NASTRAN Analysis of a Turbine Blade and Comparison with Test and Field Data

Transcription

1 75-GT-77 I Copyright 1975 by ASME $3. PER COPY author(s). $1. TO ASME MEMBERS The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printeo only if the paper is published in an ASME journal or Proceedings. Released for general publication upon presentation. Full credit should be given to ASME, the Technical Division, and the NASTRAN Analysis of a Turbine Blade and Comparison with Test and Field Data J. M. ALLEN Fellow Engineer L. B. ERICKSON Engineer Westinghouse Electric Corp., Gas Turbine Systems Division, Engineering Department, Eddystone, Pa. A NASTRAN finite element analysis of a free standing gas turbine blade is presented. The analysis entails calculation of the first four natural frequencies, mode shapes, and relative vibratory stresses, as well as deflections and stresses due to centrifugal loading. The stiffening effect of the centrifugal force field was accounted for by using NASTRAN's differential stiffness option. Natural frequencies measured in a rotating test correlated well with computed results. Areas of maximum vibratory stress (fundamental mode) coincided with the three zones of crack initiation observed in a metallographic examination of a fatigue failure. Airfoil stress distributions were found to be significantly different from that predicted by generalized beam theory, especially near the airfoil-platform junction. Contributed by the Gas Turbine Division of The American Society of Mechanical Engineers for presentation at the Gas Turbine Conference & Products Show, Houston, Texas, March 2-6, Manuscript received at ASME Headquarters December 2, 197. Copies will be available until December 1, THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, UNITED ENGINEERING CENTER, 35 EAST 7th STREET, NEW YORK, N.Y. 117

2 NASTRAN Analysis of a Turbine Blade and Comparison with Test and Field Data J. M. ALLEN L. B. ERICKSON INTRODUCTION One of the most significant advances in solid mechanics in recent years is the development of finite element computer programs. One of the most powerful and best known codes is NASTRAN (1, 2), l It is especially well suited for turbine blade analysis because of its capacity to handle dynamic problems with many degrees of freedom, and its capability to include the stiffening effect of centrifugal force (). NASTRAN became a part of the Westinghouse computer system in 197. Since then it has been used extensively in the design and analysis of steam and gas turbine blades. In this paper, results of an analysis of a free standing gas turbine blade are reported, and correlations made using test data and field failure experience. A number of comparisons between NASTRAN and generalized beam theory are made to show some of the limitations in the traditional method. CF STRESSES AND DEFLECTIONS Stresses and deflections were calculated at the design speed of 36 rpm, To account for the stiffening effect of rotation, the differential stiffness feature in NASTRAN was em- FINITE ELEMENT MODEL The subject of this analysis is the last low blade of the Westinghouse W51 family of gas turbines. The airfoil is approximately 18 in. (5.7 cm) high, and has a nearly constant chord of 6.5 in. (16.5 cm), Fig. 1. A proprietary, pre-processer, mesh generator program was used to generate the finite element model. 6 elements, connected at 26 grid (nodal) points, were used to approximate the blade geometry, Fig. 2. The element employed is identified in NASTRAN as CTRIA2 (1) and is a triangular, flat shell element with both inplane (membrane) and bending stiffness. Fig. 2 is a tengential projection of the middle surface of the elements. Fig. 3 is a cross-sectional view of the model near the base of the airfoil. 1 Numbers in parentheses designate References at end of paper. Fig. 1 W51 last row turbine blade 2

