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Probabilistic failure risk assessment for aeroengine disks considering a transient process
Aerospace Science and Technology 78 (2018) 696–707
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Probabilistic failure risk assessment for aeroengine disks considering a transient process Shuiting Ding a , Ziyao Wang a , Tian Qiu a , Gong Zhang b , Guo Li a,∗ , Yu Zhou c,∗ a
Aircraft/Engine Integrated System Safety Beijing Key Laboratory, School of Energy and Power Engineering, Beihang University, 100191 Beijing, China Airworthiness Certification Center, Civil Aviation Administration of China (CAAC), 100102 Beijing, China c Beijing Key Laboratory for High-efficient Power Transmission and System Control of New Energy Resource Vehicle, School of Transportation Science and Engineering, Beihang University, 100191 Beijing, China b
a r t i c l e
i n f o
Article history: Received 6 April 2018 Received in revised form 11 May 2018 Accepted 12 May 2018 Available online 16 May 2018 Keywords: Aeroengine disk Transient process Risk assessment
a b s t r a c t One of the key processes for safety design of aeroengines is to accurately predict the failure risk of aeroengine disks. Current risk assessment methods mostly based on a constant stress are suitable for steady-state analysis but inappropriate for dangerous transient process. This work proposes a method of probabilistic failure risk assessment for aeroengine disks considering a transient process, and the core procedure is zone definition through refinement and further partition of a constant pre-zone based on the time-varying stress in a flight cycle. An aeroengine compressor disk is analyzed, and the failure risks of the disk considering a transient process and based on a steady-state design point are compared to examine the influence of the transient process on the failure risk of the disk. Results show that the failure risk considering the transient process is approximately 3.7 times of that based on the steady-state design point because the peak stress of the disk during the transient process exceeds the steady-state stress. The proposed method obtains more accurate predictions of failure risk, and is thus valuable for the safety design of aeroengines. © 2018 Elsevier Masson SAS. All rights reserved.
1. Introduction As a key aeroengine part [1], disks play an important role in the safety of aeroengines. Disk fracture directly induces noncontainment of high-energy debris, which results in catastrophic events, such as loss of aircraft and death [1,2]. Aeroengines generally operate with high failure rates during a transient process [3] possibly because transient and complex environments can induce stresses different from those in a steady state [2,4]. A typical example of the thermal and stress responses of a turbine disk for a transport engine over a full flight is shown in Fig. 1 [5]. A large temperature difference exists between the hub and rim of the disk during take-off and climb periods, and the resulting transient thermal stress is approximately 10% larger than that in the cruise period. This example implies that a peak stress that is larger than steady-state stress exists in a disk during a transient process and seriously affects the safety of aeroengines. Thus, the effect of the transient process must be considered in research on the safety of aeroengine disks.
*
Corresponding authors. E-mail addresses: [email protected] (G. Li), [email protected] (Y. Zhou).
https://doi.org/10.1016/j.ast.2018.05.017 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.
Various safety analysis methods have been proposed involving key structural components like engine cylinder heads [6,7], aeroengine disks [8,9] and aircraft spars [10–12]. Variable amplitude loads [12,13] were considered to simulate the actual complex working conditions of aircrafts during the service period. All these methods provide effective tool for the safety design of aircraft, and are thus valuable. Specifically for aeroengine disks, the most widely used method is the conventional rotor life management methodology, which was established based on the assumption that no anomaly exists in disks prior to flight service [8]. Industrial gas turbine experience has shown that the occurrence of material and manufacturing anomalies, although rare, can potentially degrade the structural integrity of high-energy rotors [14]. The conventional methodology does not explicitly consider the occurrence of anomalies and may thus be insufficient for safety analysis. A probabilistic damage tolerance design method, namely, probabilistic failure risk assessment, was established to augment the conventional life management approach for aeroengine disks. The framework of this method is presented in Fig. 2. The fundamental philosophy behind this method is to predict the failure risk of a disk (i.e. the probability of fracture) as a function of flight cycle in consideration of the existence of an initial anomaly. The assessment process involves a zone-based probabilistic fracture analysis [9] (Fig. 3), which consists of the following three steps:
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Nomenclature a C E F G K Kc m n N p pd Pf r T
α
crack length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Paris constant Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa event of failure geometrical correction coefficient stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 fracture toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 total number of zones Paris index flight cycle; rotation speed . . . . . . . . . . . . . . . . . . . . . . r/min pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa conditional failure probability of zone probability of fracture of disk radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mm temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K thermal expansion coefficient . . . . . . . . . . . . . . . . . . . . . K−1
K
γ λ
ν ρ σ τ
range of stress intensity factor during flight cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 anomaly occurrence rate thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . W/(m K) Poisson’s ratio density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg/m3 circumferential stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s
Subscripts DP i in out TR
based on steady-state design point index of zone inlet of disk cavity; initial crack outlet of disk cavity considering transient process
Fig. 2. Framework of probabilistic failure risk assessment for disks.
Fig. 1. Typical thermal and stress responses of a turbine disk [5].
1) Zone definition. A disk is subdivided into a finite number of zones. 2) Failure risk analysis for individual zones. The probability of fracture of each zone is calculated independently. 3) Failure risk analysis for the disk. The results of the zones are summed statistically to obtain the failure risk of the disk. The basic idea of the three steps is as follows. An anomaly, if it exists, could be in any location on a disk, and the stress of the disk is generally non-uniform. The risk caused by the anomaly can vary with different locations. Thus, the disk structure is divided into a number of zones on the basis of a finite element mesh and the stress results. The stresses of all sub-regions in each zone are similar, such that each zone is an approximately uniform stress field, and the risk computed for an anomaly occurring in any sub-region
Fig. 3. Process of probabilistic failure risk assessment.
can be regarded as similar. The failure risk of the disk is the combination of the risks for all zones, that is,
P f = P [F 1 ∪ F 2 ∪ · · · ∪ Fm]
(1)
where P f is the probability of fracture of the disk, F i represents the failure of zone i and m is the total number of zones. Notably, the probability that the disk contains an anomaly is generally low. Therefore, assuming that only one anomaly exists in the disk is
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Fig. 4. Process of zone definition considering a transient process.
reasonable. The failures of the zones can thus be regarded as independent events [9]. Equation (1) can be rewritten as
Pf ≈
m
P [Fi]
(2)
i =1
where P [F i ] indicates the probability of fracture of zone i. It is decided by
P [ F i ] = p d,i γi
(3)
where p d,i is the conditional failure probability, namely, the failure risk given that an anomaly exists in the zone, and γi is the anomaly occurrence rate, i.e. the probability of having an anomaly in the disk. Considering that γi one of the material inputs, the problem is the calculation of p d,i , which can be solved by a failure risk analysis for the individual zones. After obtaining p d,i , the failure risk of the disk can be calculated with Eqs. (3) and (2). Notably, the safety level of a constant disk is determined by the design, and the failure risk is regarded one of the key design targets. An advanced design for a disk should make sure that the failure risk of the disk during its expected flight life is low enough to achieve a certain safety level of aeroengines. It is thus essential to accurately predict the failure risk of aeroengine disks in the design stage. Many studies have examined probabilistic failure risk assessment, and various analysis tools were proposed [15–19]. The sensitivity of failure risk with respect to various geometries, materials [20,21] and flow parameters of the rotating cavity [22] was investigated with these tools. The above currently-available risk assessment methods provided effective tools for failure risk predictions of aeroengine disks. However, they present common limitations. Firstly, their analysis is generally based on steady loads of disks, that is, the conventional method of zone definition is applied based on a steady-state stress or an equivalent stress from the principle of equivalent damage, and the resulting zone is constant. This zone does not account for the effect of transient stresses in a flight cycle, which are more complicated than a constant stress and possibly highly related to the safety of aeroengines. Besides, zone definition for these tools is generally manually-operated, which is inconvenient and can cause accidental errors due to human factors. Therefore, these tools may be insufficient to accurately predict the failure risk of disks, and the design of a disk based on this method may fail to meet the growing safety requirement. Research on probabilistic failure risk assessment based on a transient stress may present novel regulations that would help further improve safety.
