TRANSIENT STRESS ANALYSIS AND HIGH CYCLE FATIGUE LIFE ESTIMATION OF A GAS TURBINE SHROUDED HP COMPRESSOR BLADE
Vinayaka Nagarajaiah National Institute of Technology
Durgapur-713209, West Bengal, India [email protected] Fax No: +91-(343)-2543447
Nilotpal Banerjee National Institute of Technology
Durgapur-713209, West Bengal, India
B.S. Ajay Kumar
Bangalore Institute of Technology Bangalore, Karnataka, India
Kumar. K. Gowda Vivekananda Institute of Technology
Bangalore, Karnataka, India
Tulsidas Dalappa Vivekananda Institute of Technology
Bangalore, Karnataka, India
ABSTRACT This work is focused on developing a technique to assess high cycle fatigue of shrouded HP compressor blades subjected to thermo-mechanical loads like centrifugal stresses, vibratory stresses and thermal stresses in a gas turbine rotor. In practice, the blades are also subjected to resonance condition during steady or transient conditions of operation when passing through critical speeds. Hence, shrouds are added initially at 3/4 length along blade height which helps in reducing vibration amplitude by providing suitable stiffness to the blade and hence better structural integrity. Industrial best practice like Campbell diagram is used for the above purpose. Two approaches employed here for fatigue analysis are linear Finite element analysis (FEA) and Elasto-Plastic FEA. Fictive elastic results are recalculated using Neuber’s Rule. Strain amplitude approach is followed and Coffin-Manson Equation is used to determine the number of start-up and shut-down cycles. Design and analysis is performed using ANSYS 14.5 tool for reliable fatigue life estimation and to predict catastrophic failure due to High Cycle Fatigue. Keywords: High Cycle Fatigue, Shrouded HP Compressor, Gas Turbine Rotor, Campbell diagram, Neuber’s Rule, Coffin-Manson Equation, Fatigue Life Estimation.
INTRODUCTION Mechanical integrity of the Gas Turbine has been a key role in the design of turbines. Stresses generated in the turbines are not only due to centrifugal forces, but also due to many other factors including thermal loads, gas bending, fatigue loads and residual stresses. It is also seen that the moving blades after reaching a certain stage, undergo failure which happens due to and not restricted to hot corrosion, metallization in the blades, high cycle fatigue cracks, fractured edges and material voids. A solicitous design of the blades has to be done to meet the specific operational requirement. The turbine stages use internal cooling often whereas compressor stages have long pre-twisted blades to reduce the thermal effects. The critical regions in the compressor blades has to be found out and it has to be bolstered by the appropriate design changes.
Shrouds in the compressor blades increases both torsional and bending frequencies of the blades without materially affecting the overall weight[1]. Adding the shrouds to the system can make long blades stiffer. The shroud position, shroud angle and butting surfaces influence the system and sometimes weaken the structure.
Proceedings of the ASME 2016 Power Conference POWER2016
OBJECTIVES 1.Evaluate the mechanical integrity of the bladed disc considering the shrouds at 3/4 position and tip shroud. 2.Geometric and material non-linearity for simulations and life cycle estimation. 3.To conduct transient thermal analysis for the blades. 4.To draw Campbell diagram from the vibratory analysis of the models and check for resonance condition. 5.Fatigue life cycle evaluation using Coffin-Manson Equation to determine number of startup and shut down cycles.
NOMENCLATURE b=Fatigue strength exponent 휎 = Amplitude allowable stress 휎푓 = Stress at fatigue fracture when the material under zero mean stress cycled loading 휎 = Mean stress actual loading 휎 = Tensile strength of the material 휎 = Endurance strength R= Fatigue co-efficient HCF = High Cycle Fatigue LCF = Low Cycle Fatigue 휎 = Von-Mises Stress API = American Petroleum Institute D = Logarithmic Ductility G = Strength Co-efficient M = Ductility Co-efficient ∆∈ = Total strain ∆∈ = Plastic strain ∆∈ = Elastic strain Nf = Number of cycles E= Modulus of Elasticity CTE= Co-efficient of Thermal Expansion
HIGH CYCLE FATIGUE High cycle fatigue has low amplitude high frequency elastic strains. An airfoil subjected to repeated bending in the compressor and turbine blades are the best examples. High velocity gas pressure deflects the blades as the blade enters the gas path. The frequency of the blades are affected by the speed of the rotor. If the excitation at any point matches the blades natural frequency, it leads to catastrophic resonances. When the fatigue occurs above 103 cycles[3], it can be considered as the high cycle fatigue where the deformations in this type will be within the elastic limit. High cycle fatigue is found out by using stress life method where S-N curves are used to predict the life cycle sustained under stress before it fails [2]. The amplitude allowable stress for high cycle fatigue is given by Gerber and Goodman are given by:
휎 = 휎푓 1 − (1)
where: r=1, Goodman line which is close to the results of notched specimen r=2, Gerber parabola which better represent ductile metal. LOW CYCLE FATIGUE The applied load cycles to failure are small, when the cyclic stresses are in the neighborhood of the yield strength of the material. Early fatigue research showed that damage is dependent on plastic deformation. The total strain has elastic and plastic components.[5]
∆∈ = ∆∈ + ∆∈ (2) It can also be written as:
∆∈ = ∆ + ∆∈ (3)
∆∈ = 푀푁 + (푁 ) (4)
The M is governed by ductility and G is governed by strength. Murari.P.Singh.,etal [4] NEUBER’S RULE Material plasticity simulation in FEA are resource intensive. But the fatigue life and endurance analysis makes it expensive. In order to reduce the analysis time, Neuberisation of the results can be done to do Elasto-Plastic analysis.
