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International Journal of Turbo & Jet-Engines

Ed. by Sherbaum, Valery / Erenburg, Vladimir

IMPACT FACTOR 2018: 0.863

CiteScore 2018: 0.66

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Source Normalized Impact per Paper (SNIP) 2018: 0.625

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2191-0332
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Volume 33, Issue 1

Fatigue Life Analysis of Turbine Disks Based on Load Spectra of Aero-engines

Yan-Feng Li
• Institute of Reliability Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China
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• Other articles by this author:
/ Zhiqiang Lv
• Institute of Reliability Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China
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/ Wei Cai
• Institute of Reliability Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China
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/ Shun-Peng Zhu
• Institute of Reliability Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China
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/ Hong-Zhong Huang
• Corresponding author
• Institute of Reliability Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China
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Published Online: 2015-03-10 | DOI: https://doi.org/10.1515/tjj-2015-0004

Abstract

Load spectra of aero-engines reflect the process of operating aircrafts as well as the changes of parameters of aircrafts. According to flight hours and speed cycle numbers of the aero-engines, the relationship between load spectra and the fatigue life of main components of the aero-engines is obtained. Based on distribution function and a generalized stress–strength interference model, the cumulative fatigue damage of aero-engines is then calculated. After applying the analysis of load spectra and the cumulative fatigue damage theory, the fatigue life of the first-stage turbine disks of the aero-engines is evaluated by using the SN curve and Miner’s rule in this paper.

1 Introduction

The remainder of this paper is structured as follows. Section 2 reviews several deterministic fatigue life prediction methods that would be used in the following sections and then introduces the procedure of the fatigue life analysis. In Section 3, by analyzing the measured speed cyclic matrix of an aero-engine, the number of the main work cycles of the aero-engine is obtained. In Section 4, based on the cracks in turbine disk mainly caused by low cycle fatigue failures, the loading stress of the turbine disk is analyzed. In Section 5, by fitting the material S–N curve, the S–N curve of the turbine disk is obtained. In Section 6, by using the S–N curve of the turbine disk and Miner’s rule, the life of the turbine disk is predicted. In the last section, the conclusions are drawn.

2 Fatigue life prediction methods

According to the irreversibility and randomness of fatigue damages, fatigue life prediction methods can be classified into two groups: deterministic fatigue life prediction methods and probabilistic fatigue life prediction methods [3, 1012]. In this section, the most commonly used deterministic fatigue life prediction methods are reviewed.

2.1 Stress-based fatigue life prediction methods

Stress-based fatigue life prediction method is one of the earliest developed methods used for fatigue life prediction. The S–N curve life prediction is a classic example [4, 9]. Basquin formula is a common fatigue formula used in engineering, which can be expressed as follows [3]: ${\mathrm{\sigma }}_{f}^{{\prime }^{}}\left(2{N}_{f}{\right)}^{b}=\frac{E\mathrm{\Delta }{\mathrm{\epsilon }}_{e}}{2}={\mathrm{\sigma }}_{a}$(1)where ${\mathrm{\sigma }}_{f}^{{\prime }^{}}$ is the fatigue strength coefficient, ${N}_{f}$ is the fatigue life; $b$ is the fatigue strength exponent, $E$ is the modulus, $\mathrm{\Delta }{\mathrm{\epsilon }}_{e}$ is the elastic strain range and ${\mathrm{\sigma }}_{a}$ is the stress amplitude.

Due to the stress concentration, the stress-based fatigue life prediction method can be further divided into nominal stress method, hot spot stress method and notch stress method.

In this paper, the nominal stress method is used to calculate the low cycle fatigue life of the aero-engines. Material S–N curves are derived from material data and determined by the stress concentration factors. Also, it should be noted that different stress concentration factors correspond to different S–N curves [5].

