CC BY
70
ВЕСТНИК ПНИПУ
2014 Механика № 1
UDC 539.3
O.O. Larin, O.I. Trubayev, O.O. Vodka
National Technical University «Kharkiv Polytechnic Institute»,
Department of Dynamics and Strength of Machines, Kharkiv, Ukraine
THE FATIGUE LIFE-TIME PROPAGATION OF THE CONNECTION ELEMENTS OF LONG-TERM OPERATED HYDRO TURBINES CONSIDERING MATERIAL DEGRADATION
The work deals with development of a new approach for forecasting a high-cycle fatigue lifetime of bolted connection of hydro turbines runner. Operation of hydro turbines on normal operation condition does not lead to high stresses rates in bolted connection. However the high cycle fatigue failures have been occurred. High rates stresses occur in bolted connection in transient (start/stop) regimes of hydro turbines operation. The frequency of transient regimes occurrence depends from many factors and defined in this paper as a random function of time. Long-time bolted connection operation lead to natural degradation of material (aging). The degradation process is also a random process of time. So, this work pays attention to developing stochastic mathematical model of damage accumulation that take into account stochastic nature of degradation process and frequency of transient regimes occurrence. Application of the developed models is shown on real engineering example.
Degradation of properties has been modeled as a process of the reduction of fatigue (endurance) limit in time. Kinetics of damage accumulation is introduced in the context of the effective stress concept. Mathematical expectation, correlation function and the continuum damage parameter variance have been obtained as functions of time. Analysis of the influence of natural aging process on statistical parameters of damage accumulation as well as on the life-time has been carried out. The stress-strain state of bolted connection is determined by finite element method.
Keywords: fatigue, life-time, bolted connection, hydro turbine, material degradation, aging, damage models, probability approach, transient regimes, finite element method.
Introduction
Reliability analysis for many types of engineering structures is performed in accordance with the parameters of static and dynamic stress-strain state which occur under nominal operating conditions. The design is performed considering safety factors which provide high reliability for these operating regimes. Nevertheless, fatigue failures often occur in the various elements of different machines. The practical and theoretical issues of the fatigue strength prediction for real mechanical systems are still relevant in spite of the detailed description of the history of occurrence and investigation of the causes of fatigue failures for different machines and mechanisms represented in numerous sources (for example, [1-4]). Among all the mechanisms experiencing fatigue strength problems, structural elements of
hydro turbines represent a significant part. Depending on the specifics of construction of the rotor and the flow part, certain elements of a turbine are susceptible to fatigue.
A large number of fatigue failures, detected in various components of hydro turbines demonstrate that the models of reliability prediction of such elements are not accurate enough. It is caused by considerable complexity of the processes in hydro turbines during their considerable service life. The present work studies the elements which are most of all susceptible to fatigue in the Kaplan turbines: elements of the blades rotating mechanism [56], blades [7], drive elements [8] and the rotor on the whole [9].
An important issue is represented by fatigue strength of bolted connections, as its failures may lead to the failure of the whole structure. There are a lot of papers on the problems of analysis, design and use of threaded joints [10] (this paper provides references to more than 700 papers published between 1990 and 2002), as well as design recommendations of bolted joints are widely used [11-12], the issues of strength analysis and reliability of bolted connections forecasting remains actual [13]. Calculation of fatigue strength of rotor wheel bolted connection has a range of peculiarities such as the presence of corrosive environment [14-15]. Therefore, evaluation of reliability of bolted rotor wheel is an important engineering problem.
Despite the large number of papers on the fatigue strength of elements of hydro turbines [5-13] most approaches to the analysis of reliability and life-time consider the elements of construction as variable or random loading amplitude with fixed frequency or random fatigue endurance limit [16]. However, this leads to omission of consideration of a large class of problems of reliability theory, where the life-time is determined by the work on the transient regimes. Naturally, the life-time depends on the frequency of transient regimes, which is determined by the operation conditions, and is therefore subject to changes. So, the frequency of transient regimes occurrence is of random nature. Thus, we can distinguish a number of cases where the amplitude of loading is a deterministic value, but the frequency of transient regime occurrence is a random process.