3 X-COORDINATE, CM E AIRFOIL X COORDINATE, INCHES Fig, 3 Element model of airfoil hub cross section } -PLATFORM SHANK Fig. 2 NASTRAN finite-element model mid surface projection ployed. Differential stiffness is a first order approximation to large deformation effects, such as those considered in beam-column action. Briefly, the procedure is to first solve the problem assuming no effect of deflection on loading. Then, using the resulting displacements normal to the plate elements and the membrane forces in each element, a linearized first approximation of the stiffening influence of the membrane forces is obtained in the form of a differential stiffness matrix. It is assumed that the applied loads from which the differential stiffness is derived remain fixed in magnitude and direction during motion of the structure, and that the points of application move with the structure. The elements of the differential stiffness matrix are added to the corresponding elements in the basic stiffness matrix and the problem solved again. This is the only iteration made, though some increase in accuracy could be obtained by additional repetitions of the process. However, it is doubtful that this increase in accuracy is enough to justify the increase in computer costs. All displacements, stresses, and natural frequencies referred to in this paper are based on the differential stiffness method. Boundary conditions are such that the six degrees of freedom are suppressed for the eleven nodal points along the bottom row of the model, with the exception of the translation displacement along this line. Here only the nodal point on the leading edge was fixed from moving in this direction. In other words, the blade is clamped where the blade root and disk steeple first come into contact, but the root is free to move in the direction of the groove from a fixed point on the leading edge, The remaining grid points in the model are each allowed six degrees of freedom. Airfoil displacements due to cf loading are shown in Fig.. The overall distortion may be characterized as a general bending and untwisting of the airfoil. While the untwisting is relatively small in this instance, 1. deg at the tip, this untwisting and/or local distortion of airfoil shape can be an important design consideration in long, thin, airfoils under high cf fields. Cf maximum principal stress patterns are shown in Fig. 5. These are Calcomp plots created by running the NASTRAN stress output through a proprietary, post-processor, plotting program. In these plots, the curved middle surface is plotted flat, which explains the difference in appearance between Figs. 2 and 5. It can be seen that the stress distribution at any given cross section is far from uniform, as would be indicated by elementary beam theory (load divided by crosssectional area). This is especially true at the base of the airfoil, Fig. 6. Because of the untwisting action described previously, the hub section cannot remain plane after deformation, and at the same time, stay compatible with the heavy platform. Thus, a high degree of warping occurs in sections near the airfoil-platform junction, as indicated by the stress pattern exhibited in Fig. 6, The blade was also analyzed using a generalized beam theory (3, 5). For comparison 3

4 MAX MIN 2 1 CONVEX SURFACE SURFACE 1 1 KSI MPz Fig, 5 Maximum principal stress (ksi) due to centrifugal force Fig. Deflection due to centrifugal force purposes, the hub section stress results are plotted in Fig. 6. It can be seen that there is no similarity between the NASTRAN and generalized beam patterns. NASTRAN predicts that the maximum stress occurs on the suction side near the leading edge, and is nearly twice the nominal stress (68 versus 37). As we move away from the airfoilplatform junction, the NASTRAN calculated stress distribution changes dramatically and becomes somewhat more uniform. For example, at 2 percent airfoil height, the maximum stress occurs on the concave surface at mid-chord, and is now only 28 percent greater than the nominal, Fig. 7. Here the stress pattern is fairly similar to that predicted by generalized beam theory, Fig. 7. It should be noted that the beam analysis also included the stiffening effect of rotation, and assumed the blade to be clamped at the root. No measurement of strains due to cf loading of this blade are available to compare with these computations. However, it has been observed that a NASTRAN analysis of an axial flow compressor blade yielded cf stress patterns very similar in appearance to those measured by htsuka in his investigation of compressor blade untwist [Reference (5), Figs. 13(b) and 13(d)]. NATURAL FREQUENCIES AND MODE SHAPES The first four natural frequencies were calculated using both NASTRAN and generalized beam theory. The results are presented in Table 1. Also shown are natural frequencies obtained from strain gage output in a rotating test of this blade. Blade excitation was supplied by steam jets in an evacuated chamber. Evacuation was achieved by condensing steam. Correlation between test and NASTRAN results is good. Deviation between average test values and NASTRAN is less than six percent for all modes. Generalized beam theory is less consistent showing deviations of 17 and 27 percent for the second and fourth modes. The main reason NASTRAN predictions are high is because no root-groove flexibility was provided in the model. As pointed out in the discussion on boundary conditions, the blade shank was considered to be clamped along the bottom row of elements. In reality, this condition can never by fully attained. Any cantilever support will elastically distort and allow some rotation at the base. It is important to remember that the same clamping condition was assumed for both beam and NASTRAN models. Mode shapes are displayed in Fig. 8. These are plots of the deformed structure mid-

