# Fatigue Life Scatter Modeling and Failure Probability Prediction of Multisite Damage Structural Part under Variable Amplitude Load History

Authors: Bo Qin; Liyang Xie
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

An “index stress” (a constant amplitude cyclic stress) is introduced to characterize the scatter of the fatigue life under a variable amplitude load history. It is defined as a fatigue damage weighted average of the amplitudes of the individual stress cycles in a variable amplitude load history. The fatigue test results show that the scatter of fatigue life under variable amplitude load histories can be well characterized by such defined index stress. For probabilistic fatigue life prediction and failure probability estimation, a multiple-site damage structural part is taken as a series system, and the total probability theorem is applied to reflect the effect of load history uncertainty on failure probability.


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## Body

## 1. Introduction

For the various kinds of engineering objects such as structural parts and mechanical components, fatigue life is uncertain even under a deterministic load history. The scatter of fatigue life of a structural part under constant amplitude cyclic load is a function of stress amplitude. The scatter of fatigue life under variable amplitude load histories is jointly determined by many factors such as the stress amplitudes of the load cycles included in the load history and the respective number of the load cycles with different stress amplitudes. A structural part or a mechanical component commonly has several high stress zones where fatigue failure may occur. As a random variable, the fatigue life of a multisite damage (MSD) structural part differs from that of any of its damage site.Most of the investigations on MSD structural part fatigue life prediction were performed under deterministic framework [1–9]. Considering the large number of damage sites on a structural part, the assessment of lifetime characteristics was suggested to be performed under probabilistic framework [10–13]. Owing to the difficulty of developing theoretical models, the probabilistic failure behaviors of MSD structural parts were usually analyzed by means of numerical methods or Monte–Carlo simulation [12–18]. Especially, most of the investigations were regarding crack propagation behavior [1, 2, 7, 9, 10, 17–21]. Kim et al. [19] commented that the existing probabilistic methods coping with uncertain random variables seldom obtain efficient and reliable results.Stress-probabilistic life relationship-based methods were usually applied to predict probabilistic fatigue life, especially for single-damage-site components [22–30]. Such methods implicitly assume that the life scatter contributed by a certain stress cycle in a variable amplitude load history exclusively depends on its amplitude itself. For MSD structural parts, a system failure probability analysis method is necessary [12, 13, 31], as all the damage sites play their roles. Moreover, for system (e.g., MSD structural part) failure probability estimation, the potential dependence between elements (e.g., damage sites) is an important issue to consider. Especially, the stochastic dependence between element failures (i.e., the so called “common cause failure” resulted from the uncertainty of load history) may exist [31].Leonetti et al. considered the dependency of rivet hole failures in fatigue life estimation of shear riveted connections by considering the change of the state of stress in the connection when a ligament between two rivets fails [12, 13]. It stated that “unlike other system reliability models available in the literature, the evaluation of the probability of failure takes into account the stochastic dependence between the failures at each critical location” [12, 13], where the dependency between the failures at different critical locations was considered by updating the state of stress given that a failure had occurred. Obviously, the dependency is a type of physical dependence arisen from stress redistribution rather than the stochastic dependence (i.e., common cause failure effect) resulted from load uncertainty.Investigations on probabilistic fatigue life prediction methods under variable amplitude load histories are mostly object specific [32, 33]. In an investigation on fatigue reliability of reinforced concrete structural parts, Petryna et al. [32] illustrated the general inapplicability of local and linear fatigue models to system level of structures. Due to the increasing complexity and scale of modern engineering structures, it becomes increasingly necessary to propose an accurate and efficient approach for the assessment of uncertainties in material properties, geometric dimensions, and operating environments [10, 21, 34].Addressed to failure probability estimation and probabilistic fatigue life prediction of MSD structural parts, the present paper defines an “index stress” to characterize the scatter of fatigue life under variable amplitude load histories, develops an index stress-based fatigue failure probability prediction method, and applies the total probability theorem to incorporate the effect of failure dependence between damage sites.


