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Finite element analysis on vibration behavior and fatigue cracking prediction of a Francis type hydraulic.. Chan, Sze Bun 1999

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Finite Element Analysis on Vibration Behavior and Fatigue Cracking Prediction of a Francis Type Hydraulic Turbine by  Sze Bun Chan B . A . S c (Mechanical Engineering), The University of British Columbia, 1997 A Thesis submitted in partial fulfillment of the requirement for the Degree of Master of Applied Science in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department o f Mechanical Engineering) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A September 1999 © Copyright, Sze B u n Chan, 1999  ln  presenting this  degree at the  thesis  in  partial  fulfilment  of  the  requirements  University  of  British  Columbia,  I agree that the  for  an  advanced  Library shall make it  freely available for reference and study. 1 further agree that permission for extensive copying of  this thesis for scholarly purposes may be  granted by the head of my  department  or  understood  by  his  or  her  representatives.  It  is  that  publication of this thesis for financial gain shall not be allowed without permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  copying  or  my written  Abstract G M S turbine cracking problem was investigated using finite element analysis. Analysis results confirmed that fatigue cracking could occur due to resonance induced by pressure variation as turbine blades pass wicket gates. Analyses also showed that fatigue life could be extremely low due to extremely high cycles experienced by the turbine. Installation o f blade stiffeners and thickness modifications could change the natural frequency o f the turbine. However, the modifications could cause efficiency reduction, cavitation behavior change or structural weakness. were suggested.  ii  Therefore no feasible modifications  Table of Content  ABSTRACT  ii  TABLE OF CONTENT  iii  LIST O F F I G U R E S  iv  LIST O F T A B L E S  vi  LIST O F A P P E N D I C E S 1  vii  TECHNICAL BACKGROUND AND RESEARCH APPROACH  1.1  Introduction  1  1.2.1 1.2.2  Technical Background Technical Data Crack Locations and Cracking Pattern  2 2 3  1.3.1 1.3.2 1.3.3 1.3.4 1.3.5  Previous Work Fracture Surface Analysis Residual Stress Measurements Operational Stress Measurements Natural Frequency Tests Dynamic Response Test on Turbine Model by B C Hydro  3 3 4 5 6 7  1.4.1 1.4.2  Research Approach Preliminary Plan Finite Element Method and Computational Tools  1.2  1.3  1.4  2  1  M O D E L I N G A N A L Y S I S O F TURBINE  8 8 10  11  2.1  Finite Element Model of Turbine  11  2.2  Modal Analysis  13  2.2.1 2.2.2 2.2.3 2.3 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2  Introduction and Basic Equations Rotation Related Effects on Natural Frequency Stress Stiffening  13 13 16  Damping using Added Mass Effect  17  Estimation of Excitation Frequencies Rotational Speed Related Excitation Vortex Shedding as Excitation  19 19 20  Harmonic Response Analysis Theory of Harmonic Response Analysis Boundary Conditions for Harmonic Response Analysis  20 21 22  iii  2.5.3 2.6  23  2.6.1 2.6.2 2.6.3  Stress Intensity Factor (SIF) Introduction '. Theory of Stress Intensity Factor Determination Modeling and Boundary Conditions  24 24 25 27  2.7.1 2.7.2  Fatigue Initiation and Propagation Life Prediction Definition of Crack Initiation Fatigue Propagation Life Prediction  28 29 31  2.7  3  Displacement Superposition of Static and Harmonic Response Results  DESIGN MODIFICATIONS A N D A N A L Y S I S R E S U L T S  3.1  34  Overview of Design Modifications  34  3.2.1 3.2.2 3.2.3 3.2.4 3.2.5  Modal Analysis Mode Shapes and Natural Frequencies Distribution of Modal Stresses Modal Analysis Simulating Scaled Down Turbine Model Effect of Spin Softening and Stress Stiffening to Natural Frequencies Effect of Damping on Natural Frequencies  35 35 36 36 37 37  3.3.1 3.3.2 3.3.3  Harmonic Response Analysis Harmonic Response Analysis Results Explanation of Cracking Locations Problem of Manufacturer's Model Test  38 38 41 42  3.4.1 3.4.2 3.4.3  Fatigue Life Prediction Initiation Life Prediction Propagation Life Prediction Implications and Error Discussion from the Fatigue Life  43 44 44 45  3.5.1 3.5.2  Turbine Design Modifications Installation of Blade Stiffeners Thickness Modifications  46 47 49  3.2  3.3  3.4  3.5  4  CONCLUSION  51  5  REFERENCE  53  iv  List of Figures Figure 1: Cross Sectional V i e w of the Turbine Runner Compartment  58  Figure 2: Schematic Diagram of the Cracking Locations and the Stainless Steel Overlay 59 Figure 3: Schematic Representation of the Various Structural Regions i n Etched Macrograph o f the Blade Tip Region  60  Figure 4: Contour Measurement Data Point Arrangement  61  Figure 5: Spectrum Diagram of Turbine M o d e l  62  Figure 6: Spectral M a p of the Turbine M o d e l  .63  Figure 7: Blade F E M o d e l  64  Figure 8: C r o w n F E M o d e l  65  Figure 9: Band F E M o d e l  65  Figure 10: F u l l Turbine M o d e l  66  Figure 11: A p p l i e d Displacement Boundary Condition  66  Figure 12: Spinning Spring Mass System  67  Figure 13: Schematic Explanation for Superposition of Harmonic Response Analysis Result and Static Analysis Result  67  Figure 14: L o c a l Coordinates Measured from a 3D Crack Front  68  Figure 15: Solid F E M o d e l  69  Figure 16: Fine Meshed Solid Band for Crack Extension  69  Figure 17 Crack T i p Wedge Element  70  Figure 18: F u l l Solid-Shell M o d e l  70  Figure 19: F u l l Solid Shell M o d e l with Boundary Conditions Applied  71  Figure 20: Experimental and Theoretical Fatigue Crack Growth Rates for Both Short and L o n g Cracks  72  Figure 21: Bounding Conditions for Fatigue Limits of Materials Containing Short and L o n g Cracks  72  Figure 22: M o d a l Analysis Deformation Contour Plot: Mode-1 (Swing)  73  Figure 23: M o d a l Analysis Deformation Contour Plot: Mode-3 (2N Elliptical)  73  Figure 24: M o d a l Analysis Deformation Contour Plot: Mode-5 ( 1 Rotational)  74  Figure 25: M o d a l Analysis Deformation Contour Plot: Mode-6 (3N Triangular)  74  st  v  Figure 26: M o d a l Analysis Deformation Contour Plot: Mode-8 (2  Rotational)  75  Figure 27: M o d a l Analysis Deformation Contour Plot: Mode-9 (Translational)  75  Figure 28: M o d a l Analysis Deformation Contour Plot: Mode-11 (4N Square)  76  Figure 29: M o d a l Analysis Deformation Contour Plot: Mode 13 (Crown Bending)  76  Figure 30: M o d a l Analysis Deformation Contour Plot: Mode-15 (5N Pentagonal)  77  Figure 31: M o d a l Analysis Deformation Contour Plot: Mode-17 (6N Hexagonal)  77  Figure 32: Relative Stress Contour of Blade in Mode 5  78  Figure 33: Pressure Distribution Contour of Blade  78  Figure 34: von Mises Stress Result of H R Analysis @ 42.5 H z  79  Figure 35: von Mises Stress Result of H R Analysis @ 60 H z  79  Figure 36: von Mises Stress Result of H R Analysis @ 85 H z  80  Figure 37: von Mises Stress Result of H R Analysis @ 120 H z  80  Figure 38: von Mises Stress Result of H R Analysis @ 127.5 H z  81  Figure 39: von Mises Stress Result of H R Analysis @ 170 H z  81  Figure 40: von Mises Stress Result of H R Analysis @ 180 H z  82  Figure 41: Deformation of Turbine Blade at Mode 5 (58.5 Hz)  82  Figure 42: Nonlinear Static Analysis using H R Displacement Result @ 60 H z  83  Figure 43: Nonlinear Static Analysis using H R Displacement Result @ 180 H z  83  Figure 44: Stress Contour Result using Displacement Superposition @ 2.5% H R  84  Figure 45: Stress Contour Result using Displacement Superposition @ 5% H R  84  Figure 46: Static Analysis Result using 100% C F D pressure on Blade  85  Figure 47: SIF Range versus Crack Length @ 2 . 5 % H R Loading Case (Plain strain)  86  Figure 48: SIF Range versus Crack Length @ 5 % H R Loading Case (Plain strain)  86  Figure 49: Configuration of Stress Stiffener on Turbine  87  Figure 50: Areas of Thickness Modifications (Side V i e w of Turbine)  87  vi  List of Tables Table 1: Results o f Residual Stress Measurements at G M S  88  Table 2: Technical Details and Results on Mitsubishi Model Tests  88  Table 3: Comparison o f M H I M o d e l Runner Test and B C Hydro F u l l Size Turbine Test 88 Table 4: Holographic Test Results (Reference [9]) on Turbine M o d e l , U B C , 1975  89  Table 5: Comparison o f M o d a l Data before and after stiffener installation on Unit 4 runner  89  Table 6: Rotational Speed Related Frequencies o f Excitation (up to 180 H z )  90  Table 7: M o d a l Analysis Results (25 Modes)  90  Table 8: Natural Frequency Comparison: F E - Experimental results  91  Table 9: H i g h Stress Locations o f Each Mode  91  Table 10: Ratio o f Natural Frequencies o f F u l l F E M o d e l to 1:14 F E M o d e l and Comparison to Reference [9]  92  Table 11: Effect o f Spin Softening and Stress Stiffening on Natural Frequencies  93  Table 12: Highest Stress and Its Location for Harmonic Response Analysis  93  Table 13: Total Strain Range and Predicted Initiation Life  94  Table 14: Stress Intensity Factor Range o f 2 . 5 % H R and 5 % H R loadings  94  Table 15: Total Fatigue Life Prediction Results  95  Table 16: Effect o f Installation o f 40-70 Blade Stiffeners  95  Table 17: Effect o f Thickness Modifications  96  vii  List of Appendices Appendix A : Conversion Method from M o d e l Test Data to Prototype Data  97  Appendix B : Theory o f Modal Analysis  98  Appendix C : Derivation of Surface Panel Method for Added Mass Effect (Reference [16])  100  Appendix D : Summary of Element Properties  103  Appendix E : Computation of Initiation Life in M a t h C A D  105  viii  1 Technical Background and Research Approach Introduction  1.1  B C Hydro and Power Authority ( B C Hydro) is the major power supplier i n the Province o f British Columbia, Canada.  It has been experiencing a severe fatigue  cracking problem with five Francis-type turbine runners. These turbine runners (Units 1 to 5) were installed in Gordon M . Shram ( G M S ) Power Generating Station from September 1968 to M a y 1969 [1].  Mitsubishi Heavy Industries Limited  Takasago, Japan, manufactured all five o f these problem turbines. units were installed from 1971 to 1980. occurred i n the latter runners.  (MHI),  Five more turbine  However, no severe cracking problem has  1  The fatigue cracks were found on the blades o f all five turbines during regular inspections since 1972. Both the M H I and B C Hydro had done extensive inspections and tests on finding the source o f the fatigue cracks. However, no conclusive results can be drawn. It is postulated that the fatigue cracks could be induced by resonance o f the turbine under normal operating condition.  With sophisticated aids o f powerful computers and  finite element programs, it is thought possible to answer the above question. It would be the interest o f B C Hydro i f such research could lead to turbine modifications to eliminate or reduce the occurrence o f fatigue cracking on the turbine runners.  ' Tokyo Shibaura Electric (Toshiba) Co., Ltd., Tokyo, Japan, manufactured turbines unit 6-8. Fuji Electric Co. Ltd., Tokyo, Japan, manufactured turbines unit 9-10. 1  1.2 Technical  Background  1.2.1 Technical Data A s mentioned previously, Mitsubishi Heavy Industries Ltd., Japan manufactured the first five turbine runners. They are vertical shaft Francis wheel type runners o f diameter of 5.3 m (17.5 ft). Each runner is connected to a generator above by a hollow vertical shaft.  The turbine runners drive 235 M W generators at a speed o f 150 R P M (2.5  cycles/second). Referring to Figure 1, for general layout o f the turbine, each runner was cast as one-piece casting. The base material is A S T M A 2 7 steel, with yield strength o f 220 M P a (32 ksi) and ultimate tensile strength o f 448 M P a (65 ksi). This material is considered to be similar to A S T M A 5 5 [3] or AISI 1020 low carbon steel. Each turbine consists o f 17 blades. These blades are held between a circular disc on top, known as the crown, and a circular ring at the outer diameter o f the turbine, known as the band. The turbine runner is connected to the generator through a vertical shaft connecting to the inner r i m o f the crown.  The turbine runners are powered by head  pressure o f 165 m (540 ft). Water flows into the turbine compartment through a spiral case. Water is redirected by 24 stay vanes, followed by 24 wicket gates. Water enters the runner through passages between the blades and discharges vertically downward at the center o f the runner. O n each blade, stainless steel overlay was welded on the suction side o f the blade to prevent cavitation. The turbines were stress-relieved at various stages o f manufacture by heating i n an oven at an elevated temperature.  2  1.2.2 Crack Locations and Cracking Pattern Fatigue cracks are mainly located at two locations (Refer to Figure 2): (1) trailing edge o f the suction side o f the blade where the blade meets the crown, so called location [A]; (2) leading edge o f the suction o f the blade where the blade meets the band, so called location [C]. In the most severe case, 55 cm (22 inches) through thickness crack was found on the blade at the crown discharge end [7]. In most cases, cracking at location [A] is much more extensive than that at [C]. One turbine runner has had a total o f 28 cracks [2]. There has been no specific pattern on cracking, however, some turbines have more serious cracking problem than others do and some blades on some turbines appear to crack more than others on the same turbine.  