In practice, operating of hydro turbines units in the CIS countries led to a number of failures of bolted connections and pin joints of the turbines, however their nominal (design) life-time has not got worked out [17]. The result of metallographic appraisal has been shown that the type of fracture crack corresponds to the high cycle fatigue (Fig. 1).
Fig. 1. Hydro turbine driven wheal bolts (M 110x4) after 30 years of operation at Dnipro Hydroelectric Station
Due to the relatively low incidence of transient regimes as well as high values of safety factor, which is built into at the design stage, the lifetime of such engineering structures becomes comparable with the time of natural degradation (aging) processes in materials. Natural aging is a set of irreversible microstructural, physical and chemical changes in the material. The study of these processes is performed experimentally. The information is limited and the obtained data has a considerable variance. The latter determines the necessity of probabilistic models of life-time prediction, which taking into account degradation of mechanical properties of materials.
Thus, the actual task is to develop models and approaches to the reliability analysis and forecasting the life-time of runner bolted connection with a significant design life-time in the stochastic way, considering the strength degradation of the material and random frequency of transient regimes. The object of study is the bolted connection of the runner of the hydroturbine PL40-V-700 (Dnipro Hydroelectric Station, the Dnipro River, Ukraine). The bolts are made from material closed to 45 steel (EN steel name is C45).
1. Loading characteristics
Nominal operating regime corresponds to a low rate loading of hydro turbines elements. Hydro turbines have a significant advantage in comparison with other sources of power generation: the short period of time for
launching and connecting to electricity network. This period of time is about a minute. Therefore, hydroelectric units (HU) are regularly used to manage the daily electricity consumption peaks. This increases the number of starts / stops of HU, making the frequency of occurrence of transient regimes be a random process.
Statistics of PJSC "Ukrhydroenergo" on the number of HU launches per month for different machines of the Dnipro cascade is shown in Fig. 2.
50
40
a 30
20
I
c
10-
1 I j r
V * / \T 1 1, ^ / ► \
__ \X '< / - -"-1 -
V'' \ '*■ ) r \A < \ w v« /: •
> '"•-A"'' \ ..A > » ►/ / J • i r
mean value
5 7
t, month
11
Fig. 2. The graph of dependence of the frequency of occurrence of transient regimes on time statistics PJSC "Ukrhydroenergo"
In the current investigation we introduce the hypothesis of the station-arity of the random process of the incidence of such regimes. This means that probabilistic characteristics of the process do not depend on the starting time (on the beginning of observation). In accordance with this assumption, the probabilistic characteristics of the process can be determined through statistical data processing, namely the average frequency of starts per month (ro), their variance Var[(£>], and coefficient of variation Vra. Numerical values are given in Table 1.
1
(œ)=-Zœ '
n
i=1
1
Var [œ] = --(œ)) ,
n i=i
(1)
V (3)
CO Tr r 1 ' ^ '
Var[o>]
where n is the number of frequency in the dataset, ro, - i-th frequency.
In these formulae we introduce notation <...> - for the mathematical expectation operator and Var [...] for the operator of taking variance, V- for the coefficient of variation.
Table 1
Values of statistical parameters of the function ro(t)
< ra> Var[a] Vm
32,67 158,97 0,38
Another important characteristic of HU starts frequency as a random process is a correlation function. Its reliable determination requires much more statistical data, which results in considerable difficulties. Therefore, in this paper we propose to a priori postulate the form of the correlation function, and only define its parameters.
An exponential law [18-19] is proposed as such an approximation with variance (Var[ro]) and intensity (X^) of the incidence of the dangerous regime as its parameters
Ka(ti,t2)=Var[w]-exp(-V\t2-1,|). (4)
Exponential correlation function of ro(t) random process has a finite correlation period; it determines the time frame within which occurrences of the dangerous regimes are set as statistically independent. For example, operation of hydro turbines in Dnipro cascade (Ukraine) has number of secondary factors (e.g. weather around the country and in neighboring lands), which leads to a unique situation from month to month according to the expert judgment. So, representative period of time can be set as 1 month. Sometimes, the stricter dependences in operation of such system can be found: seasonal or yearly interactions.
