Materials Science and Engineering A 395 (2005) 218–225 Reliability analysis for low cycle fatigue life of the aeronautical engine turbine disc structure under random environment C.L. Liua , Z.Z. Lua , Y.L. Xub , Z.F. Yuec,∗ a c School of Aeronautics, Northwestern Polytechnical University, Xian 710072, China b China Aviation Research Institute 608, Zhuzhou 412002, China Department of Engineering Mechanics, Northwestern Polytechnical University, Xian 710072, China Received 12 August 2004; received in revised form 9 December 2004; accepted 9 December 2004 Abstract The low cycle fatigue life (LCFL) of the aeronautical engine turbine disc structure is related to the stress–strain level of the disc applied by cyclic load and the life characteristic of the material. The randomness of the basic variables, such as applied load, working temperature, geometrical dimensions and material properties, has significant effect on the statistical properties of the stress and the strain of the disc structure. In most cases, due to the complicated relationship between the LCFL and the basic random variables, it is very difficult to derive the statistical properties of the LCFL analytically, and to analyze the reliability directly. By use of the finite element analysis as a numerical experiment tool, a simulation method is presented to obtain the probability density distributions of the stress level and the strain level at the dangerous points in the turbine disc structure. On the basis of the Linear Damage Accumulation (LDA) law and the simulated probability density distributions of the stress and the strain, two models are presented for the reliability of the LCFL, and the randomness in the life characteristic of the material is taken into consideration in the presented models. The load-life interference and the equivalent probability transformation are developed to construct the third reliability model for the LCFL of the turbine disc as well. The aeronautical engine turbine disc is then introduced to illustrate the feasibility of three reliability models. The calculation results show that the reliability results of models 1 and 2 are in good agreement, which are different from that of model 3. By keeping agreement in the reliability results of three models, we discover an alternative method to identify the damage strength parameter in the LDA law. © 2004 Elsevier B.V. All rights reserved. Keywords: Low cycle fatigue life; Reliability analysis; Load-life interference model; Turbine disc 1. Introduction The turbine disc is an important part of the aeronautical engine, it is subjected to highly hostile conditions. In order to reduce weight and improve working life without loss of reliability, an accurate algorithm for reliability of LCFL is worthwhile to establish, which is the main purpose of this contribution. We know that the construction of limit state equation and the statistical property of LCFL are prerequisite for the reliability estimation. However, for the complicated mechanical structure, such as the turbine disc, the relationship between the LCFL and the basic random variables is ∗ Corresponding author. E-mail address: [email protected] (Z.F. Yue). 0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.12.014 numerical, i.e., the limit state equation is implicit. At this case, the probability density distribution of the LCFL cannot be analytically derived by the statistical properties of the basic random variables, and the reliability analysis cannot be completed directly by the conventional method for the explicit limit state equation. There are many methods, such as Monte Carlo method [1,2], response surface method [3,4] and improved response surface method [5–7], to solve this problem. Since a single high-fidelity simulation need take much time to compute for the complicated mechanical structure, the computational effort of Monte Carlo method is unacceptable for the small failure probability calculation in engineering generally, which requires large amount of simulation. The precision of the response surface method, even that of the improved one, still remains questionable for the highly non- C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 linear implicit limit state equation, there is no guidance for the selection of sampling points in theory. From the view of statistics in the paper, the probability density distributions of the stress and the strain in the turbine disc are firstly obtained by numerical experiment based on the finite element code. Three models are then constructed for the reliability analysis of LCFL of the turbine disc structure. By the way, an alternative method for identification of the damage strength parameter in the LDA law is discovered in the analysis of the illustration. Following a brief description of the LCFL analysis in Section 2, three models for the reliability of the LCFL are elaborated, and finally the given turbine disc structure is employed to illustrate the feasibility and validity of the presented models. Some conclusions are drawn from the discussion of the calculation results. 