1.3  Previous  Work  1.3.1 Fracture Surface Analysis In June 1973, B C Hydro had examined some fracture surface samples. It is reported that the trailing edge cracks appeared to originate on the suction side o f the blade where the stainless steel overlay terminated and blended into the parent mild steel casting [2]. It was suggested that there might have been some fabrication problems. The report stated that the manufacturer applied the harder stainless steel E301 in the first pass, and high alloy E308 stainless steel i n the second pass. Later metallurgical study confirmed that Vickers hardness value o f 440 was found in the casting/overlay interface, compared to Vickers value o f the parent material at 170 H and that o f the stainless steel close to the v  surface at 230 H . Refer to Figure 3 for a hardness distribution on the blade. The wrong v  order o f application o f stainless steel caused excess dilution and induced a more brittle constitutes at the interface [3].  3  In July o f the same year, M H I analyzed some o f the initial crack samples.  They  found that cracks originated around defects i n the transition zone between the factory applied stainless steel overlay and the mild steel parent material. Vickers hardness value in the neighboring material o f the crack initiation point was about 500-600 [2].  This  hardness value is abnormally high compared to the general maximum value o f 400. The manufacturer report confirmed the B C Hydro findings. The manufacturer suggested 'perfect welding' repairing procedures to remove the heat-induced cracks. It involved removal o f crack and partially filling the void first with a m i l d steel electrode (7018) and then applying a "butter" layer o f electrode E309 before the finish electrode E301 [2]. However, the cracking continued to re-occur even after applying the 'perfect welding' procedures. This suggested that the cracking problem is not solely material related in nature and that other factors may be involved.  1.3.2 Residual Stress Measurements Residual stress measurements were performed on the turbine runner units 2 and 3. Eight measurements were taken on parent mild steel material where no stainless steel overlay was applied (Refer to Figure 2). The residual stress is measured using the center hole technique [4-5]. The highest tensile residual stress value was found to be at the yield point. Cracking blades have higher average residual stress than that o f the noncracking blades.  However, some o f the non-cracking blades measured residual stress  value near the yield point. Moreover, both suction side and pressure side o f the blade measured high residual stresses [6]. Table 1 summarizes residual stress values obtained from testing. It is, therefore, non-conclusive that high residual stress alone can initiate and propagate the fatigue cracks.  4  The existence o f brittle layer and high residual stress constitute the possibility o f crack initiation. However, high operational stress and vibration sources seem to be also necessary  for the observed fatigue crack to propagate.  Thus, measurements o f  operational stresses and natural frequencies were also carried out.  1.3.3 Operational Stress Measurements In 1973, M H I performed tests to measure the stress on the model runner under similar operational condition [7]. The model was set to rotate at 1062 R P M to simulate power produced at 165 m (540 ft) head pressure.  Strain gages were attached to the  critical locations o f the model runner, including locations [A] and [C].  Using  dimensional analysis, the equivalent stress on the actual turbine can be determined (Appendix A discusses the derivation o f the dimensional analysis and the related operating parameters are listed under Table 2). M H I model test showed that the operational stress o f the model runner is highest at locations [A] and [C]. The stress levels at [A] and [C] were 9.8 M P a (1.4 ksi) and 66.9 M P a (9.1 ksi), respectively.  Stress fluctuation is about 10.3 M P a (1.5 ksi).  The  measured stresses seemed to be too low to cause the fatigue crack propagation. It is important to note, however, that the M H I model test found that operational stress at [C] is almost 7 times higher than that at [A]. This result does not agree with, and actually contradicts with the fact that cracking is more severe at [A] than at [C]. Later in this thesis, we w i l l discuss the reason why M H I test was unable to predict highest stress level at [A], and why it cannot detect resonance of the turbine runner. BC  Hydro also performed operational stress measurement  using strain gages  attached on the runner in 1975. The measurement was taken within 2 hours o f start-up  5  while the turbine wheel was running at maximum loading. Test results reported mean operating stress ranging from 31.7 M P a to 91.6 M P a (4.6 ksi to 13.3 ksi) and fluctuating stress ranging from 2.8 M P a to 10.3 M P a (0.4 ksi to 1.5 ksi) [3]. Comparing B C Hydro and M H I results, it is noted that the fluctuating stress is very close whereas the mean stress found by B C Hydro is significantly higher than that obtained by M H I . Comparison is summarized in Table 3.  1.3.4  Natural Frequency Tests  1.3.4.1 Holographic  Test on Turbine  Model by UBC  In 1975, the model runner was sent to the University o f British Columbia ( U B C ) for dynamic behavior analysis [9]. The study showed that the ratio o f natural frequency o f the model runner to that o f the full size turbine was about 10.1 [9]. The test found the fundamental frequency o f the model runner to be 492 H z and the last detectable frequency to be 3461 H z . Therefore, the fundamental frequency o f the full size turbine should be about 49.2 H z . O n the other hand, the study indicated that sub-harmonic forcing frequencies could induce super-harmonic responses [9]. This idea w i l l be used later to determine the presumed excitation frequencies in the harmonic response analysis of finite element method. The first six resonant frequencies were recorded by laser holographic interferometry. However, the holographs can only identify band dominant mode shapes.  It is very  difficult to identify other types o f vibrating modes by holographs. For reference, the first six natural frequencies are listed in Table 4(a).  The natural frequency shift o f the  submerged turbine model ranged from 16.4% to 36.2%. The result is also listed i n Table 4(b).  6  1.3.4.2 Actual Turbine Test and Effect of Stress Stiffeners B C Hydro continued the natural frequency testing on the full size turbine runner. A l s o , B C Hydro started investigating the effect o f installing blade stiffeners on the runner. The blade stiffeners are steel rods o f 2.22 cm (7/8 inch) in diameter, 20.3 c m (8 inches) i n length. Two stiffeners were installed near the trailing edges o f each pair o f blades. They were located at 40% and 70% o f edge length measuring from the crown. Each stiffener was on one end connected to the trailing edge and on the other end connected to the suction side o f the blade. The schematic diagram shown i n Figure 49 best represents the arrangement. The changes in natural frequencies were measured and results are listed i n Table 5. The  table shows natural frequencies recorded from the blade and from the band,  respectively. The discrepancy between the blade and band results might be originated from the curve-fitting computation carried out by the measuring instrument. numbers were compared with F E results obtained in later sections.  These  Installing stress  stiffeners appear to increase the natural frequencies of the turbine from 12.1 to 18.1%. Detailed discussion o f the effect o f the stiffeners w i l l be given in the section 3.5.1, Turbine Modifications: Installation o f Blade Stiffeners.  1.3.5 Dynamic Response Test on Turbine Model by BC Hydro In early 1999, B C Hydro had conducted harmonic response test on the turbine model.  The model was mounted vertically on a steel frame and the experiment was  carried out by using an exciter or by hammering.  The response o f the turbine was  measured by two accelerometers mounted at different positions o f the turbine.  7  The  spectrum diagram o f the turbine was also plotted and is shown i n Figure 5. It shows very high response at 1085 H z and 1789 H z . The effect o f water damping on natural frequencies was studied by gradually immersing the turbine model into water. were placed in a water tank.  The turbine model and the mounting frame  Water was filled i n the tank stepwise until water fully  covered the turbine model. Between each step, spectrum o f the turbine was recorded. A spectral map was afterwards created. Figure 6 shows a spectral map o f blade 14 that was excited by a vertical random force. The response in the map was taken at the blade close to the band. The bottom o f the map shows the spectrum curve with 0% water covering the model, and the top shows the response with 100% water covering the model. The map shows that lower natural frequencies are relatively unaffected by the water damping, but some experienced higher shift. This means that damping effect is mode dependent.  Moreover, higher frequencies tended to shift more and might merge to  together. It is also worthwhile to note that some frequencies had very high response as water was covering the model half way. This test provides significant information on the effect o f water damping mass at different frequencies.  1.4  Research  Approach  1.4.1 Preliminary Plan Previous work did not reveal high stress fluctuations that may initiate or propagate fatigue cracks on the problem turbines. Due to limitation of numbers o f data points taken and duration o f experiment, it is very difficult to study the complete vibration behavior o f the turbine using the research methods mentioned above. It is therefore proposed to use the finite element method ( F E M ) for this project to obtain stress-strain data, natural  8  frequencies, mode shapes, harmonic responses and stress intensity factors (SIF) o f the turbine.  Moreover, the overall vibration behavior o f the component may also be  simulated. In the G M S turbine-cracking problem, the source o f stress fluctuation is not known. N o particular excitation frequency could be isolated. Considering the fact that overall stress level o f the turbine was low and crack propagation rate was high, it was reasonable to suspect that resonance caused high local stress fluctuation and fatigued the turbine blades. Yet, evidences had to be found to support this theory. In order to evaluate the above hypothesis, the harmonic response o f the turbine under certain excitation frequencies has to be found.  Since excitation frequencies are not  known, a wise guess o f the excitation frequencies would be necessary.  The correct  excitation frequency should reproduce experimental findings typically in the highest stress areas.  T w o main conditions are the basic criteria for determining the 'correct'  excitation frequency; these are: highest stress was located on suction side o f the blade and stress level at location [A] was higher than that at location [C]. This analysis w i l l be preceded by calculating the turbine natural frequencies. starting point since it also facilitates checking the  This should provide a good F E model against  available  experimental data. If resonances were found to be the source o f stress fluctuation that led to fatigue, it would be o f our interest to compare the F E M predicted crack propagation rate with actual propagation rate. Finally, some turbine modifications should be suggested to reduce or eliminate further fatigue cracking.  9  1.4.2 Finite Element Method and Computational Tools Finite element method ( F E M ) starts with building a finite element (FE) model i n the computer.  In sophisticated finite element program, F E model can be built using a  graphical interface, known as pre-processing program. Boundary conditions and method o f analysis can be chosen and applied through this interface. H i g h intensity computation is then performed in the processing program using the data input from the pre-processing phase. Finally, computational results are reviewed by post-processing program. In this work, a commercially available finite element program, known as A N S Y S , is used to produce all finite element results. It provides a single integrated package o f preprocessing, processing, and post-processing programs.  The program is run using a  personal computer (PC) o f Pentium II 233 M H z with 192 M B R A M . A l s o , a spreadsheet was used to produce fatigue crack propagation life prediction. A n d M a t h C A D was used to compute fatigue initiation life.  10  2 Modeling Analysis of Turbine 2.1  Finite Element Model of Turbine The F E model is built to simulate the geometry and physical properties o f the  structure or the component.  Computational intensity and hardware requirement would  increase as a function o f the total number o f degree of freedom ( D O F ) i n the model. W h i c h is, i n turn, function o f the number o f nodes in the model. Each node may have 6 D O F that typically include translational motion and rotational motion i n 3 axes. Geometrical and material properties o f the structure are usually assigned on the element level. In this work, shell and solid elements are used to model the turbine. 