Against this background, this work proposes a method of probabilistic failure risk assessment for aeroengine disks that considers a transient process and investigates the influences of the transient process on the failure risk of disks. The rest of this paper is structured as follows. The method of probabilistic failure risk assessment considering a transient process is presented in Sec. 2. A near-real disk in a simplified flight cycle is studied. Real-time transient temperature and stress are obtained through a numerical simulation, and the failure risk of the disk is calculated using the proposed method. The computational model and inputs are introduced in Sec. 3, and the analysis results are presented and discussed in Sec. 4. The major conclusions are summarized in Sec. 5. 2. Method of probabilistic failure risk assessment considering a transient process This section introduces a method of probabilistic failure risk assessment considering a transient process. Among all the procedures of the assessment process, zone definition and failure risk analysis for individual zones are the most essential steps and are thus presented in detail. 2.1. Zone definition Zone definition divides a disk into a finite number of regions (made up of several finite elements) following certain zone definition principles. Each zone can be regarded as a uniform stress field, such that the life of a zone is approximately constant for a given initial crack size. Unlike the approach for conventional zone definition that is manually operated based on constant stress, the analytical approach for zone definition in this work considers transient stress in a flight cycle and is automatically operated. The proposed method of zone definition can be broken down into three basic steps, as presented in Fig. 4. 1) Zone partitioning based on a stress distribution at a time point A disk is divided based on a stress at a time point of a flight cycle. This division is achieved by grouping finite elements into zones at a certain stress interval according to the principle of stress similarity. Stress intervals of 34.5 MPa are practical for initial zone definition and are thus employed in this study [8]. Surface zones should be defined to consider anomalies/cracks located in nearsurface regions that generally grow faster than embedded cracks under the same load conditions. Hence, three types of zones are considered in this study: zones containing embedded cracks (ec), zones containing surface cracks (sc) and zones containing corner cracks (cc).
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Fig. 5. Analysis of the conditional failure probability of zones.
2) Stress difference analysis Given that the initial zone definition in step 1) is based on steady-state stress, several zones may fail to meet the demand of stress similarity if stresses vary with time in a flight cycle. Thus, a stress difference analysis is performed for each initial zone. The difference between the maximum and minimum stresses in a zone at different times is calculated, and the stress difference for each zone varying with time in a flight cycle is determined. 3) Further zone refinement based on transient stress Zone refinement is conducted for zones that fail to present stress similarity through a flight cycle. Each of these zones is further divided, and this subdivision is carried out until the stress difference in each zone does not exceed the given stress interval through a flight cycle. 2.2. Failure risk analysis for individual zones The calculation of conditional failure probability p d involves a crack growth analysis for a zone with an initial crack (i.e. anomaly). Notably, the zone is subject to cyclic loads, and crack length a and stress intensity factor K increase with flight cycle. Crack growth is governed by Paris’ law [23]:
da dN
= C ( K )n
(4)
where C and n are the constant and index of Paris’ law and K is the maximum change in K that can be calculated by crack length a and the circumferential stress of the zone σ . This work only considers mode-I crack because it is the most common crack in aeroengine disks. Thus, the maximum principal stress, namely, circumferential stress σ , is employed. Failure occurs when the maximum K is equal to or exceeds fracture toughness K c . Thus,
p d ( N ) = P K (a, σ , N ) ≥ K c
(5)
where N is the number of flight cycle. While K c is a material property, K can be complex, and the determination method for K is essential for a failure risk analysis. Research on the calculation model for K is abundant [24,25]. This work intends to study the method of failure risk analysis, and actually various K analysis methods can be considered. In this study, the Newman’s model [25] is employed because it is validated effective for a single crack analysis and is widely used in structural analyses, in which K is treated as a function of crack length, crack geometry and loads as follows:
K = Gσ
√
πa
(6)
where G is the geometrical correction coefficient and depends on the geometry of the zone and crack length. Theoretically, p d can be derived from Eqs. (4)–(6) if the probability density function of initial crack length is given. However, the probability density function can be highly complex, and no analytical form is available. Typically, the failure risk of each zone can be calculated via Monte Carlo simulation, which is a robust means of addressing such problems. However, the number of simulations required is generally related to the computed risk. Hence, many samples are required, and the process could become too time-consuming to carry out. Several assumptions are employed for simplicity. The initial crack length is modeled as the only random variable, and the random variable involving inspection, stress and material property is neglected for simplicity. Given that this study aims to investigate the effect of a transient process on failure risk, we believe that these simplifications would not weaken the general meaning of this work for aeroengine practical application. Owing to these simplifications, an integrated probabilistic method with high efficiency [26] can be used to calculate p d . The basic process is as follows. We let a0 be a random initial crack that exists in a certain zone. Fatigue crack growth analysis is performed, and the crack growth curve is illustrated in Fig. 5. Crack growth ceases when failure occurs, i.e., K > K c . The corresponding crack size is ac , and the corresponding flight cycle number is N c . To obtain p d at flight cycle N, we track back along the crack growth curve from N c to (N c –N), and the corresponding crack size is af,N . For initial crack ain = af,N , the crack grows to ac after N flight cycles, and failure occurs. Thus, the failure risk of the zone at flight cycle N is actually the probability of initial crack size ain > af,N , which can be acquired from the given initial crack size distribution. The conditional failure probability of each zone can be calculated using this method, and the failure risk of the disk can be obtained. The probabilistic failure risk assessment process presented above is performed by coding via Matlab 10.0, which is a core procedure in this work. The code is then validated effective using a recognized test case [8]. Therefore, the code developed in this study is sufficient to study the failure risk of a disk considering a transient process. 3. Computational model and inputs A rotating compressor disk is considered in this work. The stress and temperature of the disk are acquired through a thermoelastic analysis using the one way fluid-structure interaction (FSI) method [27]. Probabilistic failure risk analysis is then conducted to calculate the failure risk of the disk. Notably, this study intends to present a method of failure risk analysis for disks. The process of thermoelastic analysis is not introduced in detail in this paper because the stress and temperature of disks obtained by a prequel thermoelastic analysis are inputs for failure risk analysis
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Fig. 6. 3D model and radial–axial cross section of the disk.
A 3D model of a rotating compressor disk is presented in Fig. 6a. Given that the model is assumed to be axisymmetric, a one-tenth sector is shown. While the stress and temperature of the disk during a flight cycle are obtained based on a 3D model, probabilistic failure risk analysis is performed in the radial–axial cross section, as presented in Fig. 6b. The material parameters of the disk are presented in Table 1, and they are assumed to be isotropic and homogeneous. The material parameters are regarded as temperature-independent for simplicity because the variations caused by temperature changes within the temperature range in this case are small.
Similar evolution trends are also observed for the other boundary conditions, except for the temperature of the air at inlet T in , which remains unchanged throughout a flight cycle (i.e. T in = 288.15 K). Notably, actual components in aeroengines are subjected to complex working loads (mostly variable amplitude loads) during the service period [12]. The safety analysis for aeroengines should consider complex loads that can reflect actual variable amplitude loads. The boundary conditions presented in Fig. 7 are not perfectly identical to the real loads of disks. Actually, they are based on necessary simplifications in geometry and real load conditions, and these simplifications are necessary treatments for the approach of obtaining the boundary conditions (i.e. secondary air-system analysis). However, secondary air-system analysis has been proven to be reasonable and effective [29,30]. The boundary conditions calculated from secondary air-system analysis match one another, despite that the value of the boundary conditions may differ from real conditions. Besides, the difference between a transient process and a steady-state condition is reflected, although the boundary conditions are not with variable amplitudes. Furthermore, the motivation of this work is to study the effect of a transient process on the failure risk of aeroengine disks. Thus, although these conditions are different from real conditions, they can reflect the evolution of loads for an actual aeroengine during a real flight cycle. We hope that this numerical study based on near-real boundary conditions may provide some guiding significance for applied engineering problems.