Figure 1. Neuber’s hyperbola [3]
When the stresses are more than the gross yielding stress (880MPa), the material yields and gets into plastic region. Thus, linear analysis is counter intuitive. Hence, Neuber’s rule can be used to approximate the plastic stress strain state from linear analysis. A general extrapolation can be done from the linear graph from Neuber’s hyperbola to get the Neuberised stress. Neuber’s hyperbola is represented in the stress-strain curve in Figure 1. [3]
DESIGN SPECIFICATIONS Based on the explanation given above, the specifications considered for analysis are:
Temperature at blade tip= 5500C Environment temperature= 4300C Mechanical loading: Transient thermal loading
coupled with speed and fatigue Rated Speed: 10000 rpm 121% over speed as per API Static pressure of 0.1 bar Material: Ti-6Al-4V [6]
DESIGN CONSIDERATIONS Structural response under two different locations are captured to analyze the results. Gross yielding of the material is monitored to obtain stress values from Linear Analysis, Transient Analysis and Neuber’s rule[8]. A gradual load is applied to the specimen. Under this type of loading, the linear stresses are immutable, also thermal analysis is time dependent and urges for the transient analysis. But co-efficient of thermal expansion always changes with respect to temperature. Hence, the relation has been found and the same is used for the transient analysis.
CTE= (8.58+0.002287*T)* 10 - 6 (5)
An axiomatic 121% over-speed has been considered while designing the system. In additional to this, the material properties which can create uncertainty i.e grain boundaries, manufacturing uncertainties like transition and voids, calls for a reserved factor of safety as considered in equation.6 [3]
σ =( ) ∗( )
(6)
Since, the speed plays a vital role in the rotary components and centrifugal force is a function of mass, center of gravity height from the rotating axis and the square of the speed. Hence, 11% over speed due to full throw-off condition can increase the force up to 121% margin. Circumspection of the shroud relation to the force also leads to major design considerations. Upon introducing shroud at tip and at 3/4 height, the CG of the blade shifts upward which increases the force acting on the components and in-turn the stresses. Meanwhile, due to more area available for the stress distribution it reduces the overall stress acting on the blade and also plays a vital role with respect to the stiffness. This phenomenon is discussed in the later stages of this paper. Secluding the sector, the axi-symmetry conditions can be utilized to analyze the sector, i.e cyclic symmetry conditions, are given in the analysis and hence, reducing the
computational time and ensuring the mechanical integrity of the system. METHODOLOGY Here, the compressor blade is analyzed by taking an exemplar specimen with shrouds at tip and at 3/4 along blade height for a HP Compressor blade of 60mm blade height. Assuming 100% fixity conditions, the specimen is analyzed for the 4 cases as listed: Case 1: Tip shroud with straight butting Case 2: Tip shroud with angular butting Case 3: 3/4 shroud with straight butting Case 4: 3/4 shroud with angular butting Exploiting the axi-symmetric property of the model, only a sector is created and the cyclic symmetry condition is given to the exemplar. The blade and the shrouds are assumed to made of Ti-6Al-4V material with yield strength of 880MPa, Young’s Modulus of 113GPa and density 4430 Kg/m3. [6] CONVERGENCE CHECK Without checking the results to be convergent, it is not appropriate to conclude the results. Hence, in order to get the proper results the analysis has been tested for mesh converge for all cases. When the stress anomaly with respect to the mesh size stops, the mesh convergence is achieved. Figure.2 shows the results for case 3.