So, the correlation function in the exponential form shows that the frequency of occurrence of the dangerous regimes is almost the same in close periods of time; but this trend fades with the time, and the velocity of this process is determined by the correlation function intensity parameter (Xro). The intensity parameter is determined on the basis of the correlation
time (rmk), i.e. the period of time necessary for the expected statistical influence of frequency of a dangerous regime within time t1 on the frequency of this mode within time t2 to vanish. So, the correlation time is a system memory concerning its service life that can be calculated using the following equation [18-19]:
Trak fKra(TYT , T =
Var[ra]^
t2 ^,
K =
/l^rak
(5)
(6)
According to the presented example of the expert judgment about the hydro turbines in Dnipro cascade operational conditions the correlation time for the frequency of occurrence of the dangerous regimes can be set to 1, 3, 6 or 12 months.
2. Determination of loads affecting the bolted connection
Assessment of the nature of the transient process can be carried out by using the results of full-scale tests. Such tests have been performed on similar to HU in Dnipro Hydroelectric Power Plant on Chardara Dam (Kaplans turbines PL-661-VB-500, the Syr Darya river, Kazakhstan) [20-21], during which a number of parameters were measured, including the axial force and torque on the turbine shaft. Axial force and torque on shaft of HU on Chardara Dam can be scaled to HU of Dnipro power plant:
MD (t ) =
H
D C DD ^
H
Ch
D
Ch
MCh (t ),
(7)
PD (t )=
ax \ y
H
D C DD ^
H
Ch
D
Ch
pcC (t ),
(8)
where MCh(t) is torque in the hydraulic unit of Chardara Dam; MD(t) -torque in the hydraulic unit Dnipro Hydroelectric Power Plant; PaxCh(t) - the axial force in the hydraulic unit ofthe Chardara Dam; PaxD(t) - the axial force in the hydraulic unit of the Dnipro Hydroelectric Power Plant; HCh-head at the Chardara Dam; HD- head at the Dnipro Hydroelectric Power Plant; DCh-diameter of the runner at the Chardara Dam; DP -diameter of the runner at the Dnipro Hydroelectric Power Plant.
Fig. 3. Graph of dependency of torque MD(t) and axial force PaxD (t) on the time of operation of the hydraulic unit
The graph of the obtained dependences of axial force and torque on the shaft of the hydraulic turbine PL40-V-700 is shown in Fig. 3.
3. Study of stress-strain state of bolted connection.
These characteristics of the loading (fig. 3) allow estimating the bolted connection stress-strain state. Loading frequencies that are relatively low compared to rotor eigenfrequencies allow to solve problem of investigating stress-strain state in the quasistatic formulation. Since the axial force and torque on the whole rotor shaft can be considered constant, it is possible to select a piece of the flange of the rotor and runner that correspond to one bolt (Fig. 4, a) and take into account the cyclic symmetry with respect to the rotation axis (Fig. 4, 6).The finite element (FE) model has been built considering the bolt threads. The thread profile has been built according to GOST 9150-81, GOST 8742-81 and GOST 24705-81, however, with the purpose of simplifying the application of the finite element mesh, the threads are formed as a rotation body without regard to helicity (Fig. 4, c).
Boundary conditions are implemented as follows: the upper edge of the shaft is clamped, and the lower edge is exposed to application of force equivalent to the axial force and torque. It should also be noted that the contact interaction between the flange of the shaft, the runner and a bolt is considered in the FE model. For finite element analysis the existing software is used.
As a boundary condition simulating the tightness of bolts, the initial offset (tightness) of the head of the bolt relative to the support surface by the amount A is set.
a b c
Fig. 4. FE models: a - sector of shaft with one bolt; b - sector of shaft, with regard to the cyclic symmetry; c - bolt
Fig. 5 shows the distribution of displacement and stress on the bolt at the moment of time t = 90 seconds with the initial preload A = 0,15 mm, which corresponds to the tightening elongation 5 = 0,106 mm. There are two zones of stress s concentration the first one is a thread and the second one is a smaller radius fillet under the bolt head. Absolute maximum of von Misses equivalent stress has always been observed on the first turn of the thread in the calculations.