3.2. Boundary conditions in a flight cycle
3.3. Inputs for probabilistic failure risk assessment
The representative process of safety analysis for an aeroengine disk starts with aircraft and engine requirements, flight profile selection and performance analysis [2]. Load analysis in a flight cycle is then performed, followed by life and failure risk analysis. Considering that the complete process is complicated and that this work intends to investigate failure risk analysis methods of disks, the prequel procedures are not presented in detail. A simplified flight cycle, which consists of acceleration, one engine inoperative (OEI) [28] and cruise period, is considered. The details of these time-varying boundaries in a flight cycle, which are determined through a secondary air-system analysis, are presented in Fig. 7. Rotation speed N initially increases from the initial state value to a steady-state value at about 3.2 s and remains constant for a short period. Then, N rapidly increases to the maximum value within 10.3 s to 12.4 s, which indicates that the transient process of OEI is achieved. When the OEI state is achieved, N remains at the peak value for about 20 s. Then, N decreases to a steady-state value at about 26.2 s, and the steady-state design point is achieved. Finally, N decreases to the initial value, and the flight cycle ends.
Crack growth analysis is a crucial process in probabilistic failure risk assessment and strongly influences the reliability of the results. Accordingly, material properties involving crack growth are necessary for probabilistic failure risk assessment. The constant C and index n of Paris’ law are 9.25E–13 and 3.87, respectively (SIunits: da/dN, m/cycle; K , MPa m1/2 ). Initial crack length is also a key input because probabilistic failure risk assessment essentially deals with the assessment of failure risk when an initial crack exists in a disk. In this study, the initial crack size distribution is given in the form of an exceedance curve according to industrydefined data [8], as shown in Fig. 8. This distribution is regarded as the only random variable in the analysis.
Table 1 Physical parameters of the disk. Physical parameter
Value
ρ , kg/m3
4620 7.955 110 0.30 8.6
λ, W/(m K) E, GPa
ν α , 10−6 /◦ C
for disks. This section introduces the computational model and conditions of the disk and other essential inputs for probabilistic failure risk assessment. 3.1. Computational model
4. Results and discussion Given that the model and circumferential stress are axisymmetric, the radial–axial cross section is adopted to present temperature and stress in a flight cycle, zone definition and failure risk analysis. The results of the temperature and circumferential stress of
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Fig. 7. Time-varying boundary conditions of the disk cavity.
for temperature and circumferential stresses at different times are similar, although the concrete values change with time. The effect of temperature is disregarded for simplicity because the variations in material properties caused by temperature changes within the temperature range in this case are small. Thus, the time-varying temperature distributions are not considered in the following analysis. Only the stress distributions at different times are used to perform zone definition. The stress distribution of the disk indicates that the stress at the hub is higher than that in the other regions of the disk. Thus, the stress of a node at the disk hub varying with time in a flight cycle is presented to further study the stress evolution. As shown in Fig. 11, the trend of the stress varying with time is consistent with that of rotation speed N. This result indicates that the stress of a compressor disk is mainly dependent on centrifugal loads (i.e. rotation speed N) as a result of the working characteristic of the compressors of aeroengines. Fig. 8. Initial crack size distribution [8].
4.2. Zone definition the disk are presented in this section. Zone definition is acquired based on the results, and probabilistic failure risk assessment is performed. The failure risks of the disk considering a transient process and based on a steady-state design point are calculated and compared to investigate the influence of the transient process on the failure risk of the disk. 4.1. Temperature and stress in a flight cycle The temperature and circumferential stress contours of the disk at six time points (0, 2, 11, 24, 25 and 40 s) in a flight cycle are presented in Figs. 9 and 10, respectively. The distribution trends
Zone definition is performed based on the time-varying stress distribution of the disk in a flight cycle, and the resulting zone data are passed to the sequel probabilistic failure risk assessment. According to the zone definition methodology presented in Sec. 2.1, the process of zone definition is as follows. 1) Zone partitioning based on a stress distribution at a time point The disk is divided based on the stress distribution at 24 s following zone definition principles (1)–(3), namely, stress similarity, geometric continuity and near-surface zone refinement,
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Fig. 9. Temperature contours of the disk at different times of a flight cycle.
Fig. 10. Circumferential stress contours of the disk at different times of a flight cycle.
as illustrated in Fig. 12. The resulting zone definition presented in Fig. 12e is the basis for the following zone refinement considering a transient process and is thus a pre-zone. The process of pre-zone definition is actually zone generation based on a certain stress distribution. It is applicable to zone definition based on a steady-state stress distribution, which is presented in Sec. 4.5. 2) Stress difference analysis Stress difference analysis is performed for each initial zone. The stress difference for each zone varying with time in a flight cycle is acquired, and zones requiring refinement are identified.
3) Further zone refinement based on transient stress Zone refinement is conducted for zones that fail to fulfill stress similarity through a flight cycle. Each of these zones is further divided, and this subdivision process is carried out until zone definition principles (1) and (2) are fulfilled, as presented in Fig. 13. Special attention should be devoted to the fact that the number of zones increases significantly after performing zone refinement based on transient stress. A reason is that the geometric continuity requirement of zones needs to be satisfied by further subdividing zones. This “side effect” is a limit of zone definition that is automatically operated. Therefore, the proposed zone definition approach still needs to be improved in the future. The final zone
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Fig. 11. Circumferential stress of the disk hub varying with time in a flight cycle.
Fig. 12. Pre-zone definition based on stress at max rotation speed state.
definition shown in Fig. 13c is then passed to the following failure risk analysis.
4.3. Failure risk of the disk
Probabilistic failure risk analysis is conducted according to the zones generated in Sec. 4.2. The stress of the disk within a flight cycle is used to calculate the failure risk of the disk. The failure risk of the disk P f varying with flight cycle N is shown in Fig. 14. P f increases with an increase in N. Specifically, P f = 2.94E–11 at N = 20,000. Moreover, a certain cycle number N s exists, and when N < N s , P f = 0. In this case, the value of N s is about 1,500. The disk may fail after such a short service time (about 1,500 flight cycles) because of the effect of initial anomalies, although the occurrence of initial anomalies is low. Thus, the impact of initial anomalies on disk safety must be identified using probabilistic failure risk assessment.
4.4. Failure risk of hub zones The failure risk of the disk is the sum of the failure risks of all zones. To identify the most risky zones in the disk, this study considers the contribution of each zone to the disk and represents it by a risk contribution factor as follows:
y f, i =
P f, i Pf
(7)
where y f,i is the risk contribution of zone i, P f,i is the failure risk of zone i, and P f is the failure risk of the disk, namely, the sum of the failure risks of all zones. To identify the relative value of zone geometrical size, volume fraction y v is defined as
y v,i =
Vi V
(8)
where y v,i is the volume fraction of zone i, V i is the volume of zone i, and V is the volume of the disk, namely, the sum of the volumes of all zones.
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Fig. 13. Zone refinement considering transient stress.
Fig. 14. Failure risk of the disk varying with flight cycle.
Fig. 15 shows the risk contribution factors of the zones at the 20,000th cycle. As shown in Fig. 15a, the contour of risk contribution is similar to the stress contour, and the risk contributions
of zones in the hub region are higher than those of the other regions of the disk. This result may prove that the risk contribution of a zone is related to the stress level. For a quantitative analysis, the six zones with the highest risk contributions are identified from Fig. 15b. These six zones are 38, 33, 29, 36, 35 and 34, and they are all located in the hub region of the disk, as demonstrated in Fig. 16. Detailed information of the six zones is presented in Table 2. The six zones in the hub region occupy only 3.79% of the disk volume but contribute 66.54% to the disk failure risk. Specifically, the six zones at the hub contribute 17.56% risk per 1% volume, whereas the rest of the zones contribute only 0.35% risk per 1% volume. Accordingly, the zones at the hub are the most risky parts and probably strongly influence the failure risk of the entire disk. The results also show that zone type (inside or near-surface) and stress level are two key factors that decide the failure risk of a disk. For zones with a similar stress level, those in the near-surface region are more risky than those in the inside of the disk. For zones with the same type, stress level is a domi-
Fig. 15. Failure risk contributions of zones at the 20,000th flight cycle.