Figure 2. Stress variation with respect to mesh size For the frequency evaluation, the model without shroud is taken as reference and the system is solved in ANSYS, Matlab commercial packages and compared to theoretical calculations to bolster the claim of the results in the future research. The results obtained are as tabulated in Table.1 and showed close proximities with each other.
RESULTS AND DISCUSSION The Finite Element Analysis has been carried out with the loadings mentioned under design specifications. Only elastic strains are monitored. The resultant Von-Mises stress is found to be in the range of 789MPa to 798MPa.The peak stress is observed solicitously. It shows that the stresses are almost equal to the yield stress. But these can be negotiated because the Von-Mises stresses are maximum at the fixity conditions. As the peak stresses are within the elastic limit, there needs no evaluation using non-linearization of the stresses or bi-linear. It is appropriate to consider the sectional stresses along the blade which can be considered as the appropriate stress values. It is very interesting to observe that the failure of the material happens due to combined effect of the stresses in the plane, but not solely due to point stresses. Hence, sectional average stresses are considered to evaluate the results. The model is evaluated with the 50%, 100% and over-speed of 121% as per API(American Petroleum Institute) standard to ensure the stresses and increase the blade fatigue life.
Figure 3. Straight and Angular butting
Figure.3 shows the top view of the considered shroud butting configurations i.e straight and angular butting at both blade tip and 3/4 of blade height. When the centrifugal force is acting on the rotating blade, the butting surface tries to untwist it by exerting bending and torsional forces. In case of straight butting, area of contact is less, hence less thermal dissipation occurs. Whereas, angular butting provides more contact area and hence greater thermal dissipation. In angular butting there exist dislocations in the edge of contact region due to untwisting motion.
Case 1: Tip shroud with straight butting Here in this case the shroud is introduced at the blade tip with straight butting as shown in the Figure.4 below. The sectional stresses obtained are well within yield strength limit.
Figure 4. Sectional Stresses for Case 1 The stresses at the peak are trying to increase because of the reason that the shroud is introduced at blade tip. Here, AVG1 refers to the bottom plane and the AVG6 refers to the top most plane and the stress variations along the blade length is as shown in Figure.5
Figure 5. Variation of the sectional stresses
for Case 1 Case 2: Tip shroud with angular butting Figure.6 shows the average stresses obtained from FEA are less compared to the tip shroud. Consequently, as the shroud is twisted, the shroud tries to untwist itself due to large tension and creates less stresses at the blade tip compared to the straight butting case. Also, the angular butting surface increases the butting area to lock the shroud and helps in the distribution of the thermal gradients in the system due to more area compared to the straight butting.
for Case 2 Case 3: 3/4 shroud with straight butting Figure.7 shows the plot of variation of sectional stresses with respect to blade height till the shroud for 3/4 shroud positioning with straight butting. It can be clearly seen that there is no gross yielding in the blade sections. As the shroud is deceased in position about 3/4 the blade height, the stresses are reduced to about 20% compared to the tip shroud for the same configuration.
Figure 7. Variation of sectional stress for Case 3
Case 4: 3/4 shroud with angular butting Figure.8 shows the plot of variation of sectional stresses with respect to blade height till the shroud for case.4 i.e 3/4 shroud positioning with angular butting configuration. When the twist is given to the shroud, it tries to unravel itself and leads to the twisting reaction. It shows a sectional stress value of 119.644351 Mpa at blade root fillet and 45.8440003 Mpa at 40mm blade height near the shroud positioning.
Figure 8. Variation of sectional stress
for Case 4 Upon performing the sectional stress analysis of all the 4 test cases considered, now it calls for identifying the best configuration among them. Hence, a comparison plot of all the 4 test cases are plotted to conclude the obtained results.
Figure 9. Comparison of the sectional stresses of all the 4 cases considered.
Figure.9 shows the comparison plot of the sectional stresses with respect to blade height for all the 4 test cases considered. Upon comparison it can be concluded that case.3 i.e 3/4 shroud with straight butting configuration provides the least stress among all 4 test cases, which is 117.764366 Mpa at the blade root fillet and 43.6192141 Mpa at 40mm blade height near the shroud positioning.
COLLECTIVE RESULTS FROM THE FINITE ELEMENT ANALYSIS All the 4 test cases were subjected to transient analysis and their respective simulations results are tabulated in Table.2 and Table.3. The maximum principal stress occuring at the fillet area for case 2 i.e Tip shroud with angular butting obtained from the analysis is 609.7 MPa and is as shown in Figure.10 below.