5 w z I X COORDINATE, CM e X COORDINATE, INCHES 8 çdnve X NA ST A A N 5 ti y CO NOMINAL 3 N 2 N 2 1 GENERALIZED BEAM THEORY 6 3 NOMINAL w CON VE% 2 y 1 ~ C D KE Fig. 6 Stress due to cf hub airfoil section 6 3 X COORDINATE, CM ti 2 3 I w t 3 8 >.1 g X COORDINATE, INCHES 9 X MIN. MAX. I so RAX, 8 NASTRAN 5 6 NOMINAL y CONVEX D 2 B NOMINAL GENERALIZED BEAM THEORY surface superimposed on the original geometry midsurface. These plots were made using NASTRAN's plotting option (2). Since this is a free vibration solution, only relative displacements can be 3 1 CONVEX 1 Fig. 7 Stress due to of 2 percent height 3 D CONVEX SURFACE SURFACE 1 KXI E835.i1Pa Fig. 9 Relative vibratory stress (ksi) radial direction computed. Hence, one nodal displacement component must be arbitrarily pre-set and all others calculated relative to it. In this case, the largest deflection component in the global coordinate system (the cylindrical coordinate system was used) was set equal to 1 in. (2.5 cm), Fig. 8. The first mode shown in Fig. 8 is, of course, the fundamental mode which is predominently a beam type bending about the minimum inertia axis of the airfoil. The third mode exhibits a high degree of twisting, and has the appearance o.f a 5

6 Table 1 Calculated and Measured Natural Frequencies MODE AVERAGE TEST FREQUENCY NASTRAN GENERALIZED BEAM CALCULATED % ERROR CALCULATED % ERROR Hz 189 Hz Hz T 12 T X COORDINATE, CM X COORDINATE CM Q' 2 ^ 1 1 _ O f X COORDINATE, INCHES X COORDINATE. INCHES NA ST RAN 1 fi q 5 3 N -5 CONVEX I CONVEX -1 ` 36 fi 15 - GENERALIZED BEAM THETAY 9 15 GENERALIZED BEAM THEORY N x. 5 3 N N y ti 5 3 CONVEX Fig. 1 Mode 1 vibratory stress hub airfoil section 5 5B ^ 3 1 cdnvex Fig. 11 Mode 1 vibratory stress 2 percent height Boo torsional mode. This is probably why the beam program predictions are so good for the first and third modes. A considerable amount of twisting and plate-like bending is present in the second and fourth modes, particularly the fourth mode. This explains why the beam frequency predictions are so poor for these modes. VIBRATORY STRESSES Relative vibratory stresses were also calculated for each of the four modes. Since we 6

7 Fig. 1 Typical trailing edge crack (1.5 x). Arrow indicates origin ^g. Fig. 13 Typical leading edge crack (1.5 x). Arrow indicates origin only have data on the first mode to compare with the analytical results, only the first mode stress patterns are included. Fig. 9 shows the isostress contours for the stress in the radial direction (the maximum or minimum principal and radial stresses are very nearly the same for this mode). The relative stresses shown are quite large because of the arbitrarily chosen maximum deflection of 1 in. (2.5 cm). Fig. 1 is a plot of surface stress versus axial length of the base airfoil section. Surface stresses computed by the generalized beam program are also plotted on the same graph. Here again, we see that plane sections do not remain plane after deformation. This is especially noticeable on the leading edge where the maximum stress calculated by NASTRAN occurs on the convex side, which is opposite to that which is demanded by beam theory. The other peak stress points are on the convex surface at the thickest part of the airfoil and on the trailing edge con- Fig. 15 Typical convex side crack (leading edge crack on left). Arrow indicates origin cave side. This agrees with the beam prediction, but NASTRAN shows a larger gradient of stress across the trailing edge. This comes about because of local (plate-like) bending. Away from the airfoil-platform junction, the stress pattern changes and becomes more like the beam distribution. -Phis is illustrated in Fig. 11 for the 2 percent height section. Note, however, that some plate-like bending occurs here also. Direct measurements of stress patterns in this blade have never been made, but field failure data does exist to support the NASTRAN vibratory stress results. In 1972, a W51 unit experienced a severe fuel nozzle clogging problem, which resulted in a large third engine order stimulus at design speed (36 rpm). This set up a fundamental F