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Fig. 16. Six zones with the highest risk contribution factor. Table 2 Details of the six zones with the highest risk contribution factor.
a
Zone index
yv
Zone typea
σ (MPa)
yf
y f per 1% volume
38 33 29 36 35 34
0.34% 0.62% 2.36% 0.14% 0.20% 0.13%
sc ec ec sc ec sc
639.1 583.9 519.4 613.2 593.1 581.9
27.24% 13.88% 7.94% 7.72% 5.49% 4.27%
79.39% 22.56% 3.37% 55.98% 27.00% 33.81%
ec – embedded cracks; sc – surface cracks; cc – corner cracks.
nant factor, and the failure risk generally increases with the stress value. 4.5. Effects of the transient process on failure risk To study the effect of the transient process on the failure risk of the disk, the failure risk of the disk based on a steady-state design point is calculated, and the result is compared with that considering the transient process. Zone definition is performed based on the steady-state stress distribution at τ = 40 s. The process is consistent with that for the pre-zone in Sec. 4.2, and the final zone definition is presented in Fig. 17. Then, the failure risk of the disk based on a steady-state design point is obtained. For comparison, the failure risks of the disk considering the transient process and based on a steady-state design point are presented in one figure (Fig. 18). For convenience, subscripts “TR” and “DP” are used to refer to probabilistic failure risk assessment considering the transient process and based on a steady-state design point, respectively. The failure risk of the disk P f increases with an increase in N, and P f,TR is larger than P f,DP at the same flight cycle. Specifically, at N = 20000, P f,TR = 2.94E–11 and P f,DP = 7.90E–12. P f,TR is about 3.7 times the value of P f,DP . Cycle number N s exists for both cases, but the values are different. The values of N s,TR and N s,DP are about 1,500 and 2,500, respectively. These comparisons show that the failure risk of the disk is increased, and the disk may fail at an early flight number because of the effect of transient stress. Thus, the transient process greatly influences disk safety
Fig. 17. Zone definition based on stress at steady state.
Fig. 18. Failure risk of the disk considering the transient process and based on a steady-state design point.
and should be considered in the failure risk analysis for aeroengine disks. Given that the hub is the most risky part of the disk and contributes most to the failure risk of the entire disk, a zone at the
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above analysis implies that probabilistic failure risk assessment based on a steady-state design point does not address the influence of transient stress, which could exceed the steady-state stress, and may thus be insufficient to predict the failure risk of the disk accurately. The method of probabilistic failure risk assessment considering a transient process can obtain more accurate predictions of failure risk. Since the failure risk is one of the key design targets involving safety, the proposed failure risk assessment method can help achieve safer designs of aeroengine disks. 5. Conclusions
Fig. 19. Conditional failure probability of a zone at the disk hub considering the transient process and based on a steady-state design point.
disk hub is considered to further investigate the effect of the transient process on the failure risk. Figs. 13c and 17 show that although the zone definitions for probabilistic failure risk assessment considering the transient process and based on steady state are different, they have zones in common. Zone 38 in Fig. 13c and zone 34 in Fig. 17 are actually the same region of the disk. This zone contributes the most to the failure risk of the entire disk in both cases. Thus, this zone is analyzed, and the conditional probability of failure p d varying with flight cycle for both cases is presented in Fig. 19. p d increases with an increase in N, and p d,TR is larger than p d,DP at the same flight cycle. Specifically, at N = 20000, p d,TR = 0.41 and p d,DP = 0.14. p d,TR is about 2.9 times the value of p d,DP . The reason for this difference is the stress difference in the two cases. For probabilistic failure risk assessment considering the transient process, the stress is the peak stress in a flight cycle (i.e. σ TR = 639.1 MPa), whereas for probabilistic failure risk assessment based on a steady-state design point, the stress is the steady-state stress in a flight cycle (i.e. σ DP = 572.2 MPa). The value of σ TR is 11.7% larger than that of σ DP , and the failure risk is consequently raised to 190%. The failure risk is highly sensitive to the stress level. The reason for the above regulation can be explained by the analysis method of conditional probability of failure in Sec. 2.2. We let a0 be a random initial crack that exists in a certain zone. Fatigue crack growth analysis is performed for peak stress and steady-state stress cases, and the crack growth curves are illustrated in Fig. 20. Given that σ TR > σ DP , the crack growth rate under σ TR is faster than that under σ DP . As a result, af,DP > af,TR and p TR > p DP . Accordingly, because the peak stress during the transient process exceeds the steady-state stress, crack growth is accelerated, and the failure risk is consequently increased. The
On the basis of the safety demand for aeroengine disks, this work proposes a method of probabilistic failure risk assessment considering a transient process and studies the influence of the transient process on the failure risk of disks. An aeroengine compressor disk is analyzed. The temperature and stress evolutions of the disk during a simplified flight cycle are acquired through a numerical simulation based on FSI, and probabilistic failure risk assessment considering the transient process is performed to calculate the evolution of the failure risk of the disk. The result is compared with that obtained by probabilistic failure risk assessment based on a steady-state design point, and the influence of the transient process on the failure risk of the disk is investigated. The major results are summarized as follows. 1) The failure risk of the disk increases with flight cycle, and the hub is the most risky part of the disk that contributes the most to the failure risk of the entire disk. 2) Stress level and zone type (zone location in the disk) are the key factors that affect the failure risk. For the same type of zone, the failure risk increases with the stress level. For the same stress level, zones in the near surface are more risky than those in the internal region. 3) The failure risk of the disk considering the transient process is greater than that based on a steady-state design point, because the peak stress of the disk during the transient process exceeds the steady-state design point stress. To summarize, the method of probabilistic failure risk assessment considering a transient process can obtain more accurate predictions of failure risk. Although the analysis of this study is based on simplified treatments from the actual situations of real aeroengines, the proposed method provides new insights to assess the safety of aeroengine disks in design. Conflict of interest statement There is no conflict of interest regarding this journal article.
Fig. 20. Conditional failure probability analysis considering the transient process and based on a steady-state design point.
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Acknowledgements The work was funded by the National Natural Science Foundation of China [Grant No. 51676009 and No. 51775025]. The work was supported by the Innovation Team of Complex System Safety and Airworthiness of Aero Engine from the Co-Innovation Center for Advanced Aeroengine of China. References [1] US Department of Transportation Federal Aviation Administration, Advisory Circular 33.75-1A: Guidance Material for 14 CFR 33.75 Safety Analysis, Washington DC, US, FAA, 2007. [2] US Department of Transportation Federal Aviation Administration, Advisory Circular 33.70-1: Guidance Material for Aircraft Engine-life-limited Parts Requirements, Washington DC, US, FAA, 2009. [3] International Air Transport Association, Safety report 2013 IATA, Montreal, Canada, 2013. [4] S.T. Ding, Z.Y. Wang, G. Li, C.K. Liu, L. Yang, Transient stress control of aeroengine disks based on active thermal management, Appl. Therm. Eng. 103 (2016) 16–27. [5] S.A. Sattar, C.V. Sundt, Gas turbine engine disk cyclic life prediction, J. Aircr. 12 (4) (1975) 360–365. [6] B. Krstic, B. Rasuo, D. Trifkovic, I. Radisavljevic, Z. Rajic, M. Dinulovic, Failure analysis of an aircraft engine cylinder head, Eng. Fail. Anal. 32 (2013) 1–15. [7] B. Krstic, B. Rasuo, D. Trifkovic, I. Radisavljevic, Z. Rajic, M. Dinulovic, An investigation of the repetitive failure in an aircraft engine cylinder head, Eng. Fail. Anal. 34 (2013) 335–349. [8] US Department of Transportation Federal Aviation Administration, Advisory Circular 33.14-1: Damage tolerance for high energy turbine engine rotors, Washington DC, US, FAA, 2001. [9] G.R. Leverant, D.L. Littlefield, R.C. McClung, H.R. Millwater, J.Y. Wu, A probabilistic approach to aircraft turbine rotor material design, 1997, ASME Paper, 1997-GT-22. [10] D. Petrasinovic, B. Rasuo, N. Petrasinovic, Extended finite element method (XFEM) applied to aircraft duralumin spar fatigue life estimation, Teh. Vjesn. 19 (3) (2012) 557–562. [11] M.A. Grbovic, B. Rasuo, Use of modern numerical methods for fatigue life predictions, in: R. Branco (Ed.), Recent Trends in Fatigue Design, Series: Mechanical Engineering Theory and Applications, Nova Science Publishers, New York, 2014, pp. 31–74. [12] A. Grbovic, B. Rasuo, FEM based fatigue crack growth prediction for spar of light aircraft under variable amplitude loading, Eng. Fail. Anal. 26 (12) (2012) 50–64. [13] M.F. Siladic, B. Rasuo, On-condition maintenance for nonmodular jet engines: an experience, J. Eng. Gas Turbines Power 131 (3) (2009).