4node elastic shell elements are used in static analysis, modal analysis and harmonic response analysis whereas 20-node 3D solid elements and 8-node plastic shell elements are used in the sub-modeling o f the cracking area to calculate stress intensity factor (SIF). The properties o f these are summarized in Appendix D . The basic assumption made in building the finite element model includes: (1) average contour scan o f the model runner correctly represents the contour o f the full size runner, (2) every blade has the same contour and (3) the crown, blades and band are thin such that they may be modeled using shell element. The G M S turbine F E model is built based on the blade contour o f the manufacturer's turbine model. The 1:14 turbine model was sent to G E Hydro, Ontario, for a contour scan. A total o f five (5) blades were scanned. in the F E modeling. obtained [10].  2  2  The average o f these five scans was used  Contours o f both pressure and suction side o f the blade were  Contour measurements were stored in Cartesian coordinate.  G E Hydro contour scan obtained pressure and suction side contour of blade 5, 6, 7, 11, and 14.  11  O n each  surface scan, the blade was divided from the crown to the band into 16 sections. First section was located at the crown-blade intersection, whereas the sixteenth section was located at band-blade intersection. In each section, 42 data points were obtained. The first point was at the leading edge, whereas the 4 2  n d  point was at the trailing edge o f the  blade. This would create 16 by 42 grids on both sides o f the blade. The data points were denser near the leading edge and trailing edge. Figure 4 shows the arrangement o f the scanned data points. In order to obtain a reasonable aspect ratio for the elements, only 300 points were taken on each side, i.e., 15 by 20 grids were used instead. Since the turbine blades were relatively thin compared to the blade width, shell element is appropriate to use. The data points were first scaled to the actual turbine size. M i d points o f the corresponding grids on the pressure and suction side were used as corner nodes o f the shell elements. The distance between the corresponding grids was taken as corner thickness. The blade elements were generated using a simple program written by the author.  The program created all 'real constant' for the blade elements,  which stored the corner thickness o f each element on the blade.  3  A l s o this program  created data for quadrilateral shell elements for one single blade as shown in Figure 7. The crown and the band o f the turbine were generated using section 1 and section 16's data points, respectively. The data points were joined to create the outline o f the crown and the band. The lines were then rotated one seventeenth (1/17) o f a circle about the center axis o f the runner to obtain meshable areas for the crown and the band. Using the auto-mesh option o f A N S Y S , the elements o f the crown and the band were automatically generated. Since the crown does not end at the trailing edge, an extended  Read constant is the terminology used by A N S Y S . It contains thickness data for shell element. For other type elements, the real constant contains information, such as moment of inertia about different axes. 3  12  crown area was created. The extended area started from the trailing edge o f the blade and extended horizontally and radially inward by 48 cm (18.9 in), where the runner joins the shaft. Figure 8 shows the crown mesh, and Figure 9 shows the band mesh. The above procedure produced one seventeenth o f a turbine runner. B y copying the above elements 17 times about the center axis o f the turbine, the whole runner was then generated. Nodes at identical coordinates were then merged to join different parts o f the turbine runner together. The full turbine model consists o f 14008 shell elements, 13591 nodes and 80934 degrees o f freedom.  Modal  2.2  2.2 A  Analysis  Introduction and Basic Equations  M o d a l analysis is to determine the natural frequencies and mode shapes o f a structure. B y determining the natural frequencies and mode shapes o f the turbine runner, it should give insight to estimate the possible sources o f excitation. M o d a l analysis also provides basis o f verifying the accuracy o f the F E model by comparing the results to experimental findings. The performed analysis is a linear one assuming constant stiffness and mass effects. Fully fixed displacement boundary condition was applied to the inner rim o f the extended crown to simulate the rigid connection to the shaft as shown in Figure 10 and Figure 11.  The basic theory and underlying equations o f modal analysis is  discussed i n Appendix B .  2.2.2  Rotation Related Effects on Natural Frequency Modeling technique in modal analysis assumes that the turbine is stationary i n space.  N o rotation effect was taken into account. In practice, three (3) types o f rotation related  13  effects could alter natural frequencies o f the runner. In the following, we briefly discuss these factors.  2.2.2.1 Natural  Frequency  Split  The natural frequency o f a stationary model would split into two natural frequencies as the rotational speed o f the turbine increases. Typically, this effect could be visualized on the Campbell diagram . The increased natural frequency is called the forward 4  precession and the decreased one is called the backward precession [13].  The rate o f  change i n natural frequency may vary, however, and it is also mode dependent. To estimate the effect on natural frequency, it was assumed that the change i n natural frequency is equal to rotational speed o f the turbine. Since the turbine is running at fixed speed o f 150 R P M (2.5 cycles/second), we may assume that natural frequency splits into plus or minus 2.5 H z in actual operation.  2.2.2.2 Spin  Softening  The vibration o f a rotating body could cause relative circumferential and radial motion. The motion could change the direction of the centrifugal load and therefore may destabilize the structure.  Since modal analysis cannot directly account for changes i n  geometry, the effect could be accounted for by adjustment o f the stiffness matrix. This effect is called spin softening [14]. To explain the effect, consider a simple spring-mass system as shown Figure 12. The equilibrium o f the spring and centrifugal forces on the mass using small deflection assumption requires:  4  The Campbell diagram plots natural frequency against rotational speed.  14  iol[M]{r}  [K]{u} =  where  (1)  u = radial displacement of the mass from the static position r = radial rest position of the mass with respect to the axis of rotation CO = angular velocity of rotation s  For large deflection, the following equation may be written: [K]{u}  = cv*[M]{r  + u}  ( 2 )  B y rearrangement {[K]-^[M]\u}  = cv*[M]{r}  <> 3  Defining: [K] =  [K]-co [M] 2  s  (4)  and {F}  = co][M]{r}  ^  Equation (3) becomes [K]{u}  =  {F]  (6)  Equation (6) shows that small deflection solution may be used for large deflection analysis by proper adjustment o f the stiffness matrix and the forcing vector. The eigenvalue equation becomes  [K]-a>l[M] or  15  = 0  ( 7 )  (8)  {[K]-^[M])-co [M]=0 2  where  CO = the natural frequencies of the rotating body  Spin softening is applied to the modal analysis using pre-stressed option in ANSYS. The centrifugal effect on the structure was first determined using static pre-stressed analysis. And the required data are then fed into the modal analysis procedure. 2.2.3 Stress Stiffening Stress state of a component could stiffen or weaken the structure. This is known as the stress stiffening effect. The effect normally needs to be considered for thin structure with very small bending stiffness. The effect is accounted for by generating an additional stiffness matrix.  The matrix could be first produced via static analysis and then  incorporated into modal analysis. The stress stiffening effect is incorporated by adding the following matrix to the element stiffness matrix [15]. (9)  where  [Gj] is the matrix of shape function derivatives [TJ\ is a matrix of the current Cauchy stresses in the element  Applying the stress-stiffening  matrix to the eigenvalue equation (8), the new  modified equation is: {[K] + [S]-cv [M])-co [M] 2  where  2  = 0  (10)  [S] is the stress-stiffening matrix  Equation (10) is solved to obtain natural frequencies and mode shapes of a structure including both stress stiffening and spin softening effect.  16  2.3  Damping using Added Mass Effect Dynamic response o f submerged structures requires knowledge o f effect o f the fluid  on the structure.  In general, the interaction is a complex coupled problem.  In the  simplest representation, the effect o f the fluid on the structure is proportional to only the interface accelerations. It can be included as an added mass component i n the dynamic equilibrium equations o f the structure.  The added mass concept is equivalent to an  integral o f the total kinetic energy, which the body motion imparts to the surrounding fluid i n a potential flow solution [16]. Surface panel singularity distribution method originally presented by Hess and Smith [17] formulates the interaction o f fluid and structure in terms o f boundary integral. The potential i n the fluid is represented using the singularity distributions on the body surface. Different assumptions  about the fluid potential inside the body lead to  different  formulations o f the singularity based methods, namely source formulation and dipole formulation methods.  Further discussion was given by Vorus and Hylarides [18-19].  Similar equation was given by Rajasankar et al, [20] for ship hull problem and similar analysis was also applied to turboprop by N A S A [21]. Empirical evaluation o f added mass effect on plate is also available [22]. However, the method was considered to be not suitable to apply to turbine blades. To briefly explain the theory o f the added mass effect, we started by the F E structural equilibrium equations in the form: [ M ] {«} + [C] {u} + [K] {u} = {F (0 + F (t)} f  where  [M] = structural mass matrix [C] = damping matrix  17  s  (11)  [K] = structural stiffness matrix Ff = fluid force vector F - structural and body force vector s  The added mass matrix i n the case o f source formulation is given in Appendix C . [M ] = A  where  (12)  [T] [A][H][LY [T] r  l  [7] = transformation matrix for normal to global directions [H] = coefficient matrix relating source strength to velocity potential [ M A ] = added fluid mass matrix [A] = diagonal matrix of panel areas [L] = coefficient matrix relating source strength to normal velocity at panel control points  For dipole formulation: (13)  [M ] = -[T] [A][Lr[H][T] T  A  Equations (12) and (13) are considered to be equivalent. Experiments found that the dipole formulation provided, however, better predictions to the response. The combined structural equation becomes: [M + M ] {u} + [C] {u} + [K] A  where  M = {F  {ii} = nodal acceleration vector {it} = nodal velocity vector {u} = nodal displacement vector {Ff} = fluid force vector {F } = structural and body force vector s  18  s  (t)}  (14)  Although equation (12) or (13) could be evaluated and applied to finite element method, it is very difficult to obtain coefficient matrices [H] and [L]. Therefore, this method is not applied to the finite element model. Since it is known that water damping could reduce natural frequencies anywhere from 16 to 36% [9], the damping effect could still be included by shifting natural frequencies down i n 16 to 36% range. The detailed discussion on damping would be included in section 3.2.5  2.4  Estimation of Excitation  Frequencies  Resonance would be caused by excitation frequencies that are close to natural frequencies o f the turbine.  In determining the possible excitation frequencies, it was  assumed that the excitation is originated from: (1) rotational speed related source and (2) vortex shedding related source.  2.4.1  Rotational Speed Related Excitation  Rotation related source is the most common type o f excitation sources. Since there are 24 wicket gates surrounding the turbine runner and the runner has 17 blades, the excitation frequencies could be the product o f the rotational speed and the numbers o f gates or blades.  A s stated in reference [9], the sub-harmonics may excite super-  harmonics, the multiples o f the product o f the rotation speed and the gate or blade numbers could be possible excitation frequencies. The turbine runner is rotating with 150 R P M , (2.5 cycles/second). Computing the multiples o f the product o f rotational speed and number o f blades or wicket gates, the list of possible excitation frequencies up to 180 H z is given in Table 6.  The excitation  frequency would be used in the harmonic response analysis. The 'correct' excitation frequency should produce agreeable results with experimental findings.  19  2.4.2  Vortex Shedding as Excitation  Vortex shedding could also be another source o f excitation. The phenomenon is observed as the fluid flows over a bluff object and separation begins. The vorticity in the boundary layers causes them to roll into vortex spirals at regular time intervals. This was found to be the cause o f blade and stay vane vibration problems on other turbines worldwide [23-26]. The excitation frequency may be obtained by the following equation: V f = Sd where  (15)  f=frequency of vortex S = Strouhal number, experimentally determined V= velocity of approach d = largest projected cross-section dimension on approach flow [23]  It is difficult to determine excitation frequencies using equation (15) for several reasons. The first, there is no definite range for Strouhal number. Experimental data for this number is not available for these turbine blades. The second reason is that it is very difficult to define the velocity o f approach because velocity on the blade changes from inlet to outlet and from top to bottom.  