219 Eq. (3) can be used to predict the fatigue life under single level of cyclic load. In the practical engineering problem, the structure is usually subjected to multiple levels of cyclic load. There are several criterions to calculate the fatigue life at this case, and the LDA law is one of the most popular criterions. For the sake of simplification, the LDA law in Eqs. (4) and (5) is selected to obtain the LCFL of the turbine disc applied by multiple levels of cyclic load. k ni =D Nfi (4) i=1 Nf = nt 1 = k D i=1 ni /Nfi (5) where k is the number of the levels, ni the actual cyclic number of the ith level, and Nfi the corresponding life of the ith level, and D the total damage, Nf the total fatigue life, nt = ki=1 ni . 2. Analysis of LCFL The strain-based approach to fatigue characterisation is popularly used to predict the LCFL because the engineering disciplines of strain-controlled fatigue has enhanced the understanding of fatigue processes . The strain-life curve, originally proposed by Morrow  and well known as Manson –Coffin  law, is expressed in the following form: σ ε = f (2Nf )b + εf (2Nf )c 2 E (1) where ε is the total strain range, Nf the fatigue life, σf the fatigue strength coefficient, εf the fatigue ductility coefficient, E the Young’s modulus, b the fatigue strength exponent of Basquin’s law , c the fatigue ductility exponent of Coffin’s law, n = b/c the cyclic hardening exponent. The estimation of these parameters could be found in , which gave the details of the relative references. By taking the effects of mean stress σ m and mean strain εm on the fatigue life into consideration , Eq. (1) can be rewritten as following: σf − σm ε = (2) (2Nf )b + (εf − εm )(2Nf )c 2 E In order to extend Eq. (2) to the multi-axial stress condition, the equivalent criterion of Von Mises stress and strain is adopted. σf − σme εe = (3) (2Nf )b + (εf − εme )(2Nf )c 2 E where εe is the Von Mises equivalent strain range, σ me the mean of Von Mises equivalent stress, εme the mean of Von Mises equivalent strain. In theory, σf and εf in Eq. (3) are different from those in Eqs. (1) and (2), and in engineering application, they are viewed as the same approximately . 3. Reliability models of the LCFL For the ith cyclic load level, εei , εmei and σ mei (i = 1, . . ., k) are used to denote the Von Mises equivalent strain range, the mean of Von Mises equivalent strain and the mean of Von Mises equivalent stress, respectively. Obviously, the statistical properties of εei , εmei and σ mei are determined by the basic random variables, such as the applied load, temperature, geometrical dimensions and material parameters, etc. We cannot obtain these statistical properties by the analytical methods due to the numerical relations among these parameters and the basic random variables in the complicated mechanical structure. Therefore, numerical simulation method is used to obtain these statistical properties based on the finite element code, and the precision is guaranteed by enough simulation samplings. Three reliability models are then presented as following. 3.1. Model 1: strength-damage interference For each level of the cyclic load, the corresponding LCFL, Nfi , can be calculated by Eq. (3). Nfi = Nfi (εei , εmei , σmei , σf , εf , b, c) (6) The damage Di introduced by ni , the actual cyclic number of the ith level, is expressed in Eq. (7), and the total damage D accumulated from each level is expressed in Eq. (8) by use of the LDA law. ni Di = (7) Nfi (εei , εmei , σmei , σf , εf , b, c) D= k i=1 Di = k i=1 ni Nfi (εei , εmei , σmei , σf , εf , b, c) (8) a is assumed as the damage strength parameter, which is usually taken as 1. Then the following limit state function is 220 C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 Table 1 Distribution information of the basic random variables Fig. 1. Finite element mesh of turbine disc. defined by the interference between the damage strength and the actual damage. g=a− k i=1 ni Nfi (εei , εmei , σmei , σf , εf , b, c) (9) k i=1 ni =0 Nfi (εei , εmei , σmei , σf , εf , b, c) Distribution form Mean Coefficient of variation T11 T12 T13 T14 T21 T22 T23 T24 ω1 ω2 ω3 ω4 ρ σf εf b c n1 n2 n3 n4 n5 Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Logarithm normal Logarithm normal Logarithm normal Logarithm normal Logarithm normal 407.58 (◦ C) 326.83 (◦ C) 387.16 (◦ C) 396.48 (◦ C) 305.70 (◦ C) 179.29 (◦ C) 273.96 (◦ C) 285.53 (◦ C) 600 (r s−1 ) 438 (r s−1 ) 581.4 (r s−1 ) 569.4 (r s−1 ) 6.636 × 103 (kg m−3 ) 1623.2 (MPa) 0.01336 −0.0768 −0.328 995 780 118 28646 23980 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.01 0.01 0.01 0.01 0.00063 0.05 0.05 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Where Tjk (j = 1, 2 and k = 1, 2, 3, 4) denotes the temperature of the jth dangerous point, which is determined by FE analysis, at the kth load case. The limit state equation is defined as: g=a− Basic random variable (10) The limit state equation g = 0 separates the total universe into safety part, g ≥ 0, and failure part, g < 0. The reliability is the probability of g ≥ 0, while the failure probability is that of g < 0. Since the statistical properties of the random variables in Eq. (10) can be obtained from the numerical experiments and the actual physics experiments, it is easy to calculate the failure probability and the reliability by the advanced first order and second moment (AFOSM) method. 