8 mode resonant vibration. Two blades fractured and five others developed cracks near the airfoil base. Metallographic examination of the cracked blades revealed that the failures were due to high cycle fatigue, and that there were three points of crack initiation, Figs. 12, 13, 1, and 15. The most interesting point to note is that the leading edge crack, Figs. 12 and 13, did indeed start on the convex side as predicted by NASTRAN; not on the concave side as required by beam theory, Fig. 1. The other cracks started on the concave side of the trailing edge, Fig. 1, and on the convex side at the thickest part of the airfoil, Fig. 15, as predicted by both NASTRAN and beam theory. COMPUTER COSTS The results reported herein were obtained using a CDC 66 version of NASTRAN, level 15. The cost of the analysis, including mesh generation, was about 13, based on a costing rate of.2 per cs second. Since these calculations were made, Westinghouse has installed CDC 76 computers for its engineering and scientific computations. Experience with the 76 version indicates that the cost of this analysis would be about one-fourth as much, or X36O. A factor of one-fourth cannot be used as a general rule to estimate 66 versus 76 costs because it depends on the program, the problem type, and the size of the problem. The important point is that the cost of using NASTRAN as a design tool is reasonable, especially if a CDC 76 version is available to the user. CONCLUSIONS It has been demonstrated that NASTRAN is a powerful tool for the calculation of elastic stresses, displacements, and natural frequencies in turbine blades. Although strains were not measured, field failure experience shows that peak vibratory stress locations predicted by NASTRAN agree with the three crack initiation zones observed in failed blades. Of special interest is the fact that NASTRAN correctly predicts that the fundamental mode peak stress occurs at the airfoil base section on the suction side of the leading edge, rather than on the pressure side, as predicted by beam theory. In view of this correlation, one would expect the of stress and deflection patterns to be equally good. Also, a close similarity was noted between calculated of stress patterns for a Westinghouse axial flow compressor blade, and the patterns measured by Ohtsuka in his investigation of compressor blade untwist. The cf stress distribution at any given cross section was found to be far from uniform, especially at the base of the airfoil. Here the peak stress occurred on the suction side of the leading edge and is nearly twice the nominal stress. The correlation between calculated and measured natural frequencies (first four modes) was good. Deviation between average test frequency and calculation was less than six percent for all modes, whereas generalized beam theory was less consistent, showing 17 and 27 percent deviations for the second and fourth modes. NASTRAN is uniquely suited for turbine blade analysis because of its capacity to handle large dynamic problems, and its differential stiffness feature, which takes into account the approximate effect of deflection onloading. REFERENCES 1 MacNeal, R. H., ed., "NASTRAN Theoretical Manual," NASA SP-221, Sept. 197, 2 McCormick, C. W., ed., "NASTRAN Users' Manual," NASA SP-222, Sept Houbolt, J. C., and Brooks, G. W., "Differential Equations of Motion for Combined Flapwise Bending, Chordwise Bending, and Torsion of Twisted Non-Uniform Rotor Blades," NACA Report 136, Filstrup, A. W., "Finite Element Analysis of a Gas Turbine Blade," ASME Paper 7-- WA/GT-11, Nov Ohtsuka, M., "Untwist of Rotating Blades," ASME Paper 7-GT-2, Apr