Figure 10. Maximum Principal Stress in stage blade rows for Case 2- tip shroud with angular butting Table.2 Transient analysis results for tip shroud
Results of Transient analysis for Tip shroud Entity Straight Angled
Number of blades 60 60 First Principal stress 584.25 MPa 609.7 MPa
Second Principal stress 479.74 MPa 513.99 MPa Third Principal stress 85.80 MPa 183.41 MPa Equivalent Von-Mises
stress 786.62 MPa 798.85 MPa Total Deformation 1.6695 mm 1.6865 mm
Life from Strain life 1.9561 x e7 1.053 x e7 Table.3 Transient Analysis results for 3/4 shroud
Results of Transient analysis for 3/4 shroud Entity Straight Angled
Number of blades 60 60 First Principal stress 586.42 MPa 863.91 MPa
Second Principal stress 481.32 MPa 491.41 MPa Third Principal stress 87.55 MPa 287.81 MPa Equivalent Von-Mises
Stress 789.2 MPa 790.55 MPa Total Deformation 1.6695 mm 1.6865 mm Life from strain life 1.7316 x e7 1.8175 x e7
It has been seen that the angular stresses are more compared to the straight for the reason that the twisting tries to untwist the material and urges to come back to its original position during the force action. Principal stresses at the fillet are
tenuous, but are well within the elastic limits. Eventually the stresses are mitigated as it goes towards the tip of the blade. BLADE METAL FATIGUE When the material is subjected to several start up and shut down cycles, at the end of each cycle, the residual stresses are accumulated in the material which sap the energy in the material eventually and leads to catastrophe. Due to this reason, fatigue analysis is performed here to calculate the life of the blade using Coffin-Manson method. COFFIN-MANSON METHOD Coffin-Manson have derived a relationship between fatigue life and total strain which is as follows:
∆ = F (2N)b + ε′F(2N)c (7) Strain life parameters considered for Ti-6Al-4V are: Strength coefficient, σ′F = 1737 MPa Strength exponent, b = -0.085 Ductility coefficient, 휀′F = 0.396 Ductility exponent, c = -0.684 Cyclic Strength Co-efficient = 1938 MPa ΔE is total principal strain obtained from FEA simulation Using the above equation.7, the calculated fatigue life for all the considered 4 test cases are tabulated in Table.4. The results show that fatigue life for all the cases is above 107 cycles and 3/4 shroud with straight butting i.e case.3 gives the highest fatigue life among the considered test cases and its value is 1.8464741 x 107 cycles and is the most reliable structural configuration. Table.4 Fatigue life result table from Coffin-Manson equation
Test Case Fatigue life cycles Case 1 - Tip shroud with straight butting 1.6418260 x 107 Case 2 - Tip shroud with angular butting 1.6536409 x 107
Case 3 - 3/4 shroud with straight butting 1.8464741x 107 Case 4 - 3/4 shroud with angular butting 1.7270823 x 107
NEUBERISATION RESULTS The equivalent stress of the material is well within the elastic range, hence the analysis is run for 121% over-speed. The results cross the elastic limit and reaches the plastic zone. To ensure the safety the over-speed stresses are Neuberised, whose values are tabulated in Table.5
1156.64 949.67 VIBRATIONAL ANALYSIS Rotating components especially in the turbo-machinery components, the external excitations caused in the system is very widespread. In order to understand the vibrational behaviour of the blades, as the gas passes over the compressor, it creates disturbance in the system [7]. This acts like external excitations to the rotating blades. When the excitation frequency (X=N/60) is equal to the natural frequency of the blade, resonance occurs. In order to study these frequencies of blade with respect to operating speed, separation margins and over-speed, Campbell diagram has been plotted for all the 4 test cases. The separation margins considered are left margin which is 5% towards left and right margin 5% towards right of operating speed and 121% over-speed line as shown in all the Campbell diagrams plotted. The above considerations are made because compressors may or may not run at full speed always and should be able to work even at 121% over-speed condition without any blade resonance. In practice usually the blades are checked for first 6X powerful engine order excitations to ensure the best structural performance at design and off-design conditions without any resonance.
Figure 11. Campbell diagram for Case 1 (Tip shroud, straight butting)
Figure 12. Campbell diagram for Case 2 (Tip shroud, angular butting)
Figure 13. Campbell diagram for Case 3 (3/4 shroud, straight butting) Figures 11 and 12 i.e case 1 and 2 show that all the excitations from 1X to 6X pass through modal frequencies without any resonance. The 4X excitation just passes through the right separation margin of 5%. Hence, the blade is safe from resonance for these configurations. Figure.13 i.e case 3 shows that all the excitations from 1X to 6X pass through modal frequencies without any resonance. The 5X passes inside excitation region at 2.0% separation from the operating speed intersection line. Since the blade may be subjected to resonance at 5X excitation at mode 1 natural frequency intersection point, it is considered for further harmonic analysis to test for its additional factor of safety.