This change is also evident for the largest  projected cross-section o f the blade. Without proper excitation amplitude and direction, harmonic response analysis cannot be performed.  Therefore, this source o f excitation  was ignored due to lack o f information.  2.5  Harmonic Response  Analysis  Harmonic response (HR) analysis produces cyclic response i n a structure under sustained cyclic loading. The technique is to determine the steady-state response o f a linear structure to sinusoidal loads.  It assumes constant stiffness, damping and mass  20  effects.  Therefore, no nonlinearities and transient effect may be included. The basic  theory is described as below:  2.5.1 Theory of Harmonic Response Analysis The general force equation o f motion for a structural system is:  [M]{ii} + [C]{u} + [K]{u} = {F }  W  a  where  [M] = structural mass matrix [C] = structural damping matrix [K] = structural stiffness matrix {ii} = nodal acceleration vector {u} = nodal velocity vector {«} = nodal displacement vector [f}  = applied load vector  B y applying sinusoidal loads, the applied load vector could be expressed as:  {F"} = {F } = {F e'^e'  =({F } + i{F })e  a  c  where  ai  m  l  2  < > 17  {F } = complex load vector c  Q = excitationfrequency(radian/second) T = force phase shift (radian) {F\} = real component of applied load vector {F2} = imaginary component of applied load vector The general solution of the displacement can be expressed similarly:  {«.}  = {«„«"}*"'  = ({«,}+i{ui)V  21  (  1  8  )  where  {u } = complex displacement vector c  O = displacement phase shift (radian) {u\} = real component of displacement vector {uj} = imaginary component of displacement vector B y substituting equation (18) into equation (16), we have:  (-Q  2  [M] + /Q[C] + [K])  {u }  =  c  (19)  (F } c  Defining a matrix [K ] as: c  [K ] c  + iQ[C]  = (-Ct [M] 2  +  (20)  [K])  The solution for u may be obtained by solving: c  [K ]{u } c  =  c  (21)  {FJ  Solving equation (21) is known as " F u l l Solution Method".  This method uses the  same wavefront solver as for static analysis. However, complex arithmetic is used in the computation.  A N S Y S also provides a "Reduced Solution Method".  Similar to the  reduced solution method in modal analysis, the solution is solved using reduced structural matrices.  Since the full solution method solves the turbine model i n reasonable time  (about 1.5 hours) the reduced solution method is not used.  2.5.2 Boundary Conditions for Harmonic Response Analysis Excitation frequencies listed in Table 6 are related to numbers o f wicket gates or blades. Therefore the actual source o f excitation was assumed to relate to water pressure variation. C F D model was developed to find the steady state pressure o f the turbine [27]. The model used an explicit, time marching Finite Volume scheme and the equations o f continuity and momentum were solved directly using the time marching scheme.  22  The  C F D method solved steady state pressure on one passage section between two blades. Wicket gates and other details of the turbines were omitted. The computation assumes the turbine running at full power. Preliminary harmonic response analysis was to determine which excitation frequency could match the two criteria stated in Section 1.4.1.  Therefore, the sinusoidal loading  was taken to be 100% o f the steady pressure found by C F D method.  The results at  different possible frequencies were compared to the criteria to isolate the 'correct' excitation frequency. For control purpose, a simple static analysis should be performed using the same boundary condition. If the turbine is suffering from resonance, the stress results on harmonic response analysis should be significantly higher than those obtained from the static analysis.  2.5.3  Displacement Superposition of Static and Harmonic Response Results  In practice, the actual sinusoidal loading would only be a few percent o f the steady state pressure. In this work, we, therefore propose to determine the actual stress field o f the turbine using the principle o f superposition on displacement results. Assuming the pressure variation o f the turbine is about 5% o f the steady state pressure. Thus, 95% o f the steady state pressure would be applied as static loading on the turbine. Suppose the displacement results are obtained from linear analysis. It follows that the displacements from the 5% harmonic response analysis can be added to that o f the 95% static analysis as shown in Figure 13 (This loading case would be referred to as 5 % H R loading). The sum o f the displacements should be a more realistic estimate o f the turbine response at maximum loading. Stress field o f the turbine under this composition  23  o f loading was then determined by performing a materially nonlinear static analysis using the new displacement results. B y subtracting the harmonic response displacement from the static displacement, the minimum loading situation o f the turbine could be found. For comparison purposes the same procedures were carried out at 2.5% harmonic response and 97.5% static analyses (i.e., 2 . 5 % H R loading.).  The resultant displacements were used i n the fatigue life  prediction.  2.6  2.6A  Stress Intensity Factor (SIF)  Introduction In terms o f crack propagation prediction, the direction and the propagation rate have  to be found. Since the turbine was subjected to variable loading, the crack propagation direction was yet to be determined. A n approach suggested by Sih et al relates the crack propagation direction to minimum strain energy density [28 - 29].  Other researchers  suggested maximum strain energy release rate method [30] and the maximum tensile stress theory [31]. Most o f these methods have not yet been implemented in commercial finite element program. More importantly, including crack propagation directions requires redefining the geometry with the new crack direction and size and re-meshing the structure at each crack increment. Such analysis may be best solved using a new technique, known as the Arbitrary Lagrangian Eulerian ( A L E ) finite element method [32]. However, A L E finite element formulation has not been implemented yet in commercial F E codes and it is beyond the scope o f this project to investigate this area. In this analysis, and from the knowledge o f experimental data, the crack propagation direction at location [A] is  24  presumed to be parallel to the crown.  B y creating a fine meshed zone i n the preset  location, re-meshing could be greatly simplified. After completing the harmonic response analysis, one should be able to predict the fatigue life o f the turbine blade. Paris L a w requires obtaining stress intensity factor range (SIF range) i n order to predict propagation life. In A N S Y S , SIF at a crack tip could be determined using linear static analysis. Mode I, II and III stress intensity factors can be calculated using analysis post-processing program.  2.6.2 Theory of Stress Intensity Factor Determination Assuming Linear Elastic Fracture Mechanics ( L E F M ) and defining a Cartesian coordinate at the crack tip as shown in Figure 14, the actual displacement around the crack-tip expressed by [33]. The displacement, u, in x direction, parallel to crack extension direction:  AG\2n  K,  (2K - l)cos— - c o s — ' 2 2  AG  V  (22) (2K + 3)sin — + s i n — \2n ' 2 2 V  The displacement, v, in y direction, perpendicular to crack extension and crack front:  EL v=  AG\2n  (23) (2K - l l s i n ' 2 V  sin — 2  AG\2n  (2K + 3 )cos— + c o s — ' 2 2 V  The displacement, w, in z direction, tangent to the crack front: 2K  HI  G  r  . 6 sin— \2n 2  (24)  where u, v, w = displacement in a local Cartesian coordinate system as shown in Figure 14.  25  r, 0 = coordinates in a local cylindrical  coordinate system as shown in  Figure 14 G — shear modulus Kj, Ku, Km = stress intensity factors relating to fracture mode I, II and III respectively. f3 _ 4  V  3-v +v  in plane strain or axisymmetric in plane stress  v= Poisson 's ratio By evaluating equation (22) - (24) at 0= 180'  K  n  u = — -. —(1 2G \ 2n u  ^  (  2  5  )  + KT)  K, [ T , v = — — ( l + /c) 2G\2n n  (26)  ( 2 7 )  G l2f In case of an edge crack model, the stress intensity factors are expressed as: G  K, =J27T  |Av|  \ + K  '  j=r  1+  26  K  2 8  4r  M  G  r -  K„ =^2n  ( )  Vr  (29)  ,— K,„ =^2n  G  (30)  -=r  l +K  where  \Aw  Vr  Au, Av, and Aw are the motion of one crack face with respect to the other  Therefore the SIF range at a crack tip can be evaluated by calculating the difference of stress intensity factors found from applying the maximum and minimum loading to a cracked model. The SIF ranges were then applied to the crack propagation equation to obtain predicted propagation life.  2.6.3 Modeling and Boundary Conditions In order to obtain more accurate results, solid sub-models were used for this analysis. The submodel was built to simulate the crown-trailing-edge corner, labeled as the high stress location [A]. Due to cyclic symmetry o f the turbine deformation, one seventeenth o f the turbine would be sufficient to represent the turbine behavior. The solid submodel consisted o f part o f the crown and the blade, and formed a shape o f a small T as shown in Figure 15. A very fine meshed band was introduced to the model to facilitate easy crack extension. (Refer to Figure 16). In order to transfer displacement boundary conditions from the shell elements to the fine meshed solid elements, shell elements were added to the model. The shell elements extended the T-shaped both on the crown and on the blade as shown in Figure 18 and Figure 19. The boundary conditions were transferred to the solid by the interface shell elements on the solid element faces at the shell-solid transition area [34].  A t the  overlapping areas, the shell elements were set to 2.54mm (0.1 inch) i n thickness to minimize the overlapping stiffening effects. The displacement boundary conditions found  27  from the previous shell element analysis were applied only to the interfacing shell elements. A n edge crack was put in the solid sub-model, one element below the fillet o f the blade. The crack direction was pre-determined to be parallel to the crown. This crack was incrementally extended from 2.5 m m (0.098 inch) to 154.7 m m (6.09 inches).  In  each increment, the crack was extended by one element length 6.44 mm. The extension took a total o f 25 steps. The crack tip was formed by 8-quarter-point, second order wedge elements.  This  element simulates the singularity o f the stress field at the crack tip (Refer to Figure 17). B y assuming extended  crack would not change the applied boundary  conditions,  displacements at maximum and minimum loading were applied to the sub-model as the crack extends. The difference o f the SIFs from the maximum and minimum loading was the SIF range. The analysis was performed on both sets o f loading cases.  2.7  Fatigue Initiation and Propagation  Life  Prediction  Fatigue process may be divided into two stages crack initiation (crack nucleation), and crack propagation [35]. Fatigue life is usually regarded as the number o f cycles a component could last from initiation to propagation stage. However, there has been no general rule defining the transition o f crack initiation to propagation. To determine crack initiation and propagation life, three conditions have to be available: (1) a clear definition of transition from crack initiation to propagation, (2) corresponding prediction equation with necessary experimental constants, and (3) detailed stress picture around the crack area.  28  2.7.1 Definition of Crack Initiation Crack initiation generally refers to formation o f microcracks close to the surface o f the component or the formation o f a small crack with some pre-defined length.  The  nucleation site o f the microcracks may be located at (1) slip band (2) grain boundaries, or (3) surface inclusion. The microcracks may be formed by slip plane motion, plastic flow of material matrix or debonding at material matrix or inclusion [36-38]. Since there are too many different controlling factors to crack initiation, it is very difficult to conclude with a comprehensive  microscopic theory to define the transition and usually a  macroscopic approach is used. M i l l e r et al defined the transition o f crack growth rate by the initiation o f a crack with length o f 0.7 to 0.9 m m [39].  (Refer to Figure 20)  Kitagawa defined the crack  length at which the experimental data deviates from L E F M predicted stress level as initiation crack length [40]. (Refer to Figure 21) This initiation crack length is about 0.1 to 0.5 mm.  Kujawski and E l l y i n defined the transition between the initiation and  propagation stages to be at a critical microcrack size or reduction o f material's fatigue limit [41-42]. However, no typical initiation crack length was given. The above definitions do not provide a specific initiation equation, and it was not feasible to model a crack o f 0.1 to 0.9 m m in a turbine of diameter 5.3 m. Thus a more appropriate definition was used in the analysis. Socie and Artwohl [43] argued that initiation life should be calculated from strain cycle fatigue specimen. The corresponding range of crack length is approximately 2.5 mm.  Although this definition would include some propagation  distinguishes initiation and propagation.  