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Contents lists available at ScienceDirect
Aerospace Science and Technology www.elsevier.com/locate/aescte
Probabilistic failure risk assessment for aeroengine disks considering a transient process Shuiting Ding a , Ziyao Wang a , Tian Qiu a , Gong Zhang b , Guo Li a,∗ , Yu Zhou c,∗ a
Aircraft/Engine Integrated System Safety Beijing Key Laboratory, School of Energy and Power Engineering, Beihang University, 100191 Beijing, China Airworthiness Certification Center, Civil Aviation Administration of China (CAAC), 100102 Beijing, China c Beijing Key Laboratory for High-efficient Power Transmission and System Control of New Energy Resource Vehicle, School of Transportation Science and Engineering, Beihang University, 100191 Beijing, China b
a r t i c l e
i n f o
Article history: Received 6 April 2018 Received in revised form 11 May 2018 Accepted 12 May 2018 Available online 16 May 2018 Keywords: Aeroengine disk Transient process Risk assessment
a b s t r a c t One of the key processes for safety design of aeroengines is to accurately predict the failure risk of aeroengine disks. Current risk assessment methods mostly based on a constant stress are suitable for steady-state analysis but inappropriate for dangerous transient process. This work proposes a method of probabilistic failure risk assessment for aeroengine disks considering a transient process, and the core procedure is zone definition through refinement and further partition of a constant pre-zone based on the time-varying stress in a flight cycle. An aeroengine compressor disk is analyzed, and the failure risks of the disk considering a transient process and based on a steady-state design point are compared to examine the influence of the transient process on the failure risk of the disk. Results show that the failure risk considering the transient process is approximately 3.7 times of that based on the steady-state design point because the peak stress of the disk during the transient process exceeds the steady-state stress. The proposed method obtains more accurate predictions of failure risk, and is thus valuable for the safety design of aeroengines. © 2018 Elsevier Masson SAS. All rights reserved.
1. Introduction As a key aeroengine part [1], disks play an important role in the safety of aeroengines. Disk fracture directly induces noncontainment of high-energy debris, which results in catastrophic events, such as loss of aircraft and death [1,2]. Aeroengines generally operate with high failure rates during a transient process [3] possibly because transient and complex environments can induce stresses different from those in a steady state [2,4]. A typical example of the thermal and stress responses of a turbine disk for a transport engine over a full flight is shown in Fig. 1 [5]. A large temperature difference exists between the hub and rim of the disk during take-off and climb periods, and the resulting transient thermal stress is approximately 10% larger than that in the cruise period. This example implies that a peak stress that is larger than steady-state stress exists in a disk during a transient process and seriously affects the safety of aeroengines. Thus, the effect of the transient process must be considered in research on the safety of aeroengine disks.
*
Corresponding authors. E-mail addresses: [email protected] (G. Li), [email protected] (Y. Zhou).
https://doi.org/10.1016/j.ast.2018.05.017 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.
Various safety analysis methods have been proposed involving key structural components like engine cylinder heads [6,7], aeroengine disks [8,9] and aircraft spars [10–12]. Variable amplitude loads [12,13] were considered to simulate the actual complex working conditions of aircrafts during the service period. All these methods provide effective tool for the safety design of aircraft, and are thus valuable. Specifically for aeroengine disks, the most widely used method is the conventional rotor life management methodology, which was established based on the assumption that no anomaly exists in disks prior to flight service [8]. Industrial gas turbine experience has shown that the occurrence of material and manufacturing anomalies, although rare, can potentially degrade the structural integrity of high-energy rotors [14]. The conventional methodology does not explicitly consider the occurrence of anomalies and may thus be insufficient for safety analysis. A probabilistic damage tolerance design method, namely, probabilistic failure risk assessment, was established to augment the conventional life management approach for aeroengine disks. The framework of this method is presented in Fig. 2. The fundamental philosophy behind this method is to predict the failure risk of a disk (i.e. the probability of fracture) as a function of flight cycle in consideration of the existence of an initial anomaly. The assessment process involves a zone-based probabilistic fracture analysis [9] (Fig. 3), which consists of the following three steps:
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Nomenclature a C E F G K Kc m n N p pd Pf r T
α
crack length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m Paris constant Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa event of failure geometrical correction coefficient stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 fracture toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 total number of zones Paris index flight cycle; rotation speed . . . . . . . . . . . . . . . . . . . . . . r/min pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa conditional failure probability of zone probability of fracture of disk radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mm temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K thermal expansion coefficient . . . . . . . . . . . . . . . . . . . . . K−1
K
γ λ
ν ρ σ τ
range of stress intensity factor during flight cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa m1/2 anomaly occurrence rate thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . W/(m K) Poisson’s ratio density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg/m3 circumferential stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MPa time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s
Subscripts DP i in out TR
based on steady-state design point index of zone inlet of disk cavity; initial crack outlet of disk cavity considering transient process
Fig. 2. Framework of probabilistic failure risk assessment for disks.
Fig. 1. Typical thermal and stress responses of a turbine disk [5].
1) Zone definition. A disk is subdivided into a finite number of zones. 2) Failure risk analysis for individual zones. The probability of fracture of each zone is calculated independently. 3) Failure risk analysis for the disk. The results of the zones are summed statistically to obtain the failure risk of the disk. The basic idea of the three steps is as follows. An anomaly, if it exists, could be in any location on a disk, and the stress of the disk is generally non-uniform. The risk caused by the anomaly can vary with different locations. Thus, the disk structure is divided into a number of zones on the basis of a finite element mesh and the stress results. The stresses of all sub-regions in each zone are similar, such that each zone is an approximately uniform stress field, and the risk computed for an anomaly occurring in any sub-region
Fig. 3. Process of probabilistic failure risk assessment.
can be regarded as similar. The failure risk of the disk is the combination of the risks for all zones, that is,
P f = P [F 1 ∪ F 2 ∪ · · · ∪ Fm]
(1)
where P f is the probability of fracture of the disk, F i represents the failure of zone i and m is the total number of zones. Notably, the probability that the disk contains an anomaly is generally low. Therefore, assuming that only one anomaly exists in the disk is
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Fig. 4. Process of zone definition considering a transient process.