## 2. MSD Structural Part Fatigue Failure Probability Model

Owing to the dispersion of material properties, the fatigue life of a damage site on a structural part is a random variable. Consequently, the fatigue life of an MSD structural part depends on the fatigue lives of all its damage sites. For most engineering structural parts, any damage site failure means structural part failure. Therefore, for failure probability estimation, an MSD structural part havingJ damage sites is a series system consisting of J elements. Let Rj(n) denote the reliability of damage site j (j = 1, 2, …, J) over n stress cycles, i.e., the probability that the fatigue life of the damage site j is greater than n, the MSD structural part (a series system) reliability is conventionally expressed as(1)RMSDn=∏j=1JRjn,and the failure probability of the MSD structural part is(2)PMSDn=1−∏j=1J1−Pjn,where Pj(n) = 1−Rj(n) is the failure probability of the jth damage site over n cycles of stress action.Equation (1) is the traditional series system reliability model. Such a model is valid only in the condition that the failures of the individual elements are independent of each other. In other words, equations (1) and (2) are only applicable to the situations of deterministic load; since only when load is deterministic, the failures of the different elements are mutually independent in statistical sense.Generally, the load history subjected to a structural part is uncertain. Load uncertainty results in the stochastic dependence (common cause failure) between the individual damage site failures. In this situation, the probability that two or more random events occur simultaneously does not equal to the product of the occurrence probabilities of the individual events.For an engineering structural part, potential load histories can be represented by a certain number of load history samples; each has a particular probability to realize. In the situation ofK possible load histories in total, if the structural part will subject to load history k (k = 1,2, …, K) with a probability pk, the relationship between the failure probability of an MSD structural part and that of the individual damage sites is [35, 36](3)PMSDn=∑k=1Kpk1−∏j=1J1−Pj,kn,where Pj,k(n) is the failure probability of damage site j under load history k.Equation (3) is a comprehensive series system failure probability model. It is an advanced form of the traditional series system failure probability model extended by virtue of the total probability theorem (Bayes theorem), by which the stochastic dependence between element failures is inherently incorporated.


## 3. Fatigue Life Scatter under Variable Amplitude Load Histories

Test results of fatigue life under constant amplitude load histories (CAL) of riveted lap joints illustrated the standard deviation of the log life (the logarithm of life cycles) under CAL is a function of the fatigue lifeN [37–39]. However, test results under variable amplitude load histories (VAL) illustrated that the highest amplitude of the load spectrum is responsible for the amount of scatter [37, 40, 41].Since fatigue life scatter is cyclic stress level dependent, an index stress (a constant amplitude cyclic stress) must exist in terms of fatigue life scatter. Based on fatigue test results, an index stress is defined below, under which the fatigue life scatter is the same as that under the variable amplitude stress history.