.7S7E-03
. :: .818E-D3 .833E-C3 ■849E-33 .864E-03
a b
Fig. 5. The stress-stain state that occurs in the bolt at the moment of time t = 90 s: a - von Misses equivalent stress, Pa; b - summary displacement, m
Fig. 6 shows the results of the calculation of stress in the bolted connection at the point where the maximum stresses occur (first thread).
340320300280260240220 200180
launching working stopping
-/A--
50
100 14500
t, s
14525 14550
Fig. 6. A graph of the dependence of von Misses equivalent stress on the hydraulic unit operation time
4. Investigation of hydraulic turbine bolted connection fatigue life-time with respect to peculiarities of operation
Obtained date about operating parameters of HU and stress-strain state of bolted connection allow assessing the fatigue life-time. The life-time is determined by the non-localized damage accumulation rate. A lot of different models of the fatigue damage accumulation exist, and readers can find some of them, for example, in the surveys [22-24]. In this paper an approach of continuum damage mechanics [25, 26] is used. So, it is assumed that cyclic loading leads to the accumulation of the damage D which is introduced in the sense of the Rabotnov-Kachanov theory [27] in the following way:
D = A0-A, A
(9)
where A is an effective resisting area, A0 is initial undamaged area. The effective area A is the area without damaged part, i.e. it could be obtained from Ao by removing the total area of accumulated micro-defects, microcracks, cavities etc. Such definition is used as a hypothesis of the damage isotropy, which is associated with the decrease in the cross-section effective area in the vicinity of the point of the body, and is set by a scalar function of
time t, stress amplitudes ca, a number of loading cycles N, mechanical properties, such as a stress endurance limit oe etc.
D=D(a a, N, a e, t, ...), (10)
The process of damage accumulation is described by the kinetic equation
dD=f (a a, t, ...). (11)
dt
It was assumed that the stress amplitudes were presented as effective stresses within the framework of the theory of continuum damage mechanics [13, 14, 18]
a a . (12)
a 1-D
The presented relation limits the damage parameter within 0 < D < 1.
Let kinetics of damage accumulation process be according to power law [25, 27, 30]
dtD-B^-B^J. (13)
where B and c are material damage parameters, which should be determined from experiments and generally depend on fatigue properties of materials: stress endurance limit oe and etc. So, identification of these constants is carried out on the basis of the simple fatigue test with fixed frequency of loading and constant stress amplitude. In this case one will obtain
1-[1-D]C+1 )=(c+1)Ba CJ (14)
At the time of failure (fracture) Tr damage is equal to unity (by its definition, see (9) and (13)), so
Tr = B(-rV (15)
B(c+1) a a
However, for such simple case of loading fatigue process should satisfy the Wohler (S-N) curve [24]
a maN=amNo = const, (16)
where N is the number of load cycles to failure at the ca stresses amplitude, m is the Wohler (S-N) curve parameter, ce - stress endurance limit, No is the number of stress cycles with respect to the endurance limit.
The time to failure in the simple fatigue test (with fixed load cycle parameters) is found from the (S-N) curve
N amN 1
T = NTc = (17)
l l -a
Tsc is the period of stress cycle, ro is the frequency of the loading. The comparison of the equations (15) and (17) gives us the possibility to express B and c form the standard Wohler (S-N) curve parameters, which have physical sense
c=m, B = L . (18)
No --m (m+1)
According to equations (18) and (13) the process of damage accumulation could be described by the following equation
--^. (19)
dt ^ 1-D) N0--me (m+1)
The model of accumulation of fatigue damage is close to the one described in the paper [30] and to Lemaitre-Chaboche model [31] and might be considered as its special case for isotropic damage that is generated due to the high-cycle fatigue i.e. without plastic strains.