(3/4th shroud, angular butting) Figure.14 i.e case 4 shows that all the excitations from 1X to 6X pass through modal frequencies without any resonance. The 4X excitation just passes through the right separation margin of 5%. Hence, the blade is safe from resonance for this configuration. HIGH CYCLE FATIGUE ANALYSIS It is seen from Figures 11, 12, and 14 that the 5X excitation passes outside the separation margin but in the Figure.13 i.e case.3, 5X excitation is intersecting at the Mode 1 natural frequency line in the exciter box region, which may cause possible catastrophe. Hence, high cycle fatigue analysis is done in order to check the status of the possible catastrophe. A Goodman Diagram can be used to find high cycle fatigue. Here, endurance stresses are plotted along y-axis v/s yield stresses along x-axis. In this a factor of safety of 1.5 is taken and a parallel line to the Goodman gives the allowable range as shown in Figure.15. When the natural frequency of the blade is equal to the forcing frequency, resonance may occur which can cause the system to be tenuous. Hence, the 1% stimulus or 1% gas loads is considered and the harmonic analysis is performed. Reason being wake forces emerging from the blades acts as the forcing frequency, which may not be more than 1% of stimulus. From harmonic analysis, at 625Hz range, the captured maximum stress is found out to be 12.8MPa, meanwhile the stress corresponding to the maximum harmonic in transient is found to be 49.78 MPa. The additional factor of safety (y/x) obtained is 2.41. Hence, structurally the system is safe. Also, it is very important to note that the allowable line has a fatigue cycle of 107 start up and shut down cycles and hence the life of the system is of order 107cycles. The high cycle fatigue diagram obtained is shown in the Figure.15
Figure 15. High Cycle Fatigue diagram
CONCLUSIONS 1. Mechanical integrity of the system is evaluated using transient analysis coupled with structural. 2. Campbell diagram shows that the blade is safe with an exception for case 3. But the case 3 coupled with harmonic analysis and high cycle fatigue results showed that the blade is safe even when there may be a resonance at 5X excitation. 3. Fatigue life evaluation results shows that the results are converging to a point of 107 cycles from Ansys and Coffin-Manson Equation. 4. Sectional stress shows the variation of the stresses and pose no danger to the mechanical integrity of the system. 5. Eclectic result evaluation shows that the case.3 i.e 3/4 shroud with straight butting is the best blade configuration in terms of mechanical integrity, having the highest fatigue life cycles, with no resonance at operating speed of 10000 rpm and has reduced and distributed stress at the blade root fillet and near the shroud positioning with more area to dissipate the heat thermally. 6. It is also to be noted that the airfoil shape can also give the same results as the results are axiomatic in nature. REFERENCES [1] Aircraft Propulsion System Technology and Design, 1989, Gordon C. Oates, AIAA, University of Washington Seattle, Washington. pp. 487-500 [2] Understanding Fatigue, D.P DeLuca, Global Gas Turbine News, 41: 2001, pp.7-10 [3] Tulsidas.D, Shantaraja.M, Bharath.VG, Life estimation of a steam turbine blade using low cycle fatigue analysis, Procedia Materials Science, Vol. 5, (2014 ), pp. 2392 – 2401 [4] Mahesh Shankar, K. Kumar, S.L. Ajit Prasad, 2010, T-Root blades in a steam turbine rotor: A case study, Engineering Failure Analysis, Vol. 17, pp. 1205–1212
[5] P. Mestanek, 2008, Low cycle fatigue analysis of a last stage steam turbine blade, Applied and Computational mechanics, Vol. 2, pp. 71-82 [6] B.T. Lebele- Alawa, H.I. Hart , S.O.T. Ogaji and S.D. Probert, 2008, Rotor Blades profile influence on a gas turbine’s compressor effectiveness, Journal of Applied Energy, Vol.85, pp.494-505. [7] J.S Rao, 1991, Turbomachine Blade Vibration, John Wiley and Sons, Inc, NewYork. [8] Vyas, N.S, Gupta.K and J.S. Rao, 1987,Transient Response of Turbine Blade, Proc. 7th World Congress, IFToMM, Sevilla, Spain, pp.697
