the value of m in Eq. (22) must be equal to the value defined by Eq. (11), which is calculated from test data of new components. Two reasons exist for the use of the same m. It is assumed that the fatigue damage in the new component is zero because it has not experienced cyclic loading so the fatigue damage accumulated in the used component is compared with the new component. The value of m calculated from the test data on the used components contains uncertainty associated with loading. Based on Eqs. (5) and (6), the value of m should be associated only with the uncertainty of the fatigue strength.
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Since the components have been in service for time t the components have fatigue damage prior to testing. The fatigue damage accumulated in the components while in service can be determined from: DR⫽KC⫺KR
(23)
where DR is also a random variable, whose distribution parameters can be determined by using the distribution parameters of KC and KR. The loading conditions on the components have changed and can not be predicted. However, when the function of the components is determined, even when the loading conditions have changed, the loading on the components in a sufficient long statistical unit can be assumed to be an ergodic random process. The length of load time will vary with the application. For example, when considering components in airplane engine, one statistical unit may be 100 hours. In contrast, for components installed in machinery one statistical unit may be 1 years. Based on this assumption, the distribution law and parameters of the loading on the parts in each statistical unit are the same. Thus, the fatigue damage accumulated in the used components with the same time in service, t, in one statistical unit will be: 1 Duni t⫽ (KC⫺KR) t
(24)
Then, the fatigue damaged accumulated in the components during a fatigue life of T will be: T DC⫽T ⫻ Duni t⫽ (KC⫺KR) t
(25)
After DC is determined, Eqs. (16) and (17) can be used to perform the fatigue evaluation of components with unknown loading.
4. Fatigue evaluation of compressor blades In order to demonstrate the approach described, an example is presented. Compressor blades in an airplane engine are well suited to this sort of approach because of the large number of components which can be tested, and because the risks of failure are high. In addition, the loading of the blades is not well characterized. The primary reason for fatigue failure of the compressor blades in an airplane engine is stress caused by high frequency vibration of the blades [34]. Stresses due to the vibrations are a random variable. When 100 hours of service is taken as one statistical unit, vibration stress can be assumed to be an ergodic random process. The distribution law and parameters of the stresses are then assumed to be the same for each statistical unit. Two groups of blades were used to conduct the fatigue tests [34]. One group is new blades with sample size of 76
blades. Another group, with sample size of 102 blades, are the used blades which have been in service for 300 hours. Using available test data [34] and the model developed above, the lognormal distribution parameters of the fatigue data for the new blade and used blades can be calculated. The distribution parameters calculated for the new blades are: m⫽16.3322 mln K⫽118.6632 (Mpa) sln K⫽0.1995 (Mpa) The distribution parameters calculated for the used blades are: m⫽16.3322 mln K⫽118.3982 (Mpa) sln K⫽0.1168 (Mpa) Using 100 hours as one statistical unit, the fatigue damage accumulated in the blades during service time T is: T DC⫽T ⫻ Duni t⫽ (KC⫺KR) 3
(26)
Using the limit state function Eq. (16) and the iterative solution technique [33], the fatigue evaluation of the blades are shown in Tables 1 and 2. In Tables 1 and 2, the reliability index is defined by Eq. (18). If the reliability level is defined by technical requirements, the corresponding fatigue life can be calculated at the given reliability. Table 1 shows the calculation results for this application. For example, in Table 1, if the reliability requirement is 0.999 (0.0001 probability of failure is allowed), the blades which have operated for 221 hours should be replaced. It may also be required that the reliability level of the blades at a particular length of service time be known. Table 2 shows the calculation result in this situation. For example, in Table 2, after blades have been in service for 200 hours, calculations indicate that 4 out of 10 000 blades will have failed during the 200 hours in service (evaluated reliability level is 0.9996 and evaluated failure probability is 0.0004). The calculation results shown in Table 2 are reasonably close to the statistical results from in-situ testing of the blades [34].
5. Summary and discussions When the loading of the components is unknown, it is not possible to perform fatigue evaluation of the components without additional information. A method Table 1 The evaluated fatigue life at given reliability index Reliability requirement 0.9999 Failure probability 0.00001 requirement Reliability index 3.72 Evaluated fatigue life 172.5 (hours)
0.999
0.99
0.9
0.0001
0.001
0.1
3.09
2.33
1.282
221.4
306.6
511.6
X. Le, M.L. Peterson / International Journal of Fatigue 21 (1999) 603–610
609
Table 2 The evaluated reliability at given fatigue life Service time (hours) Evaluated reliability level Evaluated failure probability Reliability index
100 0.9999997 0.0000003 5.0
has been described which shows how to obtain information about the loading from the performance of used components in fatigue tests. A probabilistic fatigue model for this problem is developed and can be used to perform the fatigue evaluation of the component with unknown loading. After a component experiences cyclic loading, some amount of fatigue damage will accumulate in the component. Because fatigue damage of the component is permanent damage, it will gradually accumulate as the time in service increases. This means, that the components have lost some of their capability for resisting fatigue damage. According to the linear cumulative damage assumption, a certain fatigue damage accumulated in the component will be associated with the number of cycles (time in service) and applied loading. So the accumulated fatigue damage in the component will provide the additional information which is needed regarding the applied loading. Using components which have been removed from service after the same number of service hours, fatigue tests can be conducted which will provide the required test data. The test data can then be used to determine the residual fatigue damage resistance of the used components. Comparing the residual fatigue damage resistance of the used components with the fatigue damage resistance of the new components can indirectly express the fatigue damage accumulated in the used components during time in service. As a result, fatigue evaluation of components can be performed based on the model developed in this paper. In order to obtain the required accuracy, the used components must be carefully selected for an appropriate time in service. The used components must have been in service for at least one statistical unit in order for KR to be able to express the fatigue damage with sufficient accuracy. For example, the typical time in service must be more than 100 hours for components in an airplane engine, or one year for components of an industrial machine. If time in service for the components is too short, KR will not contain sufficient information with respect to actual loading conditions. For example, some loading conditions will not occur during the service interval if the time is too short. However, the service time cannot be so long that fatigue failures of the components have occurred. The suitable time in service for conducting fatigue tests on the used components will depend on the design and the working conditions. The required
200 0.9996 0.0004 3.945
300 0.9912 0.0088 2.375
400 0.9606 0.0396 1.752
500 0.9072 0.0828 1.325
time ion-service corresponds to a reliability level of 0.99, for the compressor blades example. The suitable time in service of the blade prior to conducting fatigue tests will be in the range of 100 to 306 hours based on the calculation results in Table 2. The potential of this approach is significant, however, important restrictions exist in the manner in which it may be applied as well as the type of application for which it is suitable. This model applies to situations where the loading can be assumed to be ergodic. In addition, a sufficient number of components must exist which can be removed from service after a sufficient service interval. If these criteria are met however, a new approach is possible for determining the fatigue resistance of an important type of service application.
Acknowledgements This work was supported in part by the RockwellAnderson Chair in the Department of Mechanical Engineering at Colorado State University. Additional support was provided by Storage Technology Corporation and Eastman Kodak Company.
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