In bolted connection, a quasi-static loading cycle with a fixed amplitude and mean stress level, but with a random frequency is observed. (Fig. 6). In this case, the process of accumulation of fatigue damage can be described by the following equation
-D=i-^Y ) ^, (20)
dt ^ 1-D) N0-m (m+1)
where ro(t) is a stationary random process, whose implementation is shown in Fig. 2, and the probabilistic characteristics are summarized in Table 1. We shall solve this equation in quadratures (21) and introduce a change of variables for further transformations (22)
D(t) _ m t
(m+1) J (1-D(t))dD(t)=tt-^KX. (21)
0 N 0Te 0
D (t)
U(t)=(m+1) J (1-D(t))mdD(t)=1-[1-D(t)]m+1. (22)
0
The mean value of the function U(t) can be easily found by directly averaging expression with regard to the hypothesis of HU launch frequency stationarity will be a linear function of time
U(t))=vRt. (23)
Variance of possible values of the function U(t) is determined by its dispersion and can be found from its correlation function
Var[U(t)] = Ku ( = t,t2 = t), (24)
which by definition is expressed through the second initial moment
Ku (t„t2)=U( )U(t2 ))-(U(t, ))-(U(t2). (25)
Taking into account (23) and (24), we obtain
/t1 t2 \ t1 t2 U(t)-U(t2))=yH \ra(t')dt'-Jra(t")dt"W2J^w(t/)-w(t")dt'df. (26)
\0 0 /00
The second initial moment of the random frequency is expressed through its correlation function and the square of the mean value equally to (25), i.e.
kkra (t1 ,t2 ) = («( )«( H®)2 (27)
whence we obtain
Uft )-U(t2 )) =V2 JJ[Kra (t',t'')]dfdt"+V2 •(©)2 t\t2. (28)
0 0
By substituting (28) into (25) and considering (23), we obtain an integral expression for the correlation function U(t)
Ku (t„t2)=V2 JJJfKra(t/,t'')dt'dt''. (29)
0 0
It is possible to obtain an analytical expression for the correlation function of the random process U(t) considering the expression for the correlation function of frequency, which is postulated in this paper as the exponential law (4). In this case, the dispersion of the function U(t) is calculated from its correlation function according to expression (24) and will have the next form
The function U(t) for each moment of time uniquely corresponds to the process of accumulation of damage in accordance with (21). The behavior assessment for damage and the function U(t) for moments of time close to the average life-time are of the greatest interest. For the considered practical problem the fatigue life of bolted connections is estimated at decades, which is significantly greater than the expected frequency of HU launches correlation time (which is estimated at several months). On the basis of these arguments and neglecting the value of Xt as compared with unity, expression (30) can be simplified
Due to additivity of the accumulation of the process U(t) one can assume that it fulfils the conditions of the central limit theorem [32], starting from a certain time value. Then the function of probability density distribution of U(t) has the form of a normal Gauss distribution with the characteristics (23) and (31).
The integral representation of the process U(t), which is given by the formula (22) shows that the process U in each moment of time is combined by the sum of the other random variables presented by the load. Therefore, if we consider the period of time near the life-time of the design, then it can be argued that the process U is the sum of a large number of uncorrelated terms with limited values of statistical moments. Then, the probability density function of U(t) has the form of a normal Gaussian distribution with the characteristics (23) and (31).
Using the relationship between the process U(t) and damage (22), the damage probability distribution density can be written as due to [32]
(30)
Var[U (t )] = 2Var[W] t.
(31)
fD D) = fu (U,t)
dU
(m+1)(1-D)m
fD ( D,t ) = , = exp
D ^2nVar[U (t )]
dD
1-(1-D )m - U (t )) ' Var[U (t )]
(32)
(33)
Then the expectation and variance of damage are determined from (30) (D(t)) = Jd • fD (D,t )dD
Var[ D(t)] = J[D-{D(t)) ]2 fD (D,t )dD
(34)
To determine the spread of damage we are using the confidence level a, for which we determine the confidence interval:
1-a Slri)
f fD(D,t)dD, 2 -L
1+a f fD (D,t)dD,
2
(35)
(36)
^r ( t )
where a = 0,9973, which corresponds to the area under the normal law which is obtained by the three-sigma rule; Si(t) and Sr(t) are the left and right borders of the spread of damage respectively (confidence interval) (Fig. 7).