reasonable. The failures of the zones can thus be regarded as independent events [9]. Equation (1) can be rewritten as
Pf ≈
m
P [Fi]
(2)
i =1
where P [F i ] indicates the probability of fracture of zone i. It is decided by
P [ F i ] = p d,i γi
(3)
where p d,i is the conditional failure probability, namely, the failure risk given that an anomaly exists in the zone, and γi is the anomaly occurrence rate, i.e. the probability of having an anomaly in the disk. Considering that γi one of the material inputs, the problem is the calculation of p d,i , which can be solved by a failure risk analysis for the individual zones. After obtaining p d,i , the failure risk of the disk can be calculated with Eqs. (3) and (2). Notably, the safety level of a constant disk is determined by the design, and the failure risk is regarded one of the key design targets. An advanced design for a disk should make sure that the failure risk of the disk during its expected flight life is low enough to achieve a certain safety level of aeroengines. It is thus essential to accurately predict the failure risk of aeroengine disks in the design stage. Many studies have examined probabilistic failure risk assessment, and various analysis tools were proposed [15–19]. The sensitivity of failure risk with respect to various geometries, materials [20,21] and flow parameters of the rotating cavity [22] was investigated with these tools. The above currently-available risk assessment methods provided effective tools for failure risk predictions of aeroengine disks. However, they present common limitations. Firstly, their analysis is generally based on steady loads of disks, that is, the conventional method of zone definition is applied based on a steady-state stress or an equivalent stress from the principle of equivalent damage, and the resulting zone is constant. This zone does not account for the effect of transient stresses in a flight cycle, which are more complicated than a constant stress and possibly highly related to the safety of aeroengines. Besides, zone definition for these tools is generally manually-operated, which is inconvenient and can cause accidental errors due to human factors. Therefore, these tools may be insufficient to accurately predict the failure risk of disks, and the design of a disk based on this method may fail to meet the growing safety requirement. Research on probabilistic failure risk assessment based on a transient stress may present novel regulations that would help further improve safety.
Against this background, this work proposes a method of probabilistic failure risk assessment for aeroengine disks that considers a transient process and investigates the influences of the transient process on the failure risk of disks. The rest of this paper is structured as follows. The method of probabilistic failure risk assessment considering a transient process is presented in Sec. 2. A near-real disk in a simplified flight cycle is studied. Real-time transient temperature and stress are obtained through a numerical simulation, and the failure risk of the disk is calculated using the proposed method. The computational model and inputs are introduced in Sec. 3, and the analysis results are presented and discussed in Sec. 4. The major conclusions are summarized in Sec. 5. 2. Method of probabilistic failure risk assessment considering a transient process This section introduces a method of probabilistic failure risk assessment considering a transient process. Among all the procedures of the assessment process, zone definition and failure risk analysis for individual zones are the most essential steps and are thus presented in detail. 2.1. Zone definition Zone definition divides a disk into a finite number of regions (made up of several finite elements) following certain zone definition principles. Each zone can be regarded as a uniform stress field, such that the life of a zone is approximately constant for a given initial crack size. Unlike the approach for conventional zone definition that is manually operated based on constant stress, the analytical approach for zone definition in this work considers transient stress in a flight cycle and is automatically operated. The proposed method of zone definition can be broken down into three basic steps, as presented in Fig. 4. 1) Zone partitioning based on a stress distribution at a time point A disk is divided based on a stress at a time point of a flight cycle. This division is achieved by grouping finite elements into zones at a certain stress interval according to the principle of stress similarity. Stress intervals of 34.5 MPa are practical for initial zone definition and are thus employed in this study [8]. Surface zones should be defined to consider anomalies/cracks located in nearsurface regions that generally grow faster than embedded cracks under the same load conditions. Hence, three types of zones are considered in this study: zones containing embedded cracks (ec), zones containing surface cracks (sc) and zones containing corner cracks (cc).
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Fig. 5. Analysis of the conditional failure probability of zones.
2) Stress difference analysis Given that the initial zone definition in step 1) is based on steady-state stress, several zones may fail to meet the demand of stress similarity if stresses vary with time in a flight cycle. Thus, a stress difference analysis is performed for each initial zone. The difference between the maximum and minimum stresses in a zone at different times is calculated, and the stress difference for each zone varying with time in a flight cycle is determined. 3) Further zone refinement based on transient stress Zone refinement is conducted for zones that fail to present stress similarity through a flight cycle. Each of these zones is further divided, and this subdivision is carried out until the stress difference in each zone does not exceed the given stress interval through a flight cycle. 2.2. Failure risk analysis for individual zones The calculation of conditional failure probability p d involves a crack growth analysis for a zone with an initial crack (i.e. anomaly). Notably, the zone is subject to cyclic loads, and crack length a and stress intensity factor K increase with flight cycle. Crack growth is governed by Paris’ law [23]:
da dN
= C ( K )n
(4)
where C and n are the constant and index of Paris’ law and K is the maximum change in K that can be calculated by crack length a and the circumferential stress of the zone σ . This work only considers mode-I crack because it is the most common crack in aeroengine disks. Thus, the maximum principal stress, namely, circumferential stress σ , is employed. Failure occurs when the maximum K is equal to or exceeds fracture toughness K c . Thus,
p d ( N ) = P K (a, σ , N ) ≥ K c
(5)
where N is the number of flight cycle. While K c is a material property, K can be complex, and the determination method for K is essential for a failure risk analysis. Research on the calculation model for K is abundant [24,25]. This work intends to study the method of failure risk analysis, and actually various K analysis methods can be considered. In this study, the Newman’s model [25] is employed because it is validated effective for a single crack analysis and is widely used in structural analyses, in which K is treated as a function of crack length, crack geometry and loads as follows:
K = Gσ
√
πa
(6)
where G is the geometrical correction coefficient and depends on the geometry of the zone and crack length. Theoretically, p d can be derived from Eqs. (4)–(6) if the probability density function of initial crack length is given. However, the probability density function can be highly complex, and no analytical form is available. Typically, the failure risk of each zone can be calculated via Monte Carlo simulation, which is a robust means of addressing such problems. However, the number of simulations required is generally related to the computed risk. Hence, many samples are required, and the process could become too time-consuming to carry out. Several assumptions are employed for simplicity. The initial crack length is modeled as the only random variable, and the random variable involving inspection, stress and material property is neglected for simplicity. Given that this study aims to investigate the effect of a transient process on failure risk, we believe that these simplifications would not weaken the general meaning of this work for aeroengine practical application. Owing to these simplifications, an integrated probabilistic method with high efficiency [26] can be used to calculate p d . The basic process is as follows. We let a0 be a random initial crack that exists in a certain zone. Fatigue crack growth analysis is performed, and the crack growth curve is illustrated in Fig. 5. Crack growth ceases when failure occurs, i.e., K > K c . The corresponding crack size is ac , and the corresponding flight cycle number is N c . To obtain p d at flight cycle N, we track back along the crack growth curve from N c to (N c –N), and the corresponding crack size is af,N . For initial crack ain = af,N , the crack grows to ac after N flight cycles, and failure occurs. Thus, the failure risk of the zone at flight cycle N is actually the probability of initial crack size ain > af,N , which can be acquired from the given initial crack size distribution. The conditional failure probability of each zone can be calculated using this method, and the failure risk of the disk can be obtained. The probabilistic failure risk assessment process presented above is performed by coding via Matlab 10.0, which is a core procedure in this work. The code is then validated effective using a recognized test case [8]. Therefore, the code developed in this study is sufficient to study the failure risk of a disk considering a transient process. 3. Computational model and inputs A rotating compressor disk is considered in this work. The stress and temperature of the disk are acquired through a thermoelastic analysis using the one way fluid-structure interaction (FSI) method [27]. Probabilistic failure risk analysis is then conducted to calculate the failure risk of the disk. Notably, this study intends to present a method of failure risk analysis for disks. The process of thermoelastic analysis is not introduced in detail in this paper because the stress and temperature of disks obtained by a prequel thermoelastic analysis are inputs for failure risk analysis
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Fig. 6. 3D model and radial–axial cross section of the disk.
