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Liu.pdf
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 [8]. The strain-life
curve, originally proposed by Morrow [9] and well known as
Manson [10]–Coffin [11] 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 [12], 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 [13], 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 [14], 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
[15].
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 [16],
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
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[2]
[3]
[4]
[5]
[6]
5. Conclusions
Three models are proposed for the reliability estimation of
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complicated mechanical structures. In the proposed method,
the reliability evaluation of the LCFL is composed by two
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based on the finite element code, another is the reliability calculation based on three models. Three models are constructed
on different basis. Models 1 and 2 are based on the LAD law,
where the damage strength parameter is usually assumed as
1, and model 3 is based on the equivalent probability transformation. Keep the conformity in the results of three models,
we can identify the appropriate damage strength parameter.
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