1,0 0,8 0,6 0,4 0,2 0,0
I Confidence interval at xk = Confidence interval at xk = 1
12
-<D( t)>
H
0
2000 4000 6000 8000 10000 t, month
Fig. 7. Dependence of damage on time
Thus there were obtained probabilistic characteristics of the accumulation of fatigue damage during operation. Using these results it is possible to estimate the probability characteristics of the life-time as a random variable.
Identification of the non-failure operation probability (often called the reliability function) can be defined as a probability of double inequality relatively to the damage parameter, i.e. as a probability, which has a damage parameter as positive and less than unity [32-34]
P(t)=Pr[D(t)e[0;1]]. (37)
P(t) is the probability of non-failure system operation (reliability function). Pr[-] is an operator of event probability.
Such a probability will be a function of time due to the dependence of the damage parameter on time, so it can be calculated via the probability density function (PDF) of the damage [34, 35] in the following way
i
P(t)=\fD(D,t)dD . (38)
0
A PDF of a life-time (q(t))can be defined as a derivative of the probability of failure Q(t). The failure and non-failure are complementary events, so the probability of failure could be obtained from the reliability function in a simple way that gives a possibility to identify PDF, mean value (< Tr >) and variance (Var[Tr]) of life-time [32-35]
q (t)=dQ(t )=-dtP(t), (39)
dt dt
(Tr)=Jtq(t)dt, Var[Tr ]=J(t-(T^j)2 q(t)dt. (40)
00
Research of the influence of exploitation parameters on life-time of bolted connections poses certain interest. Thus Fig. 8 shows the dependence of the life-time mean on the mean frequency of launches of the HU. Fig. 9 demonstrates the dependence of variance of the life-time on the correlation time. The graph of life-time probability density depending on the time to correlation is shown in Fig. 10.
Fig. 8. Dependence of the mean life-time on the mean frequency of occurrence of transient regimes <©>
1,4x10
1,2x10
1,0x10" c 8,0x104
i6,0xl0
S 4,0x10
2,0x10
-4Var[(u] -lVarfco] l/4Var[a>
i
9 10 11 12
, months
Fig. 9. Dependence of the life-time dispersion on the correlation time
Fig. 10. The life-time probability densities depending on the correlation time
It should be noted that the obtained results are significantly higher in comparison with the available data on failures. In practice, the fatigue failure of a bolted connection occurred after 40 years of operation, which is more than 15 times less than the obtained estimate. This situation is explained by the fact that it is necessary to consider the effect of degradation (aging) of the material, since the structure life-time estimate is comparable to the time of irreversible physical and chemical degradation changes in the material.
5. Modeling of degradation as a gradual reduction of fatigue endurance
There are literature works focusing on the experimental study of the natural aging processes of different materials [36-41].
Papers [38-41] are focus on the study of aging of metals and alloys. These papers determined that the natural aging of the metal has little effect on its static strength and elastic characteristics, but it significantly changes the long-term strength (fatigue resistance). For example, the results obtained in [38] for steel 45 showed that natural aging of the material in 50 years leads to changes in static strength characteristics only within the limit of 5 %, but decreases the fatigue limit by 44 %. Moreover, the article shows that the scatter of the experimental data fields for non-degraded and degraded samples do not overlap. Further detailed analysis of fracture samples allowed the authors to identify the fundamental structural changes in the processes of nucleation of fatigue cracks for the degraded material. Similar results were obtained in [39] for steel 20. In addition, it was determined that, 15-year-long aging reduces the fatigue limit by 38 %.
Summarizing the data on degradation processes presented in the literature, this article proposes to consider the process of reducing the fatigue endurance limit according to the hyperbolic dependence having the following form:
Ge(t)=Ge-9(f); 9(0=
ft--*
P2 +P3 t
p
(41)
where p, andp are parameters and the power of approximation, o*e is the fatigue limit for the non-degraded material.