life,  it clearly  Therefore, initiation life corresponding to a  crack length o f up to 2.5 m m may be determined by strain cycle fatigue equation and  29  propagation life corresponding to a growth o f crack length from 2.5 m m can be determined by Paris L a w . The M o r r o w ' s strain cycle fatigue equation [36] is an empirical formula and may be expressed as [44]:  As, Y Where  As  As  P  =  cr/  p  Y Y +  (31)  ~7 f  =  (2N  )  +  F  f' f  s  (2N  y  As — total strain range t  ASE — elastic strain range Asp = plastic strain range CT/ = fatigue strength coefficient s/ = fatigue ductility coefficient b = fatigue strength exponent c = fatigue ductility exponent E = elastic modulus Nf= fatigue initiation life Fatigue initiation life could be obtained, therefore, i f the strain distribution at the critical area is known and the fatigue coefficients are given. Since the equation requires both components o f plastic and elastic strains, nonlinear static analysis should be performed to obtain the strain distribution. The equivalent strain from the finite element analysis was used as the total strain: 1  "  e  V2 (1 + v)  {e -e } x  y  +{e -e } y  +{e -e f  z  z  30  x  +-{y] +y) y  z  +rl)  Modifications for the mean stress and stress ratio effect were suggested by various a number o f researchers.  But they have not been found to give significant changes to  initiation life [45] and therefore they were not considered in this analysis  2.7.2 Fatigue Propagation Life Prediction Fatigue propagation life prediction using Paris power law is also empirical i n nature. The Paris equation stated that is given by [46]:  ^ = c(AKy» dN where  ( 3 3 )  a — crack length Af = number o f cycles AK = K  max  - K  min  = stress intensity factor range  C, m = material constants Experiments by Verreman and Espinosa [47] modified the above equation for low carbon mild steel, in the form:  —  where  = 6.49 x 1 (T (AK f 9  mm I cycle  20  eff  (  M  AK jf= effective stress intensity factor range (MPa-Jm) e  The effective stress intensity factor range can be expressed as [48-49]: AK =U-AK  (35)  S  (36)  eff  JJ where  S  = maximum stress  S pen  =  mSK  0  Smin  =  -S  max max  crack opening stress minimum stress  31  open  mm  )  The factor U i n the above equation modifies the stress intensity range to account for crack closure effect. Since no information is available for the crack closure, the factor U was assumed to be unity for the propagation life prediction in this project. Forman [50] proposed a modified propagation life equation based on Paris L a w . The modified equation also takes into account the stress ratio and the instability o f the crack growth as the maximum SIF approaches the fracture toughness.  Including modification  for plastic zone by Willenborg et al [51-52], the modified Forman's equation states: da  C(AK )"  =  dN where  () 37  eff  (\-R )K -AK eJf  c  eff  C', n are material constants (different from Paris L a w ) K = critical stress intensity factor for fracture c  R f/= effective stress ratio e  Effective stress ratio is defined as: A',.. .,-A' ;  max  K where  K  d  require  reduced  =K  required  ,  'educed  (38)  reduced  -K  max  (39)  = stress intensity factor required to extend the boundary o f plastic  zone. It is believed that Forman's equation may provide a more accurate prediction for propagation life.  Unfortunately, it was not possible to obtain the material constants  needed for the analysis and, therefore, Paris L a w was used in this work. It should be noted that residual stress may change the crack opening stress and stress ratio [53-58] but it is necessary to use Forman's equation to account for these effects.  32  It should be also noted that equation (34) only accounts for mode I cracking. In the analysis for the turbine blades, it was found that mode II and mode III stress intensity factor ranges were relatively small compared to that o f mode I and therefore application of equation (34) was justified. Using equation (34) to calculate propagation life requires SIF range that may be calculated from equation (28) - (30).  33  3 Design Modifications and Analysis Results 3.1  Overview of Design  Modifications  Assuming that resonance is the cause of fatigue cracking, modifications should be focused on changing the excitation frequencies or changing natural frequencies of the turbine runner. Due to the fact that head pressure, rotational speed were preset in the generating station design, it is not possible to alter these factors to reduce resonance. The only choice is to alter the natural frequencies of the runner. The modifications of turbine runner can be divided into two main categories: (1) installation of blade stiffeners, (2) modifications of turbine contour and shape. It is noteworthy that any modification of the turbine could affect cavitation behavior and efficiency. resonance on another mode shape. finalize a modification.  Also, these changes may cause  Therefore, careful analysis and study is needed to  To verify the effectiveness of a modification, modal and  harmonic response analyses will be performed to assess the changes. Installing blade stiffeners to the turbine is a simple and inexpensive external modification to the turbine structure. Since stiffeners do not alter the main contour of the turbine blades, it is easier to avoid changes on cavitation behavior and turbine efficiency. However, depending on the position and size of the stiffeners, they can be blocking blade passages and changing water flow pattern. On the other hand, method of installation is of major concern. Since welded-on stiffeners could induce high residual stress, cracks could be initialized from the heat-affected zone. A number of different arrangements have been proposed and simulated. Results are presented in the later part of this chapter. Another way of changing the natural frequencies of the turbine is by altering the stiffness and inertia of the turbine using thickness modifications. Changing thickness of  34  the turbine blade would certainly change the flow pattern and cavitation behavior o f the turbine. A n alternative is to change only the thickness o f the crown and the band. B y modifying the crown thickness at the top and the band thickness at the outer diameter, the passage size and contour could remain unchanged. Detailed results and discussion o f this option w i l l also be given in this chapter.  3.2  Modal  Analysis  3.2.1 Mode Shapes and Natural Frequencies A finite element model for the full turbine (scale 1:1) is constructed and analyzed to obtain the natural frequencies and mode shapes. The first 25 mode shapes and natural frequencies were extracted. It is important to note that each cyclically symmetric mode would have produced another identical mode with the same frequency, but vibrating i n a perpendicular plane. The mode shapes were inspected and the natural frequencies are listed i n Table 7 with a description o f their corresponding mode shapes. The natural frequencies for the first 25 modes ranged from 43.4 H z for mode-1 to 188.5 H z for mode-25.  These 25  mode shapes can be briefly categorized into 5 types: (1) polygonal shape (nodal diameter) on the band, (2) band rotating relative to the crown, (3) whole turbine swinging, (4) translational motion o f band relative to the crown, and (5) irregular blade motions. Figure 22 - Figure 31 show deformation contour plots for mode-1 to mode-18. The natural frequencies from modal analysis are compared with experimental results from B C Hydro.  The comparison revealed a maximum difference o f 11%. The very  close agreement o f the results indicate that the F E model closely simulates the turbine behavior. Table 8 compares the two results. It is important to note that new modes (not  35  found by experiment) were found by the F E model. This may be attributed to sensitivity and accuracy o f measurements, the experimental set-up and the type o f excitation used.  3.2.2 Distribution of Modal Stresses Relative stress contour plots o f the blades for all modes are examined. The highest 5  stress locations i n each mode are recorded in Table 9. In the first 18 modes, critical stress areas were located on the suction side o f the blade. Only mode-5, the 1 rotational mode st  at 58.42 H z , exhibits high stress at the suction side o f location [A] and [C] (for labeling refer to Figure 2).  Figure 32 shows the relative stress contour plot o f mode 5.  It  therefore, suggested that mode 5 may be the mode and natural frequency that caused fatigue cracking at location [A] and [C]. Results from harmonic response analysis (to be discussed below) would strengthen this argument.  3.2.3 Modal Analysis Simulating Scaled Down Turbine Model In order to determine the relations of natural frequencies o f the scaled-down turbine model (scale 1:14) to the full size turbine, an F E model o f the scaled-down turbine model was also created with proper material properties.  Basically the original F E model was  shrunk down to one fourteenth o f the size. The same fully fixed boundary conditions were applied to the inner rim o f the extended crown. The scaled-down model results are tabulated in Table 10.  According to the F E  results, the natural frequency o f the scale-down turbine model is approximately 10.1 times higher than that o f the actual full size model.  The results are compared with  reference [9] and the difference between the two sets o f results is less than 7.1%. (Refer  Stress level obtained from modal analysis does not represent actual stress level of the component. The stress level only represents relative stress magnitude on the blade.  5  36  to Table 10 for details). The 10.1 ratio o f natural frequency may explain the reason why the manufacturer failed to predict possible resonance on the turbine.  Details w i l l be  discussed later.  3.2.4 Effect of Spin Softening and Stress Stiffening to Natural Frequencies The spin softening effect was included in the F E model and the rotational speed o f the turbine was set to 150 R P M (15.71 radian/second).  In addition, gravity and water  pressure effect were also included using stress stiffening option in A N S Y S . Table 11 presents two sets o f results: (1) effect o f spinning plus gravity, (2) effect o f spinning, gravity and water pressure. Both sets o f data showed insignificant changes in natural frequencies with a maximum value of 0.37%.  Therefore the effect o f gravity,  spinning and water pressure on the natural frequencies w i l l be ignored. It was surprising to find, however, that spin softening effect did not always decrease natural frequencies.  It only decreased natural frequencies in rotational and translational  mode. This may be attributed to a smaller effect of stress stiffening i n these modes. For other mode shapes, natural frequencies were increased.  Stiffness o f these other mode  shapes is highly dependent on the band and the crown stiffness. Centrifugal forces acting on the band or the crown would stiffen the component and, therefore, higher frequencies were observed.  3.2.5 Effect of Damping on Natural Frequencies Due to implementation difficulties with added mass effect on turbine (as discuss in section 2.3), the modal analysis with added mass effect was not performed.  We will,  however, rely on experimental data to estimate the damping effect on the turbine.  37  Assuming the I  s  rotational mode (590 Hz) to be the target mode shape.  Reference  [9] stated that damping effect reduced natural frequencies by 16.4% to 25.2% giving a final value o f 494 H z to 947 H z . Therefore the reduction percentage would be about 18.3%o by interpolation. This estimate is subject to two errors: (1) mode dependency o f frequency reduction, and (2) applicability o f interpolation for frequency reduction. O n the other hand, B C Hydro results found that higher frequency modes would have higher reduction. Referring to spectral diagram, Figure 6, very small shift was recorded on the low frequency mode. D r y turbine model at 460 H z showed 8.5% reduction in frequency. Therefore we estimated the natural frequency reduction for low frequency mode to range from 8.5 to 18.3%). Note that B C Hydro results were significantly lower than those o f reference [9]. Comparing experimental natural frequency results with F E modal analysis results, it may be concluded that the F E model underestimated the turbine natural frequencies by a maximum o f 11%. Therefore, natural frequency o f the submerged turbine would change by +2.5% to -7.3% o f the results from the modal analysis. Since the damping effect is mode dependent and it cannot be accurately estimated, the following analyses were performed without damping. The overall effect o f damping is discussed after each analysis.  Harmonic Response  3.3  3.3.1  Analysis  Harmonic Response Analysis Results  Comparing the natural  frequencies  found from F E model to the  excitation  frequencies calculated i n Table 6, 60 H z and 170 H z were closest to natural frequencies of mode 5 and mode 19. Therefore, it was believed that these excitations would cause the  38  most critical response. Harmonic response analysis was performed on each o f the above excitation frequencies as well as another arbitrarily chosen 5 frequencies. The F E model was fully fixed at the inner rim o f the extended crown. Gravity and spinning effect were included. Water pressure from the C F D analysis was applied to the blades, crown and band. Since the F E model and C F D computation used different grids, a small computer program was used to interpolate the pressure data. Pressure distribution contour o f the blade is shown in Figure 33. The figure shows a gradual reduction o f pressure from the inlet to the outlet o f the blade. Seven (7) harmonic response analysis were performed using a total o f 7 excitation frequencies.  