### 3.1. Experimental Results

To study fatigue life scatter under variable amplitude load histories, experimental tests were performed under programed cyclic loads with smooth specimens of T6-aluminum alloy [42]. The mechanical properties of the material are listed in Table 1. The stress histories applied in the experiments include three constant amplitude cyclic stress histories, a two-level cyclic stress history composed of 100 cycles of high stress s3, and subsequent cycles of low stress s1 (denoted by s3(100C)-s1, as shown in Figure 1(a)), a two-level cyclic stress history composed of 1,000 cycles of s3 and subsequent cycles of s1 (denoted by s3(1000C)-s1, as shown in Figure 1(b)), and a three-level program stress spectrum composed of load blocks consisting of “1000 cycles of s3 - 1000 cycles of s2 - 1000 cycles of s1” (denoted by s3-s2-s1…, as shown in Figure 1(c)).Table 1
Mechanical properties of T6-aluminum alloy (MPa).
Tensile strengthYield strengthFatigue limit260.4238.885.2Figure 1
Variable amplitude load histories. (a) Load historys3(100C)-s1. (b) Load history s3(1000C)-s1. (c) Load history s3-s2-s1.
(a)(b)(c)Test results are summarized in Tables2 and 3, where s stands for stress amplitude, N− means life, σ standard deviation of fatigue life, and c variation coefficient, i.e., the ratio of standard deviation to mean; N−logN, σlogN, and clogN stand for mean, standard deviation, and variation coefficient of the log life, respectively. The equations shown in Table 4 describes the relationships between mean/standard deviation of fatigue life and stress amplitude.Table 2
Fatigue life under constant amplitude cyclic stress.
s1 = 110 MPas2 = 120 MPas3 = 130 MPaN¯=1000104N¯=593274N¯=364320σ = 407158 c = 0.407σ = 167256 c = 0.282σ = 78484 c = 0.215N−logN=5.972,N−logN=5.756,N−logN=5.552,σlogN = 0.154, clogN = 0.026σlogN = 0.126, clogN = 0.022σlogN = 0.093, clogN = 0.017Table 3
Fatigue life under variable amplitude load history.
s3(100C)-s1s3(1000C)-s1s3-s2-s1…N¯=1243122N¯=1190072N¯=1190072σ = 467292 c = 0.376σ = 526229 c = 0.443σ = 139138 c = 0.262N−logN=6.068,N−logN=6.048,N−logN=5.711,σlogN = 0.158, clogN = 0.026σlogN = 0.146, clogN = 0.024σlogN = 0.111, clogN = 0.019Table 4
Relationships fitted with experimental results.
RelationshipRegression equationCyclic stress-mean lifes=1332.6×2N−−0.173Cyclic stress-mean log lifes=−47.6N−log+394.4Cyclic stress-std of log lifeσlogN=0.49−0.003sThe test results demonstrate that, for the load historys3-s2-s1…, the life scatter 0.262/0.019 (measured in variation coefficient of life or variation coefficient of log life) is between 0.282/0.022 (the life scatter measurement under the constant amplitude cyclic stress s2) and 0.215/0.017 (the life scatter measurement under s3). It is the same in terms of the standard deviation of the log life or that of the life (0.126/167256–0.111/139138–0.093/78484). Although the cycle number of the high-level cyclic stress s3 is quite large (one-third of the total cycle number) in the load history, the life scatter is not exclusively determined by the high-level cyclic stress, since it is considerably greater than that under the constant amplitude cyclic stress s3. As to the load history s3(100C)-s1 and s3(1000C)-s1, the effect of the limited number of high-level stress cycles on fatigue life scatter is very slight. These experimental results illustrate that the effect of the highest-level stress cycles on fatigue life scatter is not so great as described in literature [37–41].