The power of approximation p determines the character of fatigue limit fall. For metals and alloys in the initial moments there is no decrease in characteristics observed, and when a critical time value is reached, degradation gains in intensity. The corresponding approximation can be obtained by increasing the order p. It should be noted that with an increase in the order of approximation, there is a significant variation observed for the values of the quantities pi, which can produce a negative effect on the numerical procedures when using expression (37) for solving practical problems. The results given in [38] were used as the initial data. Fig. 11 shows the dependence of the degradation function on time with p = 2.
It is necessary to mention the necessity of considering the scatter in the values of the fatigue endurance limit. It is obvious that during the material aging process, this uncertainty remains or even increases. Therefore, we assume that the fatigue limit is the product of the normalized to unity function of time approximating the degradation process 9 (t) by the value of the fatigue limit at the initial time which is a random variable, which is assumed
(as recommended in [42-43] ) as following the log-normal law of probability density distribution:
f K)
1
ae,v2n
exp
f ln (oe )-i 2 ^
V 4is y
(42)
where s and ^ are distribution parameters, determined on the basis of the values of the expectation and variance or coefficient of the fatigue limit variation.
exp
,2\
|l + -
Var[a*e ]=[exp(s2 2, V
^Var[g*e ] : fa)
exp(s2 )-1
(43)
Fig. 11. Graph of the 9(0 function (the approximation has been built for the steel 45 characteristics according to data given in [38])
Thus, changing the fatigue limit is a random non-stationary process.
One-dimensional probability density of the process appears as a linear func-
*
tional transformation (32) of the random variable a e and has the form:
f (<O e,t ) =-T2= exp
ln(Ge )-||-ln9(t)'
V2s
(44)
It is reasonable to normalize the fatigue limit by the mathematical expectation. For this purpose we shall introduce a change of variables
=(<) %
(45)
where x is a random variable following the log-normal distribution, and has unit mean and variance equal to the coefficient of fatigue endurance variation in the non-degraded state
(%)=1, Var[x]=V2. (46)
6. Determination of probability characteristics of damage
The natural aging of the material affects their mechanical properties, especially fatigue resistance [38, 39, 44]. Thus, the kinetics of the damage accumulation for the analyzed class of problem in this paper is described by the equation (19), where the right hand side depends on the two time processes: frequency of the dangerous regimes occurrence and the process of the endurance stress limit reduction. It is also should be claimed that the real operation of the engineering designs, which can be considered as the mechanical systems investigated in the paper, leads to randomness of the dangerous regimes occurrence; similarly, the process of the natural material aging has some uncertainty due to dependence of this process on a large number of various secondary factors. So, (14) it could be represented in the following way
d / \ ( t ^m ra(t) „ ,
—D(t )= ----V---, (47)
dt W ^ 1-D(t)) N0 tm(t)-(m+1)
where ro(t) is a variable in time random frequency of the loading, oe(t) is the fatigue (endurance) limit set as a randomly decreasing function of time and all other parameters are known as deterministic values.
In the paper ro(t) and oe(t) are assumed to be statistically uncorrelated random functions of time. Thus, the aging process does not influence the frequency of occurrence of the dangerous regimes directly and vice versa. Our point here is that occurrence of the dangerous regimes during all the life-time of the system is defined only by the operational conditions. On the other hand, the natural aging in the current paper is described as a separate self-contained process of physical-chemical transformations in a material
that are not caused by the mechanic load. Mechanical loading applied to a system leads to accumulation of the continuum damage, which reduces material stiffness [25, 26] without influencing the strength characteristics. The natural aging on the contrary directly affects the strength characteristics, especially the fatigue strength, as it was experimentally shown by the other authors [38, 39, 44]. The theory presented in the paper is also limited in its application to the isothermal mechanical systems operating in the room temperature. In addition, it should be noted that these processes have different time scales.