2.2 Strain-based fatigue life prediction methods

For a heavy service load, during the operation of a component, it is better to use strain–life curve to predict the life of the component. In the 1950s, a second common and more recent approach based on a local strain was established. Manson–Coffin’s formula is one of the most commonly used fatigue life prediction methods in engineering. Manson–Coffin’s formula describes a relationship between the plastic strain and the fatigue life as follows: $\frac{\mathrm{\Delta }{\mathrm{\epsilon }}_{p}}{2}={\mathrm{\epsilon }}_{f}^{\prime }\left(2{N}_{f}{\right)}^{c}$(2)where $\mathrm{\Delta }{\mathrm{\epsilon }}_{p}$ is the elastic strain range, ${\mathrm{\epsilon }}_{f}^{\prime }$ is the fatigue ductility coefficient, ${N}_{f}$ is the fatigue life and $c$ is the fatigue ductility exponent. From eq. (2), it is obvious that $\mathrm{\Delta }{\mathrm{\epsilon }}_{p}$ determines the value of ${N}_{f}$.

Mason and Hirschberg [6] put forward an improved formula to express the fatigue life by considering the total strain range: $\frac{\mathrm{\Delta }{\mathrm{\epsilon }}_{t}}{2}=\frac{{{\mathrm{\sigma }}^{\prime }}_{f}}{E}\left(2{N}_{f}{\right)}^{b}+{\mathrm{\epsilon }}_{f}^{\prime }\left(2{N}_{f}{\right)}^{c}$(3)where $\mathrm{\Delta }{\mathrm{\epsilon }}_{t}$ is the total strain range, ${\mathrm{\sigma }}_{f}^{\prime }$ is the fatigue strength coefficient, $E$ is the modulus, ${N}_{f}$ is the fatigue life, ${\mathrm{\epsilon }}_{f}^{\prime }$ is the fatigue ductility coefficient, $b$ is the fatigue strength exponent and $c$ is the fatigue ductility exponent.

2.3 Miner’s rule

Suppose that there are L levels of stress in the load spectra [7, 8]. They are denoted as ${S}_{1}$, ${S}_{2}$, $\cdots$, ${S}_{L}$. The number of cycles of each level is denoted as ${n}_{1}$, ${n}_{2}$, $\cdots$, ${n}_{L}$, and the numbers of the destruction cycles of each level are denoted as ${N}_{1}$, ${N}_{2}$, $\cdots$, ${N}_{L}$. Miner’s rule assumes that the fatigue damage can be obtained by corresponding circulation ratios. The equation is given as follows: $\sum _{i=1}^{L}\frac{{n}_{i}}{{N}_{i}}=1$(4)where L is the number of stress levels, ${n}_{i}$ is the number of cycles at a given stress level, which can be got from load spectra, ${N}_{i}$ is the number of cycles to failure at a given stress level.

A procedure of the fatigue life analysis used in this paper is shown in Figure 1.

Figure 1:

A procedure for the fatigue life analysis of the turbine disk.

First, the S–N curve of the structure should be known. In this paper, the S–N curve of the structure is obtained by fitting the material data. Second, based on the structure analysis, the nominal stress of each cycle can be represented by $S$, ${S}_{a}$, ${S}_{m}$ and ${S}_{ai}^{\ast }$. Then the equivalent damage corresponding to each cycle can be calculated. Finally Miner’s rule is used to get the fatigue damage and life of the turbine disk.

3 Analysis of speed cyclic matrix data

The measured speed cyclic matrix of an aero-engine of an aircraft is shown in Table 1. The main work cycles can be calculated by analyzing the data in Table 1. Because the data are counted in 2,000 hours many times through the same flight mission, the number of cycles may not be integer. From Table 1, the number of speed cycles of the engines is obtained. Peaks can be defined as four states: take off (>97%), rated take off (94.5–97%), valleys are zero and the ground idle. The numbers of cycles of “0-take off-0,” “ground idle-take off-ground idle,” “0-rated take off-0” and “ground idle-rated take off-ground idle” are 500.1, 304.8, 221.2 and 420.4. The total number of cycles “0-97%” and “94.5-97%-0” is 721.3, which reflects the flight subjects starting from switch off to take off. The number of cycles of “ground idle-take off state-ground idle” is 725.2, which reflects the flight subjects of landing (but not switching off) to take off again. The number of cycles of “flight idle-take off state-flight idle” is high, which reflects the low frequency using this cycle. Then, the number of the main work cycles of the aero-engine can be obtained and are shown in Table 2.