3.2. Model 2: load-life interference We assume nt , the total cyclic number, as the load subjected to the turbine disc. nt = k ni (11) i=1 By use of the LDA law in Eq. (5), the total fatigue life, Nf , can be calculated. Then the limit state equation is proposed Table 2 Fitted distribution parameters of Von Mises equivalent strain and stress at No. 71 node Load case Von Mises equivalent strain Von Mises equivalent stress Figure Mean Standard variance Figure Mean Standard variance 1 2 3 4 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 3.84 × 10−3 2.14 × 10−3 3.45 × 10−3 3.62 × 10−3 8.27 × 10−5 4.09 × 10−5 7.79 × 10−5 7.79 × 10−5 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 868.69 502.52 787.47 821.08 16.23 8.56 14.54 16.22 Table 3 Fitted distribution parameters of Von Mises equivalent strain and stress at No. 4616 node Load case Von Mises equivalent strain Von Mises equivalent stress Figure Mean Standard variance Figure Mean Standard Variance 1 2 3 4 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 4.55 × 10−3 2.61 × 10−3 4.14 × 10−3 4.31 × 10−3 8.75 × 10−5 4.74 × 10−5 7.78 × 10−5 8.72 × 10−5 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 1077.63 628.52 983.83 1022.05 20.28 11.14 18.11 20.30 C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 by the load-life interference as following: g = Nf − nt = k nt i=1 ni /Nfi (εei , σmei , εmei , σf , εf , b, c) −nt = 0 (12) Analogously, the limit state equation g = 0 separates the total universe into safety part, g ≥ 0, and failure part, g < 0, and the AFOSM method can be applied to calculate the reliability and the failure probability of the limit state Eq. (12). 3.3. Model 3: recursive load-life interference Both models 1 and 2 are based on the criterion of the LDA law, while model 3, called recursive load-life interference, is founded on the equivalent probability transformation. The recursive load-life interference model was presented in , and it is developed to analyze the reliability of the LCFL in the paper. For the ith level of cyclic load, the fatigue life, Nfi , can be calculated by Eq. (3), and the ith limit state function, gi , can be constructed similarly as Eq. (12). Further, 221 the corresponding reliability, Ri , is obtained by the AFOSM method for the ith level. Based on the equal reliability, ni , the load of the ith level, is transferred to nie , the equivalent load for the (i + 1)th level. The details of the recursive load-life interference model are given step by step as following: (1) For the first level, the limit state function g1 and reliability R1 are established in Eqs. (13) and (14). g1 = Nf1 − n1 P(Nf1 (σf , εf , b, c, εe1 , σme1 , εme1 ) (13) > n1 ) = R1 (14) The cyclic number n1 is transferred to the first equivalent cyclic number, n1e , for the second level based on the equal reliability. By solving the following equations, we can obtain n1e . Then the cyclic number of the second level is converted to (n1e + n2 ). g1e = Nf2 − n1e (15) P(Nf2 (σf , εf , b, c, εe2 , σme2 , εme2 ) > n1e ) = R1 Fig. 2. Histograms and the fitted PDF curves of Von Mises equivalent strain at No. 71 node. (16) 222 C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 (2) For the second level, the limit state function g2 and reliability R2 are given in Eqs. (17) and (18). g2 = Nf2 − (n1e + n2 ) (17) P(Nf2 (σf , εf , b, c, εe2 , σme2 , εme2 ) > (n1e +n2 )) = R2 (18) The cyclic number (n1e + n2 ) is transferred to the second equivalent cyclic number, n2e , for the third level based on the equal reliability. By solving the following equations, we can obtain n2e . Then the cyclic number of the third level is converted to (n2e + n3 ). (3) Analogously, for the kth level, the limit state function gk and reliability Rk are given in the Eqs. (21) and (22). gk = Nfk − (n(k−1)e + nk ) (21) P(Nfk (σf , εf , b, c, εek , σmek , εmek ) > (n(k−1)e + nk )) = Rk (22) (4) Repeat the above steps until the last level of the cyclic load, we can calculate the final reliability. 4. Illustration 4.1. General requirement g2e = Nf3 − n2e (19) P(Nf3 (σf , εf , b, c, εe3 , σme3 , εme3 ) > n2e ) = R2 (20) The above reliability models are used to analyze the failure probability for the given turbine disc, its finite element mesh is shown in Fig. 1, and the required working life nt is 54519 cycles under the given multiple levels of cyclic load. Fig. 3. Histograms and the fitted PDF curves of Von Mises equivalent stress at No. 71 node. C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 4.2. Load cases and the distribution information of the basic random variables There are four load cases for the given turbine disc structure during the total working life, which are indicated by cases 1, 2, 3 and 4 in the following tables, where case 1 = take off, case 2 = maximum continue, case 3 = maximum cruise, case 4 = idle. Four load cases constitute five levels of cyclic loads, which are 0-take off-0, 0-maximum continue-0, idle-take off-idle, idle-maximum continue-idle, and cruise-maximum continuecruise, and they are denoted by levels 1, 2, 3, 4 and 5, respectively. In