### 3.2. Fatigue Life Scatter Characterization

To predict the mean or median fatigue life under a variable amplitude load history, the linear cumulative fatigue damage rule, i.e., Miner’s rule, is widely applied. By virtual of a cumulative fatigue damage rule, a variable amplitude load history can be transformed into a mean damage/life equivalent constant amplitude load history. For instance, according to Miner’s rule, the load history containingn1 cycles of stress s1 and n2 cycles of stress s2 is equivalent to the load history consisting of n1N−3/N−1+n2N−3/N−2 cycles of stress s3, where N−i (i = 1,2,3) stands for the mean life under cyclic stress si. However, mean/median life equivalence does not necessarily mean life scatter equivalence. For the purpose of probabilistic fatigue life prediction under a variable amplitude load history, another constant amplitude cyclic stress can be identified, under which the fatigue life scatter is the same as that under the variable amplitude load history. Such an “index stress” is defined below.Generally, a variable amplitude stress history contains many low-level stress cycles that produce minor or no fatigue damage. To avoid ineffective cumulative fatigue damage calculation, the low-level stress cycles in a variable amplitude load history are normally deleted in advance. For the variable amplitude load history used below, only the stress cycles with amplitudes greater than material fatigue limit are considered for cumulative fatigue damage calculation.The experiment results illustrate that the scatter of fatigue life under a variable amplitude load history depends on not only the amplitudes of the stress cycles but also the numbers of the stress cycles with different amplitudes. Therefore, a damage weighted average of the cyclic stress amplitudes is appropriate as the amplitude of the index stress. For the situation of a variable amplitude load history or a load block consisting ofm effective stress cycles denoted by si, s2, …, sm, respectively, the index stress sI is defined as(4)sI=∑i=1msisi−1/b∑j=1msj−1/b.The original expression is(5)sI=∑i=1msi2si/σf−1/b∑j=1m2sj/σf−1/b,where 2si/σf−1/b is the fatigue damage caused by one cycle of stress si, and ∑j=1m2si/σf−1/b is the total fatigue damages caused by the m stress cycles.For the load historys3(100C)-s1 used in the experiment, the two cyclic stress amplitudes are s3 = 130 MPa and s1 = 110 MPa, respectively. The number of the cycles of the cyclic stress s3 equals to 100, and the number of the cycles of the cyclic stress s1 is 1,243,022 (the mean residual life). By equation (4), sI = 110.0 MPa. Therefore, based on the index stress concept, the standard deviation of the log life under this load history is predicted as 0.49−0.003×110.0=0.160 (Ref. the cyclic stress - std of log-life equation listed in Table 4), that is close to the test result 0.158.For the load historys3(1000C)-s1 used in the experiment, the number of the cycles of stress s3 equals to 1000, and the number of the cycles of s1 is 1,190,072 (the mean residual life). By equation (4), sI = 110.0 MPa. Based on the index stress, the standard deviation of the log life is also predicted as 0.160, that is also not far from the test result 0.146.For the load historys3(1000C)-s2(1000C)-s1(1000C) used in the experiment, there are three cyclic stress levels as 130 MPa (s3), 120 MPa (s2), and 110 MPa (s1), respectively, and each has the same cycle number. By equation (4), sI = 123.1 MPa. Therefore, the standard deviation of the log life is predicted as 0.49−0.003×123.1=0.121, that is close to the test result 0.111.These results show that the relative errors of the log-life scatters (standard deviations) characterized by the index stress defined in equation (4) are less than 10%. In the following, such an index stress is applied to predict probabilistic fatigue life under variable amplitude load histories.