Table 1:

N2 cyclic matrix (times, standardization to 2,000 hours).

Table 2:

The number of main work cycles.

In this section, the first-stage turbine disk is regarded as objects and their fatigue life is estimated. According to actual turbine disk fracture and related metallographic analysis, the cracks in turbine disk are mainly caused by low cycle fatigue failures.

As shown in Table 1, the centrifugal forces are the main load, which are considered in this paper.

Assuming that the blade’s centrifugal forces are evenly distributed on the contact surface of the mortises of the turbine disk, the centrifugal forces are expressed as follows: $F=m{\mathrm{\omega }}^{2}r$(5)where $m$ is the mass of the blades, $r$ is the radius of the center of mass of the blades and $\mathrm{\omega }$ is the angular velocity of the turbine disk. It means that the loading stress of the turbine disk is proportional to the square of the angular velocity.

5 The fitting of S–N curve of the first-stage turbine disk

In this paper, the material S–N curve is used to estimate the low cycle fatigue life of the turbine disk for a rough calculation to verify the feasibility of this method. The material used in the test is GH4698 super alloy. The mechanical and fatigue properties of the tested material under the room temperature are listed in Tables 3 and 4, respectively. Considering the influence of stress concentration on turbine disk, the notched fatigue specimen and notched radius are chosen as follows: $\mathrm{\rho }=0.5\phantom{\rule{thickmathspace}{0ex}}\mathrm{m}\mathrm{m}$, ${K}_{t}=2.33$ for high cycle fatigue; and ${\mathrm{\rho }}_{H}=0.25\phantom{\rule{thickmathspace}{0ex}}\mathrm{m}\mathrm{m}$, ${K}_{t}=3.35$ for low cycle fatigue.

Table 3:

Mechanical properties of a GH4698 super alloy.

Table 4:

Fatigue strength of vacuum smelting plus vacuum arc remelting forging.

According to the material’s fatigue limit ${S}_{\mathrm{\infty }}=1.18$ $×{10}^{8}$ Pa, the cycle in the experiment will not cause damage when the stress is less than or equal to 118 MPa. Through the curve fitting, $A=17.9056$, $\mathrm{\alpha }=0.1945$. Then, the S–N curve can be expressed as follows: $S=1.18\phantom{\rule{thickmathspace}{0ex}}×\phantom{\rule{thickmathspace}{0ex}}{10}^{8}\left(1+\frac{17.9056}{{N}^{0.1945}}\right)$(6)The S–N curve is shown in Figure 2 [13].

Figure 2:

The S–N curve of a turbine disk.

6.1 Estimation of cyclic stress

The high-pressure rotor speed (${N}_{2}$) cyclic matrix is shown in Table 1. Assuming that the nominal stress of the first-stage turbine disk is S, it is proportional to the square of the speed ${N}_{2}$: $S=k×{N}_{2}^{2}$(7)where k is the conversion factor. Based on the above assumption, the cyclic stress S is shown in Table 5.

6.2 Estimation of symmetric cyclic stress

After getting the cyclic stress, from the cyclic matrix we can obtain the cyclic stress amplitude ${S}_{a}$ as shown in Table 5. After applying the modified Goodman diagram and the S–N curve to the experiment data, the stress can be converted into symmetric cyclic stress ${S}_{a}^{\ast }$, the modified equation can be expressed as ${S}_{a}^{\ast }=\frac{{S}_{a}}{\left(1-\frac{{S}_{m}}{{S}_{b}}\right)}=\frac{{S}_{a}{S}_{b}}{{S}_{b}-{S}_{m}}$(8)where ${S}_{a}^{\ast }$ is the symmetric cyclic stress, ${S}_{a}$ is the amplitude of stress, ${S}_{m}$ is the mean stress, ${S}_{b}$ is the ultimate strength of the material. To simplify the calculation, we define that ${S}_{b}=878\mathrm{M}\mathrm{P}\mathrm{a}$ and the maximum stress of first-stage turbine disk is ${S}_{max}=7.885×{10}^{8}\mathrm{P}\mathrm{a}$. Then, we can get the corresponding symmetrical cyclic stress, as shown in ${S}_{ai}^{\ast }$ of Table 5.