order to analyze the reliability, the probability density distributions of the stress and the strain must be obtained firstly, and they are determined by the basic random variables. In the paper, the working temperature T, rotating speed ω and the material density ρ are selected as the basic random variables, which have effect on the probability density distributions of the stress and the strain. The parameters, such as σf , εf , b and c in the strain-life curve of the material, are selected as basic random variables as well, their distribution 223 information can be obtained by the actual physics experiment, and it is assumed in the paper. Further, the cyclic number ni (i = 1, 2, 3, 4, 5) corresponding to the ith level are selected as basic random variables, and the distribution information can be obtained by statistics of the engineering measurement. Table 1 lists the distribution parameters of the selected basic random variables. 4.3. Simulation for the probability density distributions of the stress and the strain The finite element model is constructed as a numerical experiment tool for obtaining the probability density distribution of the stress and the strain. In FE analysis, the basic stress–strain behaviour of the material is taken from experiment. By sampling the basic random variables, we can calculate the samplings of the stress and the strain at the dangerous nodes by the standard finite element software, and Nastran is selected in the paper. Once we obtain the samplings of the stress and the strain, conventional statistical method is used to fit the empirical probability density function from the distribution histogram. Fig. 4. Histograms and the fitted PDF curves of Von Mises equivalent strain at No. 4616 node. 224 C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 Fig. 5. Histograms and the fitted PDF curves of Von Mises equivalent stress at No. 4616 node. The dangerous nodes of the turbine disc are at No. 71 node and No. 4616 node. Tables 2 and 3 list the fitted distribution parameters of these two dangerous nodes, their corresponding histograms and empirical probability density functions are shown in Figs. 2–5, where PDF is the abbreviation of probability density function. 4.4. Reliability analysis Three reliability models presented in Section 3 are used to analyze the reliability of the LCFL of the turbine disc structure. Two failure modes at the dangerous nodes on the turbine disc are enumerated, and they are considered as series in the reliability analysis of the turbine disc system because any failure of the two modes can make the turbine disc system fail. The failure probabilities of each mode and the system are given in Table 4. 4.5. Discussion of the results From Table 4, we conclude that the failure probability of the turbine disc system is determined by the failure mode 2, Table 4 Results of the failure probability No. 71 node (failure mode 1) No. 4616 node (failure mode 2) system Model 1 Model 2 Model 3 2.01 × 10−16 2.01 × 10−16 1.15 × 10−16 4.52 × 10−8 4.52 × 10−8 2.45 × 10−8 4.52 × 10−8 4.52 × 10−8 2.45 × 10−8 because the failure probability of the mode 1 is much smaller than that of the mode 2. The results of models 1 and 2 are the same, and they are different from that of model 3. The phenomenon is introduced by the different basis on which three models are constructed. Models 1 and 2 are based on the assumption of LDA law, where the damage strength parameter a = 1, but model 3 is based on the equivalent probability transformation. Fig. 6 gives the relation of the failure probability and the damage strength parameter a in models 1 and 2 for mode 2. When a = 1.15, the results of three models are in good agreement, which can be used as an alternative method to identify appropriate damage strength parameter in C.L. Liu et al. / Materials Science and Engineering A 395 (2005) 218–225 225 Compared with model 3, models 1 and 2 have smaller computation efforts. The disadvantage of models 1 and 2 is that an appropriate damage strength parameter must be selected on the basis of experiment. The turbine disc structure is used to illustrate the method. The results of the example show that the system reliability of the turbine disc is determined by one significant failure mode. Acknowledgements Fig. 6. Relation of failure probabilities and damage strength parameter at No. 4616 node. the LDA law. Models 1 and 2 are simpler than model 3, and the computation efforts of model 3 are bigger than those of models 1 and 2. The advantage of model 3 over models 1 and 2 is without selection of damage strength parameter a, which must be done by models 1 and 2. The selection of damage strength parameter a should be based on the experiment, although it is suggested to take 1 generally. Supports provided by Aviation Base Science Foundation (00B53010), Aerospace Science Foundation (N3CH0502) and Shanxi Province Natural Science Foundation (N3CS0501) are gratefully appreciated. References       5. Conclusions Three models are proposed for the reliability estimation of the LCFL, and they have wide-ranging potential values for the complicated mechanical structures. 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