## 3.1. Experimental Results

To study fatigue life scatter under variable amplitude load histories, experimental tests were performed under programed cyclic loads with smooth specimens of T6-aluminum alloy [42]. The mechanical properties of the material are listed in Table 1. The stress histories applied in the experiments include three constant amplitude cyclic stress histories, a two-level cyclic stress history composed of 100 cycles of high stress s3, and subsequent cycles of low stress s1 (denoted by s3(100C)-s1, as shown in Figure 1(a)), a two-level cyclic stress history composed of 1,000 cycles of s3 and subsequent cycles of s1 (denoted by s3(1000C)-s1, as shown in Figure 1(b)), and a three-level program stress spectrum composed of load blocks consisting of “1000 cycles of s3 - 1000 cycles of s2 - 1000 cycles of s1” (denoted by s3-s2-s1…, as shown in Figure 1(c)).Table 1
Mechanical properties of T6-aluminum alloy (MPa).
Tensile strengthYield strengthFatigue limit260.4238.885.2Figure 1
Variable amplitude load histories. (a) Load historys3(100C)-s1. (b) Load history s3(1000C)-s1. (c) Load history s3-s2-s1.
(a)(b)(c)Test results are summarized in Tables2 and 3, where s stands for stress amplitude, N− means life, σ standard deviation of fatigue life, and c variation coefficient, i.e., the ratio of standard deviation to mean; N−logN, σlogN, and clogN stand for mean, standard deviation, and variation coefficient of the log life, respectively. The equations shown in Table 4 describes the relationships between mean/standard deviation of fatigue life and stress amplitude.Table 2
Fatigue life under constant amplitude cyclic stress.
s1 = 110 MPas2 = 120 MPas3 = 130 MPaN¯=1000104N¯=593274N¯=364320σ = 407158 c = 0.407σ = 167256 c = 0.282σ = 78484 c = 0.215N−logN=5.972,N−logN=5.756,N−logN=5.552,σlogN = 0.154, clogN = 0.026σlogN = 0.126, clogN = 0.022σlogN = 0.093, clogN = 0.017Table 3
Fatigue life under variable amplitude load history.
s3(100C)-s1s3(1000C)-s1s3-s2-s1…N¯=1243122N¯=1190072N¯=1190072σ = 467292 c = 0.376σ = 526229 c = 0.443σ = 139138 c = 0.262N−logN=6.068,N−logN=6.048,N−logN=5.711,σlogN = 0.158, clogN = 0.026σlogN = 0.146, clogN = 0.024σlogN = 0.111, clogN = 0.019Table 4
Relationships fitted with experimental results.
RelationshipRegression equationCyclic stress-mean lifes=1332.6×2N−−0.173Cyclic stress-mean log lifes=−47.6N−log+394.4Cyclic stress-std of log lifeσlogN=0.49−0.003sThe test results demonstrate that, for the load historys3-s2-s1…, the life scatter 0.262/0.019 (measured in variation coefficient of life or variation coefficient of log life) is between 0.282/0.022 (the life scatter measurement under the constant amplitude cyclic stress s2) and 0.215/0.017 (the life scatter measurement under s3). It is the same in terms of the standard deviation of the log life or that of the life (0.126/167256–0.111/139138–0.093/78484). Although the cycle number of the high-level cyclic stress s3 is quite large (one-third of the total cycle number) in the load history, the life scatter is not exclusively determined by the high-level cyclic stress, since it is considerably greater than that under the constant amplitude cyclic stress s3. As to the load history s3(100C)-s1 and s3(1000C)-s1, the effect of the limited number of high-level stress cycles on fatigue life scatter is very slight. These experimental results illustrate that the effect of the highest-level stress cycles on fatigue life scatter is not so great as described in literature [37–41].