6.3 Estimation of cumulative damage

The life curve is usually described by the following equation, which is called Goodman diagram: ${\mathrm{\sigma }}_{a}={\mathrm{\sigma }}_{a0}\left(1-\frac{{\mathrm{\sigma }}_{m}}{{\mathrm{\sigma }}_{m0}}\right)$(9)where ${\mathrm{\sigma }}_{a}$ is the stress amplitude, ${\mathrm{\sigma }}_{a0}$ is the stress amplitude when the mean is 0, ${\mathrm{\sigma }}_{m}$ is the mean stress and ${\mathrm{\sigma }}_{m0}$ is the mean stress when the amplitude is 0.

The stress cycles of each level can be represented by $\left({\mathrm{\sigma }}_{ai},{\mathrm{\sigma }}_{mi}\right)$ or $\left({\mathrm{\sigma }}_{maxi},{R}_{i}\right)$. Generally, $\left({\mathrm{\sigma }}_{maxi},{R}_{i}\right)$ is accepted in engineering and ${\mathrm{\sigma }}_{maxi}$ can be shorted by ${\mathrm{\sigma }}_{i}$. According to Goodman diagram, $\left({\mathrm{\sigma }}_{i}^{\ast },{R}_{i}\right)$, which corresponds to $\left({\mathrm{\sigma }}_{i},{R}_{i}\right)$ and called symmetrical cyclic stress, can be calculated by the following equation: ${{\mathrm{\sigma }}_{i}}^{\ast }=\frac{\left(1-{R}_{i}\right){\mathrm{\sigma }}_{m0}{\mathrm{\sigma }}_{i}}{{\mathrm{\sigma }}_{m0}\left(1-{{R}_{i}}^{\ast }\right)+{S}_{i}\left({R}^{\ast }-{R}_{i}\right)}$(10)Then S–N curve and eq. (10) are used to calculate the fatigue life which corresponds to $\left({\mathrm{\sigma }}_{i},{R}_{i}\right)$, the fatigue life is ${N}_{i}={\left(\frac{{S}_{\mathrm{\infty }}A}{{{\mathrm{\sigma }}_{i}}^{\ast }-{S}_{\mathrm{\infty }}}\right)}^{\frac{1}{\mathrm{\alpha }}}$(11)If ${S}_{i}$ is known, the corresponding fatigue life can be calculated by ${N}_{i}={\left(\frac{A{S}_{\mathrm{\infty }}}{{S}_{i}-{S}_{\mathrm{\infty }}}\right)}^{\frac{1}{a}}$(12)Substituting parameters into eq. (12) with ${S}_{\mathrm{\infty }}=1.18×$ ${10}^{8}\mathrm{P}\mathrm{a}$, $A=17.9056$, $\mathrm{\alpha }=0.1945$ ${N}_{i}={\left(\frac{2.1128608×{10}^{9}}{{S}_{i}-1.18×{10}^{8}}\right)}^{5.141388}$(13)The corresponding cyclic life by eq. (13) is shown in Table 5.

Then by using the Miner’s rule, we substitute parameters into eq. (12) ${S}_{\mathrm{\infty }}=1.18×{10}^{8}\mathrm{P}\mathrm{a}$, $A=17.9056$, $\mathrm{\alpha }=0.1945$, and instead of ${S}_{i}$ by ${S}_{a}^{\ast }$, the ${D}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ is $\begin{array}{c}{D}_{\text{total}}=\sum _{i=1}^{k}\frac{{n}_{i}}{{N}_{i}}\\ =\sum _{i=1}^{k}\frac{{n}_{i}}{{\left[\frac{2.1128608×{10}^{9}}{{S}_{ai}^{*}-1.18×{10}^{8}}\right]}^{5.141388}}\end{array}$(14)Cyclic life ${N}_{i}$ and equivalent damage ${D}_{i}$ are shown in Table 5, and ${D}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=0.018471$.