A 3D model of a rotating compressor disk is presented in Fig. 6a. Given that the model is assumed to be axisymmetric, a one-tenth sector is shown. While the stress and temperature of the disk during a flight cycle are obtained based on a 3D model, probabilistic failure risk analysis is performed in the radial–axial cross section, as presented in Fig. 6b. The material parameters of the disk are presented in Table 1, and they are assumed to be isotropic and homogeneous. The material parameters are regarded as temperature-independent for simplicity because the variations caused by temperature changes within the temperature range in this case are small.
Similar evolution trends are also observed for the other boundary conditions, except for the temperature of the air at inlet T in , which remains unchanged throughout a flight cycle (i.e. T in = 288.15 K). Notably, actual components in aeroengines are subjected to complex working loads (mostly variable amplitude loads) during the service period [12]. The safety analysis for aeroengines should consider complex loads that can reflect actual variable amplitude loads. The boundary conditions presented in Fig. 7 are not perfectly identical to the real loads of disks. Actually, they are based on necessary simplifications in geometry and real load conditions, and these simplifications are necessary treatments for the approach of obtaining the boundary conditions (i.e. secondary air-system analysis). However, secondary air-system analysis has been proven to be reasonable and effective [29,30]. The boundary conditions calculated from secondary air-system analysis match one another, despite that the value of the boundary conditions may differ from real conditions. Besides, the difference between a transient process and a steady-state condition is reflected, although the boundary conditions are not with variable amplitudes. Furthermore, the motivation of this work is to study the effect of a transient process on the failure risk of aeroengine disks. Thus, although these conditions are different from real conditions, they can reflect the evolution of loads for an actual aeroengine during a real flight cycle. We hope that this numerical study based on near-real boundary conditions may provide some guiding significance for applied engineering problems.
3.2. Boundary conditions in a flight cycle
3.3. Inputs for probabilistic failure risk assessment
The representative process of safety analysis for an aeroengine disk starts with aircraft and engine requirements, flight profile selection and performance analysis [2]. Load analysis in a flight cycle is then performed, followed by life and failure risk analysis. Considering that the complete process is complicated and that this work intends to investigate failure risk analysis methods of disks, the prequel procedures are not presented in detail. A simplified flight cycle, which consists of acceleration, one engine inoperative (OEI) [28] and cruise period, is considered. The details of these time-varying boundaries in a flight cycle, which are determined through a secondary air-system analysis, are presented in Fig. 7. Rotation speed N initially increases from the initial state value to a steady-state value at about 3.2 s and remains constant for a short period. Then, N rapidly increases to the maximum value within 10.3 s to 12.4 s, which indicates that the transient process of OEI is achieved. When the OEI state is achieved, N remains at the peak value for about 20 s. Then, N decreases to a steady-state value at about 26.2 s, and the steady-state design point is achieved. Finally, N decreases to the initial value, and the flight cycle ends.
Crack growth analysis is a crucial process in probabilistic failure risk assessment and strongly influences the reliability of the results. Accordingly, material properties involving crack growth are necessary for probabilistic failure risk assessment. The constant C and index n of Paris’ law are 9.25E–13 and 3.87, respectively (SIunits: da/dN, m/cycle; K , MPa m1/2 ). Initial crack length is also a key input because probabilistic failure risk assessment essentially deals with the assessment of failure risk when an initial crack exists in a disk. In this study, the initial crack size distribution is given in the form of an exceedance curve according to industrydefined data [8], as shown in Fig. 8. This distribution is regarded as the only random variable in the analysis.
Table 1 Physical parameters of the disk. Physical parameter
Value
ρ , kg/m3
4620 7.955 110 0.30 8.6
λ, W/(m K) E, GPa
ν α , 10−6 /◦ C
for disks. This section introduces the computational model and conditions of the disk and other essential inputs for probabilistic failure risk assessment. 3.1. Computational model
4. Results and discussion Given that the model and circumferential stress are axisymmetric, the radial–axial cross section is adopted to present temperature and stress in a flight cycle, zone definition and failure risk analysis. The results of the temperature and circumferential stress of
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Fig. 7. Time-varying boundary conditions of the disk cavity.
for temperature and circumferential stresses at different times are similar, although the concrete values change with time. The effect of temperature is disregarded for simplicity because the variations in material properties caused by temperature changes within the temperature range in this case are small. Thus, the time-varying temperature distributions are not considered in the following analysis. Only the stress distributions at different times are used to perform zone definition. The stress distribution of the disk indicates that the stress at the hub is higher than that in the other regions of the disk. Thus, the stress of a node at the disk hub varying with time in a flight cycle is presented to further study the stress evolution. As shown in Fig. 11, the trend of the stress varying with time is consistent with that of rotation speed N. This result indicates that the stress of a compressor disk is mainly dependent on centrifugal loads (i.e. rotation speed N) as a result of the working characteristic of the compressors of aeroengines. Fig. 8. Initial crack size distribution [8].
4.2. Zone definition the disk are presented in this section. Zone definition is acquired based on the results, and probabilistic failure risk assessment is performed. The failure risks of the disk considering a transient process and based on a steady-state design point are calculated and compared to investigate the influence of the transient process on the failure risk of the disk. 4.1. Temperature and stress in a flight cycle The temperature and circumferential stress contours of the disk at six time points (0, 2, 11, 24, 25 and 40 s) in a flight cycle are presented in Figs. 9 and 10, respectively. The distribution trends
Zone definition is performed based on the time-varying stress distribution of the disk in a flight cycle, and the resulting zone data are passed to the sequel probabilistic failure risk assessment. According to the zone definition methodology presented in Sec. 2.1, the process of zone definition is as follows. 1) Zone partitioning based on a stress distribution at a time point The disk is divided based on the stress distribution at 24 s following zone definition principles (1)–(3), namely, stress similarity, geometric continuity and near-surface zone refinement,
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Fig. 9. Temperature contours of the disk at different times of a flight cycle.
Fig. 10. Circumferential stress contours of the disk at different times of a flight cycle.
as illustrated in Fig. 12. The resulting zone definition presented in Fig. 12e is the basis for the following zone refinement considering a transient process and is thus a pre-zone. The process of pre-zone definition is actually zone generation based on a certain stress distribution. It is applicable to zone definition based on a steady-state stress distribution, which is presented in Sec. 4.5. 2) Stress difference analysis Stress difference analysis is performed for each initial zone. The stress difference for each zone varying with time in a flight cycle is acquired, and zones requiring refinement are identified.
3) Further zone refinement based on transient stress Zone refinement is conducted for zones that fail to fulfill stress similarity through a flight cycle. Each of these zones is further divided, and this subdivision process is carried out until zone definition principles (1) and (2) are fulfilled, as presented in Fig. 13. Special attention should be devoted to the fact that the number of zones increases significantly after performing zone refinement based on transient stress. A reason is that the geometric continuity requirement of zones needs to be satisfied by further subdividing zones. This “side effect” is a limit of zone definition that is automatically operated. Therefore, the proposed zone definition approach still needs to be improved in the future. The final zone
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Fig. 11. Circumferential stress of the disk hub varying with time in a flight cycle.
Fig. 12. Pre-zone definition based on stress at max rotation speed state.
definition shown in Fig. 13c is then passed to the following failure risk analysis.
4.3. Failure risk of the disk
Probabilistic failure risk analysis is conducted according to the zones generated in Sec. 4.2. The stress of the disk within a flight cycle is used to calculate the failure risk of the disk. The failure risk of the disk P f varying with flight cycle N is shown in Fig. 14. P f increases with an increase in N. Specifically, P f = 2.94E–11 at N = 20,000. Moreover, a certain cycle number N s exists, and when N < N s , P f = 0. In this case, the value of N s is about 1,500. The disk may fail after such a short service time (about 1,500 flight cycles) because of the effect of initial anomalies, although the occurrence of initial anomalies is low. Thus, the impact of initial anomalies on disk safety must be identified using probabilistic failure risk assessment.