## 3.2. Fatigue Life Scatter Characterization

To predict the mean or median fatigue life under a variable amplitude load history, the linear cumulative fatigue damage rule, i.e., Miner’s rule, is widely applied. By virtual of a cumulative fatigue damage rule, a variable amplitude load history can be transformed into a mean damage/life equivalent constant amplitude load history. For instance, according to Miner’s rule, the load history containingn1 cycles of stress s1 and n2 cycles of stress s2 is equivalent to the load history consisting of n1N−3/N−1+n2N−3/N−2 cycles of stress s3, where N−i (i = 1,2,3) stands for the mean life under cyclic stress si. However, mean/median life equivalence does not necessarily mean life scatter equivalence. For the purpose of probabilistic fatigue life prediction under a variable amplitude load history, another constant amplitude cyclic stress can be identified, under which the fatigue life scatter is the same as that under the variable amplitude load history. Such an “index stress” is defined below.Generally, a variable amplitude stress history contains many low-level stress cycles that produce minor or no fatigue damage. To avoid ineffective cumulative fatigue damage calculation, the low-level stress cycles in a variable amplitude load history are normally deleted in advance. For the variable amplitude load history used below, only the stress cycles with amplitudes greater than material fatigue limit are considered for cumulative fatigue damage calculation.The experiment results illustrate that the scatter of fatigue life under a variable amplitude load history depends on not only the amplitudes of the stress cycles but also the numbers of the stress cycles with different amplitudes. Therefore, a damage weighted average of the cyclic stress amplitudes is appropriate as the amplitude of the index stress. For the situation of a variable amplitude load history or a load block consisting ofm effective stress cycles denoted by si, s2, …, sm, respectively, the index stress sI is defined as(4)sI=∑i=1msisi−1/b∑j=1msj−1/b.The original expression is(5)sI=∑i=1msi2si/σf−1/b∑j=1m2sj/σf−1/b,where 2si/σf−1/b is the fatigue damage caused by one cycle of stress si, and ∑j=1m2si/σf−1/b is the total fatigue damages caused by the m stress cycles.For the load historys3(100C)-s1 used in the experiment, the two cyclic stress amplitudes are s3 = 130 MPa and s1 = 110 MPa, respectively. The number of the cycles of the cyclic stress s3 equals to 100, and the number of the cycles of the cyclic stress s1 is 1,243,022 (the mean residual life). By equation (4), sI = 110.0 MPa. Therefore, based on the index stress concept, the standard deviation of the log life under this load history is predicted as 0.49−0.003×110.0=0.160 (Ref. the cyclic stress - std of log-life equation listed in Table 4), that is close to the test result 0.158.For the load historys3(1000C)-s1 used in the experiment, the number of the cycles of stress s3 equals to 1000, and the number of the cycles of s1 is 1,190,072 (the mean residual life). By equation (4), sI = 110.0 MPa. Based on the index stress, the standard deviation of the log life is also predicted as 0.160, that is also not far from the test result 0.146.For the load historys3(1000C)-s2(1000C)-s1(1000C) used in the experiment, there are three cyclic stress levels as 130 MPa (s3), 120 MPa (s2), and 110 MPa (s1), respectively, and each has the same cycle number. By equation (4), sI = 123.1 MPa. Therefore, the standard deviation of the log life is predicted as 0.49−0.003×123.1=0.121, that is close to the test result 0.111.These results show that the relative errors of the log-life scatters (standard deviations) characterized by the index stress defined in equation (4) are less than 10%. In the following, such an index stress is applied to predict probabilistic fatigue life under variable amplitude load histories.


## 4. Fatigue Failure Probability Estimation of a Damage Site

The variable amplitude load history associated with the fatigue life prediction is composed by the load block, as shown in Figure2. It contains 669 peak/valley values; the maximum peak/valley stress is 140 MPa. Rain-flow counting shows that there are 334 load cycles. This load block will be applied to the damage site repeatedly until fatigue failure occurs. To predict fatigue life, the 334 stress cycles are transformed into damage equivalent fully reversed stress cycles through Gerber formula. After deleting the ineffective stress cycles of which the amplitudes of the fully reversed stress cycles are less than 85.2 MPa (material fatigue limit), the reserved 300 stress cycles are listed in Table 5.Figure 2
Variable amplitude load block.Table 5
Composition of the load block (amplitudes and cycle numbers).
No.StresssiCyclesni11405213823137141361513516134171321813129130110128111127312125413124614123515122216121317120718119419118520117421116222115423114624113625112726111527110428109329108103010753110663210513331049341031335102936101837100838998399874097124196542957439412449314459264691154790104889549889508775186552851To estimate fatigue failure probability, the index stress is obtained by equation (4) as sI = 114.9 MPa for the variable amplitude load block. According to the cyclic stress-mean log-life equation and the cyclic stress-log-life standard deviation equation, the mean log life under the index cyclic stress equals to 5.87, and the standard deviation under the index stress equals to 0.145. Therefore, the log-normal probability density function of the fatigue life under the index stress is(6)fn=10.145n2πe−1/2logn−5.87/0.1452.As mentioned above, the index stress is only a life scatter equivalent stress but not a mean life equivalent stress. Mean life equivalence means mean damage equivalence. Miner’s rule is applied to transform the variable amplitude load block containing 300 stress cycles into a damage equivalent constant amplitude load block containingne − 300 stress cycles with the amplitude of the index stress 114.9 MPa. At this stress level, the damage equivalent load cycle number ne − 300 is(7)ne−300=∑i=1300N¯sIN¯si=∑i=1300sIsi−1/0.173,where N¯sI is the mean fatigue life under cyclic stress sI, N¯si is the mean fatigue life under cyclic stress si (i = 1,2, ..., 300), and 0.173 is the fatigue strength exponent, as shown in Table 4.For the variable amplitude load block shown in Figure2, it turns out that ne − 300 = 217. It means that, in terms of mean damage equivalence, the 300 stress cycles in the variable amplitude load block is equivalent to 217 stress cycles of the constant amplitude cyclic stress with the amplitude of 114.9 MPa (the index stress). Therefore, the fatigue failure probability over 300 × m variable amplitude stress cycles is estimated as(8)PN<217×m=∫0217×m10.145n2πe−1/2logn−5.87/0.1452dn.