The von Mises stress results for these analyses are shown i n Figure 34 to  Figure 40. The results confirmed that the 60 H z and the 170 H z give the most critical response. Other excitation frequencies gave relatively low deformation and stresses. Critical stress values and its location are tabulated in Table 12. A t 60 H z and 170 H z , stresses were about 2170 M P a (315 ksi) and 7100 M P a (1030 ksi), respectively. For other excitation, stresses range from 93 M P a (13 ksi) to 730 M P a (105 ksi). Comparing with result o f static analysis, the maximum stress was 200 M P a (29 ksi) located at [C]. Refer to Figure 46 for details. The comparison shows that i f the excitation frequency does not coincide with a modal frequency, the resulting stresses w i l l be comparable to static analysis results. Comparing stresses at 60 H z and 170 H z , high stress concentration area (in the 60 H z excitation) is located at the suction side, at [A] and [C].  O n the contrary, stress  concentration for the 170 H z excitation is located on the pressure side, surrounding [ A ] . See Figure 42 and Figure 43 for details. Recalling the criteria set forth on determining 'correct' excitation frequency, the 60 H z is chosen to be the 'correct' frequency.  39  O n the other hand, the 60 H z is the product o f wicket gate number and rotational speed.  This means that the excitation was originated from pressure variation as each  blade passes by the wicket gates. The 170 H z excitation is the 4  th  multiple o f the product  o f blade numbers and rotational speed. This excitation, i f existed, would be an indirect excitation. Since maximum stress result in the harmonic response analysis o f 60 H z was much higher than the yield strength, nonlinear static analysis was performed to determine the effect o f yielding. Bilinear material properties were used a yield stress value o f 220 M P a (32 ksi). The von Mises stress results o f nonlinear static analysis are shown in Figure 42. It is interesting to note that large area on the blade is stressed up to yield point and, therefore, strain hardening is believed to occur [37].  Based on 60 H z excitation frequency, the  turbine experiences more than 10 cycles per year and a fair portion o f the blade would 9  strain harden and may be subjected to Bauschinger effect [37]. This may explain the high residual stress measured on the base metal by B C Hydro. It should be also pointed out that the F E analysis predicted that the natural frequency of rotational mode would to change from - 2 . 5 % to 7.5% due to damping effect. estimate is based on experimental results from B C Hydro and reference [9]. damping may increase the difference between rotational frequency  The Water  and excitation  frequency and, therefore, harmonic response o f the turbine may be lower. However, since the obtained results from 60 H z excitation corresponded exactly to the observed cracking locations and matched the preset criteria, it may be concluded that the rotational mode natural frequency is close to 60 H z i n the actual turbine and that the  40  damping effect is not significant. Since accurate quantitative estimate was not possible, the damping effect would not be included i n following analysis. In considering the harmonic response for 60 H z excitation, only a few percent o f the total water pressure is assumed to contribute to the harmonic excitation. The rest o f the pressure would be acting as static pressure. Therefore the procedure o f displacement superposition o f harmonic response and static results was employed. The sum o f displacements was fed into a nonlinear static analysis using bilinear material properties.  The analysis result showed a more realistic  stress contour o f the blade under excitation o f 2.5% and 5% o f the total static pressure. Figure 44 and Figure 45 show the stress contour results o f the blade at 2.5% and 5% excitation, respectively. The highest stresses at [A] at maximum loading were 214.4 M P a (31.1 ksi) and 229.6 M P a (33.3 ksi) at 2.5% and 5%, respectively. M i n i m u m loading case is obtained and the results are later be used in the SIF range computation.  3.3.2 Explanation of Cracking Locations The above analysis indicates that the fatigue cracks are stimulated by tangential excitation at 60 H z . However, the runner is oscillating i n a rotational mode and outer diameter should experience higher deformation and therefore high strain.  This means  that high stress areas should be expected at [B] and [C]. (Referring to Figure 2, the four corners o f the blade were labeled clockwise using alphabets [A] to [D] starting from the crown-trailing-edge corner.)  It is important to note, however, that by examining the  deformation diagram o f a blade at mode 5, it is found that high strain did occur at [A] and [C] because the nearby areas were unable to relieve stress by deformation. Referring to Figure 41, two important phenomena may be observed.  41  First is that, only very small  deformation was observed at [A] compared to the deformation at [D]. However the curvature o f the blade above [D] allowed room for deformation to relieve strain at this area whereas the crown at the inner edge was very stiff to allow such stress relief for point [A] and, therefore, it caused high stress at [A]. Secondly, similar phenomenon occurred at [C] where the leading edge of the blade is straight and allowed no room for deformation due to band movement.  In contrary, strain at [B] was relieved by  deformation o f outer diameter o f the crown.  These two observations suggested that  Francis type turbine is subjected to high stress at [A] and [C] when the turbine is excited at rotational mode.  3.3.3  Problem of Manufacturer's Model Test  Mitsubishi Heavy Industry Company had performed operational stress measurement on the turbine model. However, the measured results were very low. The stress level at [A] and [C] were 9.8 M P a (1.4 ksi) and 66.9 M P a (9.1 ksi), respectively. Based on the F E model results, 60 H z excitation may cause the turbine to resonate. That is, the pressure variation as the blade passes the wicket gate excites the turbine. Recall that the manufacturer used 1062 R P M on the turbine model, which is 7.08 times o f the rotational speed o f the actual turbine whereas the natural frequency ratio o f the model to actual turbine is 10.1 times. The pressure variation experienced by the model would, therefore, be 424.8 H z . Noting that the rotational mode frequency o f the model is 587.8 6  H z , the excitation source would not match the rotational mode frequency i n the model test. I believe that this is the reason why the manufacturer failed to predict resonance and produced a l o w operational stress measurement.  6  Excitation frequency = number of wicket gate x rotational speed = 24 x (1062/60) = 424.8 Hz  42  Operational stress test on a small size model should be handled with care with predicting resonance behavior. Even i f the ratio o f rotational speeds and the ratio o f natural frequencies are equal, the model may still experience lower response.  It is  because the difference between the excitation frequency and natural frequency would be increased by the same scaling factor in the model test. inversely proportional to difference  Since resonance response is  of the square o f the natural and excitation  frequencies , the response is expected to be lower and more difficult to detect i n model 7  test.  3.4  Fatigue Life  Prediction  Fatigue Life o f a component consisted of crack initiation life and crack propagation life. A s discussed in the theory section 2.7, the initiation life would be determined using strain life equation, Equation (31); whereas the propagation life would be determined using Paris L a w , Equation (34). The 3 D solid submodel was used for this computation. Due to cyclic symmetry o f the turbine response at 60 H z excitation, one seventeenth o f the turbine is sufficient to represent the full structure. To produce an accurate estimate o f the SIF range, a finer 3D mesh was used.  The model was composed from shell and solid element as shown in  Figure 18. The purpose o f the shell element was to transfer the displacement boundary condition to the solid submodel. A l l displacement boundary conditions were applied to the shell elements.  N o displacement constraint was applied to the solid model. Strain  range and SIF range was calculated using the difference o f the results from maximum and minimum displacement conditions.  7  System response x\j{co  2  — O) ) where co = excitation frequency; &j, = natural frequency 2  43  3.4.1  Initiation Life Prediction  A nonlinear static analysis for a no-crack solid submodel was used to produce total strain result. The strain difference between maximum and minimum loading conditions was considered as the total strain range. The analyses were performed on both 2 . 5 % H R loading and 5 % H R loading case.  The total strain range results and the predicted  initiation life o f the two loading cases were tabulated in Table 13. Life computation was done using M a t h C A D , (details are shown in Appendix E). The total strain range for 2 . 5 % H R and 5% H R loading was 6.5E-4 and 1.2E-3, respectively.  The strain ranges  were small, and therefore the predicted were found to be 2.06E+9 cycles and 2.36E+7 cycles for 2 . 5 % H R and 5 % H R , respectively. These cycles are only equivalent to 397.7 • 8 and 4.6 days, respectively. .  3.4.2  Propagation Life Prediction  A series o f cracked solid submodels were used to produce SIFs for propagation analysis. A through-thickness edge crack was modeled on the blade and was extended from 2.5 m m (0.098 in) to 154.7 m m (6.09 in) using 25 models or steps. A total o f 25 SIF ranges were computed using the difference between maximum and minimum stresses. To include the 2 . 5 % H R and 5 % H R loading cases, a total o f 100 linear static analysis were performed. In A N S Y S , SIF computation is based on plain strain or plain stress conditions. Results from both approaches are calculated and presented in Table 14. The predicted propagation life from plain stress formula is 35% higher than that o f plain strain. Based on the plain strain formulation, for 2 . 5 % H R loading case, the SIF ranged from 11  8  Assumes full power operation for one year. Annual cycles = 60x3600x24x365.25 = 1.90E+9 cycles 44  M P a V m to a peak o f 25.6 M P a V m and then dropped to about 18 M P a V m . For the 5 % H R loading case, the SIF ranged from 21 M P a V m to a peak of 54.3 M P a V m and then dropped to 36 M P a V m .  The SIF values were expected to drop linearly as the crack tip moved  away from high stress concentration point as shown in Figure 47 and Figure 48. Using plain stress conditions, the SIF range ( A K ) results are about 10% lower.  Plain strain  results were used i n the following analysis to obtain conservative predictions. Utilizing Equation (42), the crack growth rate was determined and the propagation life was then obtained using trapezoidal rule. The propagation life results are listed in Table 15.  It was found that the propagation lives were quite low compared to the  initiation life.  Based on plain strain formulation, propagation only contributed to  1.16E+6 cycles and 1.26E+5 cycles for 2 . 5 % H R and 5 % H R loading, respectively. The propagation lives were 0.05% and 0.5% o f the total life for 2 . 5 % H R and 5 % H R loading, respectively. Therefore, the total fatigue lives were about 398 days for 2 . 5 % H R loading, and about 4.6 days for 5 % H R loading.  3.4.3  Implications and Error Discussion from the Fatigue Life  The above fatigue life prediction results indicate several important aspects o f the turbine cracking problem. These are summarized and discussed below. (1) From the above results, it is shown that the percentage o f the head pressure used in the harmonic analysis significantly affects the response and the life o f the turbine.  The 2.5% and 5% are more or less, arbitrary chosen values.  It is  therefore, concluded that some analysis, experimental testing and/or more study leading to more accurate prediction o f this percentage is crucial in calculating the blade fatigue life.  45  (2) The F E model prediction under the assumed loading condition is conservative. B C Hydro has found a crack o f 560 m m (22 in) after 3 years o f operations. O n the other hand, F E model predicted 154.7 m m (6.09 in) crack in 1.09 years.  It  seems that the actual crack growth rate in the propagation stage is lower than the F E predicted rate. Therefore, the turbine runner may have experienced a similar or lower magnitude o f resonance at the 2 . 5 % H R loading case. (3) Sensitivity to pressure variation may indicate that the turbine was running i n an unstable conditions. This, also, may have been the cause o f inconsistent cracking behavior. Pressure variation is also sensitive to individual blade contour and clearance between blade and wicket gates.  This may also explain why some  blades always crack, whereas some blades never crack. (4) In addition to pressure variation, underestimation o f fatigue life could be related to the following reasons: (i) rotational mode natural frequency being too close to the excitation frequency, (ii) lack o f damping effect i n the F E model, and (iii) empirical errors in life prediction equations. (5) Residual stress was not considered in the above prediction. crack propagation prediction.  