4.4. Failure risk of hub zones The failure risk of the disk is the sum of the failure risks of all zones. To identify the most risky zones in the disk, this study considers the contribution of each zone to the disk and represents it by a risk contribution factor as follows:
y f, i =
P f, i Pf
(7)
where y f,i is the risk contribution of zone i, P f,i is the failure risk of zone i, and P f is the failure risk of the disk, namely, the sum of the failure risks of all zones. To identify the relative value of zone geometrical size, volume fraction y v is defined as
y v,i =
Vi V
(8)
where y v,i is the volume fraction of zone i, V i is the volume of zone i, and V is the volume of the disk, namely, the sum of the volumes of all zones.
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Fig. 13. Zone refinement considering transient stress.
Fig. 14. Failure risk of the disk varying with flight cycle.
Fig. 15 shows the risk contribution factors of the zones at the 20,000th cycle. As shown in Fig. 15a, the contour of risk contribution is similar to the stress contour, and the risk contributions
of zones in the hub region are higher than those of the other regions of the disk. This result may prove that the risk contribution of a zone is related to the stress level. For a quantitative analysis, the six zones with the highest risk contributions are identified from Fig. 15b. These six zones are 38, 33, 29, 36, 35 and 34, and they are all located in the hub region of the disk, as demonstrated in Fig. 16. Detailed information of the six zones is presented in Table 2. The six zones in the hub region occupy only 3.79% of the disk volume but contribute 66.54% to the disk failure risk. Specifically, the six zones at the hub contribute 17.56% risk per 1% volume, whereas the rest of the zones contribute only 0.35% risk per 1% volume. Accordingly, the zones at the hub are the most risky parts and probably strongly influence the failure risk of the entire disk. The results also show that zone type (inside or near-surface) and stress level are two key factors that decide the failure risk of a disk. For zones with a similar stress level, those in the near-surface region are more risky than those in the inside of the disk. For zones with the same type, stress level is a domi-
Fig. 15. Failure risk contributions of zones at the 20,000th flight cycle.
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Fig. 16. Six zones with the highest risk contribution factor. Table 2 Details of the six zones with the highest risk contribution factor.
a
Zone index
yv
Zone typea
σ (MPa)
yf
y f per 1% volume
38 33 29 36 35 34
0.34% 0.62% 2.36% 0.14% 0.20% 0.13%
sc ec ec sc ec sc
639.1 583.9 519.4 613.2 593.1 581.9
27.24% 13.88% 7.94% 7.72% 5.49% 4.27%
79.39% 22.56% 3.37% 55.98% 27.00% 33.81%
ec – embedded cracks; sc – surface cracks; cc – corner cracks.
nant factor, and the failure risk generally increases with the stress value. 4.5. Effects of the transient process on failure risk To study the effect of the transient process on the failure risk of the disk, the failure risk of the disk based on a steady-state design point is calculated, and the result is compared with that considering the transient process. Zone definition is performed based on the steady-state stress distribution at τ = 40 s. The process is consistent with that for the pre-zone in Sec. 4.2, and the final zone definition is presented in Fig. 17. Then, the failure risk of the disk based on a steady-state design point is obtained. For comparison, the failure risks of the disk considering the transient process and based on a steady-state design point are presented in one figure (Fig. 18). For convenience, subscripts “TR” and “DP” are used to refer to probabilistic failure risk assessment considering the transient process and based on a steady-state design point, respectively. The failure risk of the disk P f increases with an increase in N, and P f,TR is larger than P f,DP at the same flight cycle. Specifically, at N = 20000, P f,TR = 2.94E–11 and P f,DP = 7.90E–12. P f,TR is about 3.7 times the value of P f,DP . Cycle number N s exists for both cases, but the values are different. The values of N s,TR and N s,DP are about 1,500 and 2,500, respectively. These comparisons show that the failure risk of the disk is increased, and the disk may fail at an early flight number because of the effect of transient stress. Thus, the transient process greatly influences disk safety
Fig. 17. Zone definition based on stress at steady state.
Fig. 18. Failure risk of the disk considering the transient process and based on a steady-state design point.
and should be considered in the failure risk analysis for aeroengine disks. Given that the hub is the most risky part of the disk and contributes most to the failure risk of the entire disk, a zone at the
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above analysis implies that probabilistic failure risk assessment based on a steady-state design point does not address the influence of transient stress, which could exceed the steady-state stress, and may thus be insufficient to predict the failure risk of the disk accurately. The method of probabilistic failure risk assessment considering a transient process can obtain more accurate predictions of failure risk. Since the failure risk is one of the key design targets involving safety, the proposed failure risk assessment method can help achieve safer designs of aeroengine disks. 5. Conclusions
Fig. 19. Conditional failure probability of a zone at the disk hub considering the transient process and based on a steady-state design point.
disk hub is considered to further investigate the effect of the transient process on the failure risk. Figs. 13c and 17 show that although the zone definitions for probabilistic failure risk assessment considering the transient process and based on steady state are different, they have zones in common. Zone 38 in Fig. 13c and zone 34 in Fig. 17 are actually the same region of the disk. This zone contributes the most to the failure risk of the entire disk in both cases. Thus, this zone is analyzed, and the conditional probability of failure p d varying with flight cycle for both cases is presented in Fig. 19. p d increases with an increase in N, and p d,TR is larger than p d,DP at the same flight cycle. Specifically, at N = 20000, p d,TR = 0.41 and p d,DP = 0.14. p d,TR is about 2.9 times the value of p d,DP . The reason for this difference is the stress difference in the two cases. For probabilistic failure risk assessment considering the transient process, the stress is the peak stress in a flight cycle (i.e. σ TR = 639.1 MPa), whereas for probabilistic failure risk assessment based on a steady-state design point, the stress is the steady-state stress in a flight cycle (i.e. σ DP = 572.2 MPa). The value of σ TR is 11.7% larger than that of σ DP , and the failure risk is consequently raised to 190%. The failure risk is highly sensitive to the stress level. The reason for the above regulation can be explained by the analysis method of conditional probability of failure in Sec. 2.2. We let a0 be a random initial crack that exists in a certain zone. Fatigue crack growth analysis is performed for peak stress and steady-state stress cases, and the crack growth curves are illustrated in Fig. 20. Given that σ TR > σ DP , the crack growth rate under σ TR is faster than that under σ DP . As a result, af,DP > af,TR and p TR > p DP . Accordingly, because the peak stress during the transient process exceeds the steady-state stress, crack growth is accelerated, and the failure risk is consequently increased. The
On the basis of the safety demand for aeroengine disks, this work proposes a method of probabilistic failure risk assessment considering a transient process and studies the influence of the transient process on the failure risk of disks. An aeroengine compressor disk is analyzed. The temperature and stress evolutions of the disk during a simplified flight cycle are acquired through a numerical simulation based on FSI, and probabilistic failure risk assessment considering the transient process is performed to calculate the evolution of the failure risk of the disk. The result is compared with that obtained by probabilistic failure risk assessment based on a steady-state design point, and the influence of the transient process on the failure risk of the disk is investigated. The major results are summarized as follows. 1) The failure risk of the disk increases with flight cycle, and the hub is the most risky part of the disk that contributes the most to the failure risk of the entire disk. 2) Stress level and zone type (zone location in the disk) are the key factors that affect the failure risk. For the same type of zone, the failure risk increases with the stress level. For the same stress level, zones in the near surface are more risky than those in the internal region. 3) The failure risk of the disk considering the transient process is greater than that based on a steady-state design point, because the peak stress of the disk during the transient process exceeds the steady-state design point stress. To summarize, the method of probabilistic failure risk assessment considering a transient process can obtain more accurate predictions of failure risk. Although the analysis of this study is based on simplified treatments from the actual situations of real aeroengines, the proposed method provides new insights to assess the safety of aeroengine disks in design. Conflict of interest statement There is no conflict of interest regarding this journal article.
Fig. 20. Conditional failure probability analysis considering the transient process and based on a steady-state design point.
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