## 5. MSD Structural Part Fatigue Failure Probability Estimation

Analyzed below is a structural part with six high stress zones (damage sites); the load history applied to the structural part is the same as that described in Section4. The peak stresses at the six damage sites are 154, 154, 147, 140, 133, and 126 (MPa), respectively. The respective index stresses, calculated by equation (4), are 126.3, 126.3, 120.6, 114.9, 109.1, and 103.3 (MPa), respectively (listed in Table 6). As mentioned above, the 300 stress cycles in one load block is equivalent to 217 cycles of the index stress in sense of damage equivalence.Table 6
Peak stress and index stress at high stress zones.
Damage site123456Peak stress154154147140133126Index stress126.3126.3120.6114.9109.1103.3Under the action of the deterministic load history, the failures of the individual damage sites are stochastically independent of each other. Therefore, the failure probability of the structural part can be estimated by the multiplication of the failure probabilities of the individual damage sites (ref. equation (2)). The failure probabilities of the individual damage sites can be estimated by the procedure described in Section 4. The failure probability estimation results of the MSD part as well as the individual damage sites are shown in Figure 3.Figure 3
Fatigue failure probability number of load cycles.If the load history subjected to the structural part is uncertain, the failure probability of the MSD structural part cannot be simply calculated from the failure probabilities of the individual damage sites. When all the possible load histories are available, the MSD structural part failure probability can be estimated by equation (3).For the situation ofK possible load histories in total, if the probability that the structural part subjects to the kth load history equals to pk, the failure probability of the MSD structural part under the kth load history (denoted by PMSD,k) can be estimated by equation (2), and the MSD structural part failure probability under the uncertain loading environment equals to(9)PMSD=∑k=1KpkPMSD,k.The above analysis and estimation results show that the failure probability of an MSD structural part is considerably higher than that of its most dangerous damage site (i.e., the damage site with the highest stress level). The difference between the failure probability of an MSD structural part and that of its most dangerous damage site depends on the number of damage sites, and the differences between the stress levels of the damage sites and the life scatters of the damage sites as well. The difference increases not only with the increase of the number of damage sites on a structural part but also with the increase of the fatigue life scatters of the individual damage sites.


## 6. Conclusions

Under a variable amplitude load history, fatigue life scatter is jointly determined by all the stress cycles. To characterize the scatter of fatigue life under a variable amplitude load history, an index stress is defined as the average of the stress amplitudes of all the effective stress cycles in the load history, weighted by the relative values of the fatigue damages associated with the individual stress cycles. The applicability of such defined index stress is testified by fatigue experimental results of 6T-aluminum alloy. Under a variable amplitude load history, the failure probability of a damage site on a multisite damage structural part is predicted according to the fatigue life distribution under index stress and a mean/median damage equivalent cycle number with respect to the index stress.To predict the fatigue failure probability of a multisite damage structural part, a comprehensive series system reliability model is introduced in which the total probability theorem is incorporated to reflect the effect of failure dependence between damage sites (i.e., the effect of common cause failures) resulting from the uncertainty of load history.



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