This may affect  However, such computation was not possible  because o f the lack o f accuracy o f material constants and quantitative values for residual stresses.  3.5  Turbine Design  Modifications  Owing to the fact that head pressure, rotational speed and wicket gate numbers were preset, modifications o f the turbine were focused on altering natural frequency o f the rotational mode shape.  Altering inertial mass and/or stiffness o f the system may offset  46  natural frequencies o f the turbine to* a more desirable range.  Structural inertia o f the  turbine at rotational mode is mainly affected by moment o f inertia o f the band whereas structural stiffness is affected by both crown and blade thickness. To alter natural frequency at rotational mode, two approaches were used: (1) installation o f blade stiffeners, and (2) modifications o f thickness o f turbine blade. Modifications were targeted at changing the natural frequency by 15% or higher. It was believed that this percentage change could introduce a sufficiently large difference between natural and excitation frequencies, so that resonance could be significantly reduced.  3.5.1 Installation of Blade Stiffeners 3.5.1.1  40-70  Blade  Stiffeners  The first blade stiffener design was suggested by B C Hydro. The blade stiffeners are steel rods o f 2.22 cm (7/8 inch) in diameter and 20.3 cm (8 inches) i n length.  Each  stiffener is connected at one side to the trailing edge o f a blade, located at 70% o f the edge length measuring from the crown, and connected on the other side to the surface o f suction side o f a blade. The other blade stiffener was connected i n a similar manner but located at 40% o f the vertical distance from the crown. The stiffener arrangement is shown in Figure 49. referred to as 40-70 blade stiffeners.  This arrangement would be  These stiffeners were easy to install and it  minimized blockage o f the water passage between blades. In Table 16, the modal analysis results are compared with the experimental results obtained by B C Hydro. F E model reproduced very agreeable results compared to experimental findings. Based on F E model results, the 40-70 blade stiffeners were not  47  effective in increasing natural frequency of the rotational mode shape. increase was obtained for the rotational mode.  Only 0.5%  This result was expected because the  arrangement o f 40-70 stiffeners did not increase the structural stiffness o f the turbine to prevent rotation o f the band relative to the crown. Instead, it prevents relative motion o f blades. Another concern with this stiffener arrangement is that mode 9 and 10 frequency was changed from 115.59 to 121.44 H z . Also mode 19 frequency was changed from 170.72 to 178.77 H z . Therefore the 60 H z excitation could stimulate these two super-harmonics and create a more severe cracking situation.  3.5.1.2 Crossbar Stiffeners The most direct way to increase the rotational stiffness is to install cross bars from crown-blade intersection o f one blade to band-blade intersection o f another blade. Crossbars were simulated using beam or bar element and placed inside the water passage to produce maximum effect on structural stiffness. considered, including multiple bars.  Several different arrangements were  F E results from modal analysis show that such  arrangement could increase the rotational natural frequency by 15% or more. However, there were several problems with this arrangement. (1) Increasing rotational natural frequency by 10% or higher would require two or more 1" diameter solid bar installed in the water passage between blades. This arrangement would block water flow and decrease efficiency o f the turbine. (2) Crossbars had low natural frequencies. Their frequencies ranging from 70 to 90 H z depending on total length and diameter o f the bar.  48  There is a concern,  therefore, that vortex shedding may stimulate these natural frequency  and  produce more serious damage to the turbine. (3) Installation o f crossbars was intended to stiffen the structure o f the turbine, share harmonic loading and experience very high tensile or compressive stress. Results from F E models confirmed this concern. Therefore shear or buckling failure is possible. (4) Installation o f crossbar required welding. Therefore more areas o f the turbine might be affected by high residual stress. Crossbar stiffener was, therefore, not considered to be a feasible solution to alter rotational natural frequency.  3.5.2 Thickness Modifications Thickness modification is another method o f altering natural frequency o f the turbine. Reducing band thickness, and increasing crown and blade thickness would raise the natural frequency and vice versa. If modifications are done from the top o f the crown and the outer surface o f the band, the water passage may be kept unaffected making this method to be a desirable one. The area o f modifications included the middle section o f the band, and the entire crown excluding the extended crown.  Refer to Figure 50 for details. The effects o f the  modifications were analyzed using modal analysis and the results are tabulated in Table 17. Results showed that reducing the band thickness by 55.7% could increase rotational mode natural frequency by 5.3%. O n the other hand, increasing the crown thickness by 60% could increase the natural frequency by 9.8%.  49  Therefore, the modifications were  combined. B y reducing the band thickness by 60% and increasing the crown thickness by 60%, the total natural frequency increase was 15.5%.  However, such large  modifications to the turbine could cause structural weakness and therefore, it was not considered as a feasible solution.  50  4 Conclusion The turbine-cracking problem o f G M S power generating station started i n 1972. Previous research did not determine the causes of such fatigue cracks. Finite element analysis provided insightful results and suggested the cause and the mechanism o f cracking o f the turbine runners. M o d a l analysis o f the turbine runner indicated that mode shape at 58.42 H z produces high stress concentrations at the cracking areas, suction side o f location [A] and [C]. Spinning and gravity were considered in the analysis but they were found to be insignificant. Harmonic response analysis confirmed that excitation frequency at 60 H z causes high strain at the cracking locations. Therefore, it was concluded that the fatigue cracks were induced by the excitation originated from pressure variation as the turbine blades pass by the wicket gates. O n the other hand, modal analysis on the F E model o f the manufacturer turbine model suggested that manufacturer model test may not predict resonance. This was due to the different scaling factors that were used in power conversion and size conversion. In addition, nonlinear static analysis suggested that strain hardening could induce high residual stress on the turbine blade and this explains the existence o f high residual stress measured outside the stainless steel overlay on non-cracking blades. B y examining the deformation plot o f a turbine blade at 58.42 H z , it was found that the high strain located at [A] and [C] may be attributed to the fact that their surrounding areas are too stiff to deform. Fatigue crack prediction using strain life equation and Paris L a w indicated very short fatigue life of the turbine. The results also suggested that the fatigue life is very sensitive to the amplitude o f pressure variation used in harmonic  51  response analysis.  This might explain the inconsistent cracking behavior between  different turbine runners. Finally, modifications o f the turbine were suggested and analyzed using modal analysis. 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C , Mechanics of Fracture Initiation Propagation:  surface and volume energy  density applied as failure criterion, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. 29. Gdoutos, E . E . , Problems of Mixed Mode' Crack Propagation,  Martinus Nojhoff  Publishers, The Hague, Netherlands, 1984 30. Nuismer, R. J., " A n Energy Release Rate Criterion for M i x e d M o d e Fracture", Int. J. of Fracture, V o l . 11, 1975, pp. 245-250 31. Erdoggan, F . and Sih, G . C . , " O n the Crack Extension o f Plates Under Plane Loading and Transverse Shear", J. Basic Eng., V o l . 85, 1963, pp. 519-527 32. Wang, J, "Arbitrary Lagrangian-Eulerian Method and its Application i n Solid Mechanics", Ph.D. Thesis, Department o f Mechanical Engineering, The University o f British Columbia, Canada, 1998 33. Paris P. C , Sih, G . C , "Stress Analysis of Cracks, Fracture Toughness and Testing and its Applications", American Society for Testing and Materials, Philadelphia, S T P 381,pp.30-83 (1965) 34. Fok, S. C . M . , " N e w Design for Pressure Washer Drums" M . A . S c . Thesis, Department o f Mechanical Engineering, The University o f British Columbia, Canada, 1997 35. Lukas, Petr, Fatigue Crack Nucleation and Microstructure, Fatigue and Fracture o f Metals, ASM handbook.  55  36. Kocanda, S., Fatigue Failure of Metals, Sijihoff & Noordhoff International Publishers, Warsaw, Poland, 1978 37. Hertzberg, R . W . , Deformation  and Fracture Mechanics of Engineering  Materials, 4  th  Edition, John Wiley & Son Inc., 1996, pp.294-295 38. Shih, T., A r a k i , T., Trans. Iron Steel Inst., Japan, v.13, 1973, pp.11-19 39. M i l l e r , K . J., "Initiation and Growth Rates o f Short Fatigue Cracks", Fundamentals of Deformation  and Fracture, Eshelby Memorial Symposium, England, 1984, pp.477-  500 40. Kitagawa, H . , Takahashi, S., Int. Conf. Meek Behavior of Materials, A S M , 1976, pp.627-631 41. Kujawski, D . , E l l y i n , F., "Crack Initiation and Total Fatigue Life o f a Carbon Steel i n Vacuum and A i r " , Journal of Testing and Evaluation,  v.20, N o v . 1992, pp.391-395  42. Kujawski, D . , E l l y i n , F., " A Cumulative Damage Theory for Fatigue Crack Initiation and Propagation", International  Journal of Fatigue, v.6, n.2, 1984, pp. 169-192  43. Socie, D . F., Artwohl, P. J., "Effect of Spectrum Editing on Fatigue Crack Initiation and Propagation i n a Notched Member", ASTM STP 714, 1979, pp.3-23 44. Landgraf, R . W . , " H i g h Fatigue Resistance in Metals and A l l o y s " , ASTM STP 467, Philadelphia, 1970 45. Kaynak, C , Ankara, A . , Baker, T. J., "Initiation and Early Growth o f Short Fatigue Cracks at Inclusion", Material Science and Technology, v. 12, M a y 1996, pp.421-426 46. Paris, P. C , Bucci, R. J., Wessel, E . T., Clark, W . G . , Mager, T. R., "Extensive Study of L o w Fatigue Crack Growth Rate i n A533 and A508 Steels", ASTM STP 513, 1972, pp.141-176 47. Verreman Y . , Espinosa, G . , "Mechanically Short Crack Growth from Notches i n a M i l d Steel", Fatigue Fracture of Engineering  Material Structures, v.20, n.2, 1997,  pp.129-142 48. Elber, W . , "Fatigue Crack Propagation", Ph.D. Thesis, University o f N e w South Wales, Australia, 1968 49. Elber, W . , Engineering  Fracture Mechanics, v.II, n . l , Pergamon Press, July 1970  50. Forman, R . G . , Kearney, V . E . , Engle, R . M . , "Numerical Analysis of Crack Propagation i n Cycle Loaded Structures", Journal of Basic Engineering, of the A S M E , v.89, n.3, 1967, pp.459-464  56  Transaction  51. Willenborg, J., Engle, R. M . , Wood, H . A . , " A Crack Growth Retardation M o d e l using an Effective Stress Concept", AFFDL-TM52. Broek, D . , Elementary Engineering  71-1 -FBR, 1971  Fracture Mechanics,  Martinus Nijhoff Publisher,  The Hague, 1982 53. Harrison J. D . , "The Effect of Residual Stress on Fatigue Behavior, Residual Stresses and Their Effect", The Welding Institute, Cambridge, 1981, pp.9-16 54. Maddox, S. J., "Influence of Tensile Residual Stresses on the Fatigue Behavior o f Welded Joints i n Steel", Residual Stress Effects in Fatigue, ASTMSTP  776, 1981,  pp.63-96 55. Itoh, Y . Z . , Suruga, S., Kashiwaya, H . , "Prediction of Fatigue Crack Growth Rate in Welding Residual Stress Field", Engineering  Fracture Mechanics,  v.33, n.3, 1989,  pp.397-407 56. Gurney, T. R., "The Effect of Residual Stress, Fatigue of Welded Structures, 2  n d  Edition", The Welding Institute, Cambridge, 1979, pp.226-243 57. Parker, A . P., "Stress Intensity Factors, Crack Profiles, and Fatigue Crack Growth Rates in Residual Stress Fields", Residual Stress Effects in Fatigue, ASTMSTP  776,  1981, pp.13-31 58. Nelson, D . V . , "Effects of Residual Stress on Fatigue Crack Propagation", Stress Effects in Fatigue, ASTMSTP  776, 1981, pp. 172-194  57  Residual  58  Point B  Blade  Point C  Band  Figure 2: Schematic Diagram of the Cracking Locations and the Stainless Steel Overlay  59  E308?  MAJ6R  8%Ni l89bCr  CONSTITUENT  E30I  6'5%Ni 14-5% C r  BASE  500  400 to  to UJ UJ * 2  <  300  200H  100 X  X'  Y  Y'  Figure 3: Schematic Representation of the Various Structural Regions in Etched Macrograph of the Blade Tip Region. A B F M P  = = = = =  Austenite Bainite Ferrite Martensite Pearlite  60  XR ROAT DA IMETER =j 132 .0^)0  I \  j  /  I // / / / / / I I / / / // // / / / /! / / ' i/ / / / / / /  / // / / / I I  I  /  /  /  /  /  /  /  /  Figure 4: Contour Measurement Data Point Arrangement  61  Bi  RANGE: -15 dBV  MAG  START: 0 Hz X. 570.3125 Hz  Y:  BW: 7.8125 Hz 1.323 mVrms  STATUS. PAUSED RMS:50  STOP:  3 125 Hz  Figure 5: Spectrum Diagram of Turbine Model (The turbine model was shaken tangentially at the band, and response was measured at blade 15 where it intersects the band.)  62  RANGE: 1/50  - 1 3 dBV  A : MAG  STATUS: RMS: 4 0  PAUSED  Figure 6: Spectral Map of the Turbine Model (The turbine model was shaken vertically by random excitation, response was measured at blade 14 where it intersects the band.)  63  AN  Figure 7: Blade FE Model  64  Figure 8: Crown FE Model  Figure 9: Band FE Model  65  AN  (h  z>—  -.AAA /uV v v v VV  ( M  u  r  Figure 12: Spinning Spring Mass System 5% Harmonic Response  Pressure  95%  T  Static  -> Time  100%Harmonic Response Figure 13: Schematic Explanation for Superposition of Harmonic Response Analysis Result and Static Analysis Result.  