Table 5:

Fatigue limit and damage based on the rotor speeds of aero-engines.

According to the calculation, the cumulative fatigue damage is 0.018471 after 2,000 flight hours. It indicates that although the main failure mechanisms of the turbine disc are low cycle fatigue, the total damage of the turbine disk is small which is caused by the centrifugal forces in 2,000 flight hours. It should be noted that, in the operation of the turbine disk, it also endures high thermal stresses. But because of the lack of experimental data, in this paper, only the centrifugal forces are used to estimate the life of the turbine disk. So we will do further study about the life of the turbine disk in the future.

7 Conclusions

In this paper, based on the actual measured load spectra speed cyclic matrix, the fatigue life of the first-stage turbine disk was estimated. The experimental results demonstrated that the centrifugal forces are the main load of the turbine disk. Also, it is clear that the S–N curve and Miner’s rule can be used to estimate the fatigue life of the turbine disk. Similarly, this method can be used to estimate the fatigue life of the other components of the aero-engine. But it should be noted that, when just considering the centrifugal forces of the loading stress, the total damage of the turbine disk is small. In order to have a high precision of fatigue life estimation of the turbine disk, more experimental data about the loading stress are need. Also, the life estimation method can be further improved if there are more accurate parameters.

References

• 1. Su QY. Guide of determining life of aviation turbojet, turbofan engine main parts. Beijing: Aviation Industry Press, 2004. Google Scholar

• 2. Raiher VL. Some consequences from the two parameters model of durability scatter. Uchenve Zapiski CAHI 1982;13:130–3. Google Scholar

• 3. Zhu SP. Research on hybrid probabilistic physics of failure modeling and fatigue life estimation of high-temperature structures. Ph.D. Dissertation, University of Electronic Science and Technology of China, 2011.

• 4. Cui W. A state-of-the-art review on fatigue life prediction methods for metal structures. J Mar Sci Technol 2002;7:43–56. Google Scholar

• 5. Yao WX. The fatigue life analysis of structure. Beijing: National Defense Industry Press, 2003. Google Scholar

• 6. Mason SS, Hirschberg MH. Fatigue: an interdisciplinary approach. Syracuse, NY: Syracuse University Press, 1964. Google Scholar

• 7. Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;12:159–64. Google Scholar

• 8. Hashin Z, Rotem A. A cumulative damage theory of fatigue failure. Mater Sci Eng 1978;34:147–60. Google Scholar

• 9. Wirsching PH. Fatigue reliability for offshore structures. J Struct Eng 1984;100:2340–56. Google Scholar

• 10. Zhu SP, Huang HZ, Wang ZL. Fatigue life estimation considering damaging and strengthening of low amplitude loads under different load sequences using fuzzy sets approach. Int J Damage Mech 2011;20:876–99.

• 11. Zhu SP, Huang HZ, Ontiveros V, He LP, Modarres M. Probabilistic low cycle fatigue life prediction using an energy-based damage parameter and accounting for model uncertainty. Int J Damage Mech 2012;21:1128–53.

• 12. Zhu SP, Huang HZ, He LP, Liu Y, Wang Z. A generalized energy-based fatigue-creep damage parameter for life prediction of turbine disk alloys. Eng Fract Mech 2012;90:89–100.

• 13. Fu N. An aircraft engine turbine disk and blade strength analysis and calculation of life. M.D. Dissertation, Northwestern polytechnic University, 2006.

Accepted: 2015-02-08

Published Online: 2015-03-10

Published in Print: 2016-04-01

Funding: The authors would like to acknowledge the partial support provided by the National Natural Science Foundation of China under contract number 11272082 and the Fundamental Research Funds for the Central Universities under contract number E022050205.

Citation Information: International Journal of Turbo & Jet-Engines, Volume 33, Issue 1, Pages 27–33, ISSN (Online) 2191-0332, ISSN (Print) 0334-0082,

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