The research of the accumulation process of damage considering the natural degradation of the material will be carried out in accordance with the study performed for the same process but without considering aging. We shall solve the equation (47) in quadrature according to the expression (20) and introduce the change of variables (21). Then the equation (47) takes the form
U (t W • f—^-L-rft'. (48)
w J0 xm-r (0
The mean of the function U(t) is determined in this case by averaging the expression (48) taking into account the introduction of the hypothesis of statistical independence of loading and degradation
(U(t))=v(4 (x-m)jVm (t')dt'. (49)
0
To determine the variance of function U(t), we consider its correlation function, which by definition is expressed through the second initial moment (27) and taking into account the integral representation of U(t), (48) takes the form
(50)
and considering the hypothesis of independence of the processes of degradation and loading, we obtain the expression linking its second initial moment with the probabilistic characteristics of frequency and random component fatigue limit:
where
(x-2a)=Jx-2a •fx(%)dX, (52)
0
and fx(x) - log-normal distribution with mean and variance, which are determined from the expression (43), (46).
Second initial moment of random frequency is expressed in terms of its correlation function and the square of the mean value which is constant within the framework of assumption of stationarity
^)=v JJ-Wwr)-dtdt • (53)
Variance of function U(t) is calculated from its correlation function, see (24). In this case, to find an analytic expression for the correlation function and the variance is not possible and further analysis is carried out within the framework of numerical procedures. Thus, the integral expressions obtained for the probability characteristics of the function U(t), which, as before, satisfies the conditions of the central limit theorem, and therefore is subject to the normal Gauss's law, but with the characteristics (49) and (53).
Using the link between the process U(t) and damage (21), the probability density function of damage can be represented in the form (33). Expectation and variance of damage are determined from (34) and the confidence interval from (35) and (36). Probabilistic characteristics of the lifetime can be found from the expressions (39)-(40).
7. Numerical results
On the basis of developed model, provided in the paper, numerical calculations of the expectation and the confidence interval of damage in runners bolted connections considering degradation have been carried out. As the initial data the results of stress-strain state calculations (Fig. 6) is used. Fig. 12 represents the dependence of the mean of life-time on mean of frequency of transient regimes. Fig. 13 shows the variance of the life-time on the correlation time at different rates of the frequency variance of the transient process occurrence. The basic characteristics of damage and lifetime have been specified (Fig. 14): probability of failure of a bolted joint,
probability density of the life-time and damage accumulation with its confidence interval. Comparing the results obtained with and without degradation, the life-time obtained for the model taking into account degradation are 12 times less smaller than those defined for the model without degradation.
-l/4Var[co] ----lVar[<o]
.................4Var[co]
X/.. months
Fig. 12. Graph of the life-time dependence on Fig. 13. Graph show in the variation of the the mean of frequency of occurrence of life-time dependence on correlation time transient regimes for different variances on the frequency
of transient regimes occurrence
Fig. 14. Graphs of probability density of the life-time, probability of non-failure operation and the process of accumulation of damage depending on time
Conclusions
This paper discusses the issues of application of the developed probabilistic model by the example of the hydroturbine runner bolted connection life-time prediction. The developed model takes into account material degradation and can provide more realistic estimation of the life-time of bolting connection. The paper provides with assessment of the impact of model parameters on the main probabilistic characteristics. The comparison of lifetime by models with and without degradation of the material is done.
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About the authors
Oleksiy O. Larin (Kharkiv, Ukraine) - Ph.D. in Technical Sciences, Assoc. Professor, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: ale-xeya.larin@gmail.com).
Oleksandr I. Trubayev (Kharkiv, Ukraine) - Ph.D. in Technical Sciences, Assoc. Professor, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: trubayev@gmail.com).
Oleksii O. Vodka (Kharkiv, Ukraine) - Ph.D. student, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: a_vodka@mail.ru).
Получено 28.01.2014
Просьба ссылаться на эту статью в русскоязычных источниках следующим образом:
Larin O.O., Trubayev O.I., Vodka O.O. The fatigue life-time propagation of the connection elements of long-term operated hydro turbines considering material degradation // Вестник Пермского национального исследовательского политехнического университета. Механика. - 2013. - № 1. - С. 167-193.
Please cite this article in English as:
Larin O.O., Trubayev O.I., Vodka O.O. The fatigue life-time propagation of the connection elements of long-term operated hydro turbines considering material degradation.
PNRPUMechanics Bulletin. 2013. No. 1. P. 167-193.