67  Figure 14: Local Coordinates Measured from a 3D Crack Front  68  Figure 16: Fine Meshed Solid B a n d for C r a c k Extension  69  Figure 18: F u l l Solid-Shell M o d e l  70  AN  Figure 19: Full Solid Shell Model with Boundary Conditions Applied  71  Figure 20: Experimental and Theoretical Fatigue Crack Growth Rates for Both Short and Long Cracks  L O G CRACK L E N G T H Figure 21: Bounding Conditions for Fatigue Limits of Materials Containing Short and Long Cracks  72  ANSYS 5.5.2 SEP 7 1999 15:52:28 MODAL SOLUTION STEP=1 SUB =1 FREGF43 .414 USUM (AVG| RSYS=0 Power Graphics EFACET=1 AVRES—Mat DMX =.110749 SMX =.110749 0 ™ 1  1  1  ™  .012305 .024611 .036916 .049222 .061527 .073833 .086138 .098444 .110749  Figure 22: Modal Analysis Deformation Contour Plot: Mode-1 (Swing)  AJJSYS 5.5.2 SEP 7 1999 15:54:04 NODAL SOLUTION STEP=1 SUB =3 FREQ=46.567 USUM (AVG| RSYS=0 PowerGraphics EFACET=1 AVRES=MaC DMX =.137094 SMX =.137094 0 .015233 .030465 .045698 .060931 ^ .076163 .091396 .106629 ** .1218 61 ™ .137094  Figure 23: Modal Analysis Deformation Contour Plot: Mode-3 (2N Elliptical)  73  ANSYS 5 . 5 . 2 SEP 7 1999 15:55:18 NODAL SOLUTION STEP=1 SOB =5 rREO=58.47 OSUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.092558 SMX =.092558 0 JJ  * IS 6  2  .010284 .020568 .030853 -041137 .051421 .061705 .07199 .082274 .092558  Figure 24: Modal Analysis Deformation Contour Plot: Mode-5 (1 Rotational) st  AHSYS 5.5.2 SEP 7 1999 15:57:08 NODAL SOLUTION STEP=1 SOB =6 FRI0=87.955 OSUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.173 673 SMX =.173 673 0 j  [ ^ —  ^  .019297 .038594 .057891 .077188 .096485 .115782 .135079 .154376 .173 673  Figure 25: Modal Analysis Deformation Contour Plot: Mode-6 (3N Triangular)  74  ANSYS 5.5.2 SEP 7 1999 15:58 :17 NODAL SOLUTION STEP=1 SUB =8 FRE 0=100. 119 tJSOM (AVG) RSYS=0 Pouter Graphics EFACET=1 AVRES=Mat DMX =.107333 SMX =.107333 •i Hi  •i  •  .011926 .023852 .035778 .047704 .05963 .071556 .083482 .095407 .107333  Figure 26: Modal Analysis Deformation Contour Plot: Mode-8 (2  Rotational)  ANSYS 5.5.2 SEP 7 1999 15:59:34 NODAL SOLUTION STEP=1 SUB =9 FRE0=115.58 USUM (AVG) RSYS=0 Powe rGrapru.cs EFACET=1 AVRES=Mat DMX =.235199 SMX =.235199 0 ^ .026133 .052266 .0784 .104533 .130666 .156799 .182933 .209066 ™ .235199  Figure 27: Modal Analysis Deformation Contour Plot: Mode-9 (Translational)  75  ANSYS 5 . 5 . 2 SEP 7 1999 16:00:42 NODAL SOLUTION STEP=1 SUB =11 FREQ=124.093 USUM (AVG) RSYS=0 PouerGraphj.es EFACET=1 AVRES=Mot DMX =.258526 SMX =.258526 0 .028725 .05745 .08 6175 .1149 • .143 625 • • .172351 .201076 .229801 .258526  Figure 28: M o d a l Analysis Deformation C o n t o u r P l o t : Mode-11 (4N Square)  ANSYS 5.5.2 SEP 7 1999 16:01:42 NODAL SOLUTION STEP=1 SUB =13 FREQ=140.31 USUM (AVG) RSYS=0 PowerGraphics EFACET= 1 AVRES=MaC DMX =.28641 SMX =.28 641 0 ^ -031823 .063647 .09547 .127293 • ( .159117 .19094 .222764 ^ .254587 ™ .28641 a  Figure 29: M o d a l Analysis Deformation C o n t o u r Plot: M o d e 13 ( C r o w n Bending)  76  ANSYS 5.5.2 SEP 7 1999 16:02:36 NODAL SOLUTION STEP=1 SUB =15 FRECF150.129 USUM (AVG) RSYS=0 PowerGraphics EFACET=1 AVRES=Mat DMX =.331508 SMX =.331508 0 .03 6834 .073669 . 110503 .147337 . 184171 .221006 .25784 .294674 .331508  Figure 31: M o d a l Analysis Deformation C o n t o u r P l o t : Mode-17 (6N Hexagonal)  77  Figure 32: Relative Stress C o n t o u r of Blade i n M o d e 5  78  9  ANSYS 5.5.2 SEP 7 1999 17:15:28 NODAL SOLUTION STEP=1 SOB =1 FRE0=425 . SEQV (AVG) PowerGraphi.es EFACET=1 AVRES=Mat DMX =.120989 844.791 SMN= 46483 SMX=8 44.791 5916 10987 16057 21128 26199 31270 36341 41412 46483  Figure 34: von Mises Stress Result of HR Analysis @ 42.5 Hz  ANSYS5.5.2 SEP 7 1999 17: 18 :49 NODALSOLUTION STEP=1 SOB 1 = FRE 0=60 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX 1 =.055 SMN 3 =028 SMX 3 =14857 028 §• 3 37676 72324 • i 106971 141619 • i 176267 210914 245562 280209 314857  Figure 35: von Mises Stress Result of HR Analysis @ 60 Hz  79  ANSYS 5.5.2 SEP 7 1999 17:20:13 NODAL SOLUTION STEP=1 SUB 1 = FRECF85 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mae DMX .0 =61942 SMN 3 =14.704 SMX 1 =3541 314.704 • i 1784 • i 3254 • i 4724 6193 7663 9132 10602 • i 12072 13541  Figure 36: von Mises Stress Result of H R Analysis @ 85 H z  ANSYS 5.5.2 SEP 7 1999 17:24:39 NODAL SOLUTION STEP=1 SUB =1 FRECF120 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX =.0598 63 =173.668 =24424 173.668 2868 5563 8257 10952 13646 16341 19035 21730 24424 SMN SMX  Figure 37: von Mises Stress Result of H R Analysis @ 120 H z  80  ANSYS 5.5.2 SEP 7 1999 17:25:40 NODAL SOLUTION STEP=1 SUB =1 FREQ=127.5 SEQV (AVG) PowexGraphics EFACET=1 AVRES=Mat DMX =.064062 SMN =203.657 SMX =27253 203.657 • 3209 1  I  51 1  —  6215 9220 12226 15231 18237 21242 24248 27253  F i g u r e 38: von Mises Stress Result of H R Analysis @ 127.5 H z  ANSYS 5.5.2 SEP 7 1999 17:26:55 NODAL SOLUTION STEP=1 SUB =1 FRE 0=170 SEQV (AVG) PowerGraphic3 EFACET=1 AVRES=Mat DMX =3.932 SMN =4911 SMX =.103E+07 4911 ^ 119146 233380 347615 461849 576084 690318 804553 918787 ™ .103E+07 1  1  1  i  H  B  1  Figure 39: von Mises Stress Result of H R Analysis @ 170 H z  81  ANSY3 5.5.2 SEP 7 1999 17:27 :58 NODAL SOLUTION STEP=1 SUB = 1 FREQ=180 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX = .33761 SMN = 433.899 SMX = 106542 433.899 12224 24013 35803 47593 59383 71173 82962 94752 106542  •1 •i  F i g u r e 40: von Mises Stress Result of H R Analysis @ 180 H z ANSYS 5.5.2 SEP 7 1999 16:23:47 NODAL SOLUTION STEP=1 SUB =5 FRECF58 ..47 USUM (AVG) RSYS=0 PowerGraphlcs EFACET=1 AVRES=Mat DMX =.092558 SMN .005498 .092558 SMX .005498 .015171 ,024845 ,034518 ,044191 ,053865 .063538 ,073211 .082885 .092558  Figure 41: Deformation of T u r b i n e Blade at M o d e 5 (58.5 H z )  82  ANSYS 5.5.2 SEP 7 1999 19:23:58 NODAL SOLUTION STEP=1 SUB =100 TIME=1 SEQV (AVG) PowerGraphica EFACET=1 AVRES=Mat DMX =1.055 SMN =3961 SMX =40622 3961 • 8034 12108 I 16181 I 20255 ^ 24328 28402 32475 ' 3 6549 ^ 40622 1  Figure 42: Nonlinear Static Analysis using HR Displacement Result @ 60 Hz ANSYS 5.5.2 SEP 7 1999 19:25:26 NODAL SOLUTION STEP=1 SUB =100 TIME=1 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX =3.932 SMN =9198 SMX =58871 9198 14717  • I  20236  1  25756 31275 36794 42313 47832 53352 58871  1  —  1  ™  Figure 43: Nonlinear Static Analysis using HR Displacement Result @ 180 Hz  83  ANSYS 5.5.2 SEP 8 1999 13:38:44 NODAL SOLUTION STEP=1 SUB =1 TIME=1 SEQV (AVG) PowerGraphics EFACET=1 AVRES-Mac DMX =.089439 SMN =648.907 SMX =31118 648.907 4034 7420 10805 14191 1  1  1  5  20962  ™  27733  —  31118  17576  24347  Figure 44: Stress Contour Result using Displacement Superposition @ 2.5% HR ANSYS 5.5.2 SEP 8 1999 13:41:15 NODAL SOLUTION STEP=1 SUB =1 TIME=1 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mat DMX =.122374 SMN =887.784 SMX =33340 887.784  M  4 4 9 4  I 1  1  8  8  8099 11705 15311 18917 22522  9  1  1  26128 29734 33340  Figure 45: Stress Contour Result using Displacement Superposition @ 5% HR  84  ANSYS 5.4 HOV 4 1998 16:35:24 NODAL SOLUTION STEP=1 SUB =500 TIME=1 SEQV (AVG) PowerGraphics EFACET=1 AVRES=Mot DMX =.056899 SMN =228.7 SMX =22318 _ 228.7 2683 5137 7592 10046  m, _  12501  14955 17409 198 64 22318  Figure 46: Static Analysis Result using 100% C F D pressure on Blade  85  AK vs a @ HR 0.025 + Static 0.975 (Range of Stress Intensity Factor vs Crack Length) [Plain Strain]  25000.00  3  20000.00  -AKI AK.II  - - * - AKJII  a  5000.00  0.00 0.0000  1.0000  2.0000  3.0000  4.0000  5.0000  -5000.00 •  Crack Length (in)  Figure 47: SIF Range versus Crack Length @ 2.5%HR Loading Case (Plain strain)  AK vs a @HR 0.05 + Static 0.95 (Range of Stress Intensity Factor vs Crack Length) [Plain Strain] 60000  50000  • --AKII •A - AKIII  bO  1.00  4.00  2.00  Crack Length (in)  Figure 48: SIF Range versus Crack Length (£> 5%HR Loading Case (Plain strain) 86  Figure 49: Configuration of Stress Stiffener on Turbine  Modified Areas  Figure 50: Areas of Thickness Modifications (Side View of Turbine)  87  Table 1: Results of Residual Stress Measurements at GMS Test  Unit  1 2 3 4 .5 6 7 8  2 2 2 "2 3 3 3 3  | Blade  Edge  Side  T 7 S T 7 S T 11 S ! Band Between 11 & 12 1 13 S | 4 T S L P 1 16 16 L S  0~max  0~min  (MPa) 151.7 124.1 48.3 -206.8 Y+ 234.4 Y+ Y+  (MPa) 41.4 62.1 -34.5 -110.3 Y+ 144.8 206.8 220.6  Comments  Repeat Test 1 N o cracks, N o repair Through SS overlay N o cracks, N o repair  Key: T = Trailing Edge  P = Pressure Side  L =Leading Edge  Y + = Stress E x c e e d s t h e Y i e l d P o i n t  S = Suction Side  Table 2: Technical Details and Results on Mitsubishi Model Tests Diameter o f Runner (mm) Pressure Head (m) Rotational Speed ( R P M )  Actual Turbine Runner  M H I M o d e l Runner  5137 165 150  357.8 40 1062  N o t e : F o r C o n v e r t i n g M o d e l Test D a t a to Prototype D a t a refer to A p p e n d i x A  Table 3: Comparison of MHI Model Runner Test and BC Hydro Full Size Turbine Test (MPa)  B C Hydro Turbine Test* (1975)  M H I M o d e l Runner Test ** (1973)  Recorded Stress Fluctuating Stress  31.7-91.6 2.8-10.3  9.8 - 66.9 a 10.3  S t r e s s v a l u e w a s o b t a i n e d from s t r a i n g a g e m e a s u r e m e n t w h i l e t h e r u n n e r w a s s l o w l y a c c e l e r a t e d t o t h e m a x i m u m l o a d i n g . R a n g e o f m e a n stress w a s r e c o r d e d . * * S t r e s s v a l u e s f o r t h e M H I M o d e l R u n n e r T e s t w e r e s c a l e d t o c o m p a r e w i t h a c t u a l stress m e a s u r e d o n t u r b i n e stress.  88  Table 4: Holographic Test Results (Reference [9]) on Turbine Model, UBC, 1975 Table 4(a): First Six Resonant Frequencies |  1  Freq. (Hz) | 493-496 Mode 2-N Shape  |  2  | 946-950 3-N  3  4  5  6  1346-1348 4-N  1612-1619  1730-1734  1786-1797  ~  ~  N = Nodal Diameter  Table 4(b): Natural Frequency Shift o f Fully Submerged Turbine Model Dry  Resonant Frequencies (Hz) Wet  494 947 1347  Percent Reduction Compared to D r y Data  413 708 860  16.4 25.2 36.2  Table 5: Comparison of Modal Data before and after stiffener installation on Unit 4 runner. Mode N o .  Blade Mode* Before Blade Stiffeners (Hz)  Blade Modes After Blade Stiffeners (Hz)  Band Modes* Before Blade Stiffeners (Hz)  Band Modes After Blade Stiffeners (Hz)  1 2 3 4 5 6 7 8 9  44.22 90.83 127.32 135.18 153.8 171.8 182.5 218.3 234.5  47.06 100.39 144.53 148.85 179.02 222.46 -  43.58 90.69 127.60 152.93 170.77 185.51 211.3 -  48.88 104,55 149.78 180.56 - . -  *Experimental resonant frequencies of the same mode shape are different when measured on blade and measured on band. Therefore, two sets of data were collected. Since the Band Mode provides more definite mode shapes and repeatability, it would be used for comparison purpose.  89  Table 6: Rotational Speed Related Frequencies of Excitation (up to 180 Hz) Multiple o f  Speed (2.5 H z ) x Wicket Gates (24)  Speed (2.5 H z ) x Blades (17)  Excitation Frequency (Hz)  xl  60.0  ' 42.5  x2  120.0  85.0  x3  180.0  127.5  x4  -  170.0  42.5 60.0 85.0 120.0 127.5 170.0 180.0  Table 7: Modal Analysis Results (25 Modes) Mode 1  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Frequency (Hz) 43.37 43.37 46.40 46.40 58.42 87.79 87.79 100.12 115.59 115.59 123.95 123.95 140.30 140.30 150.02 150.02 169.29 169.29 170.73 177.06 177.06 182.20 182.20 188.48 188.48  Mode Shape Description Swinging: The whole turbine swing about center o f crown 2 N (Nodal Diameter = 2: Elliptical) Band exhibits elliptical shape 1 Rotational: Band Rotates wrt Crown (Half Sine Wave) 3 N (Nodal Diameter = 3: Triangular) Band exhibits triangular shape 2 Rotational: Band Rotates wrt Crown (One Sine Wave) Translational: Band moves sideways wrt Crown 4 N (Nodal Diameter = 4: Square) Band exhibits square shape Crown Bending: Sides o f crown bend downwards 5 N (Nodal Diameter = 5: Pentagonal) Band exhibits pentagonal shape st  n d  6 N (Nodal Diameter = 6: Hexagonal) Band exhibits hexagonal shape Not clear (Possibly 3 Rotational) Group blade movement (No specific pattern can be identified) 7 N (Nodal Diameter = 7: Heptagonal) Band exhibits heptagonal shape Alternating blade moves (No specific pattern can be identified) r d  90  Table 8: Natural Frequency Comparison: FE - Experimental results (Frequency Measured from G M S turbine Unit #4) M o d e Shape  F E Frequency (Hz)  Frequency Measured from Blade (Hz)  % Error  Frequency Measured from Band (Hz)  % Error  2 N Elliptical 3 N Triangular 4 N Square 5 N Pentagonal  46.40 87.79 123.95 150.02  44.22 90.83 127.32 135.18  +4.9 -3.3 -2.6 +11.0  43.58 90.69 127.60 152.93  +6.5 -3.2 -2.9 -1.9  Table 9: High Stress Locations of Each Mode Mode  Frequency Description Side o f Blade H i g h Stress Location* (using labeling o f Figure 2) (Hz) A 1 43.37 Swing S 2 43.37 A+D 3 46.40 2 N Elliptical S 4 46.40 5 58.42 1 Rotational S A+C 3 N Triangular D+A 6 87.79 S 7 87.79 A 8 100.12 2 Rotational S A 9 115.59 Translational S 10 115.59 11 4 N Square D 123.95 S 12 123.95 A 13 140.30 Crown Bending s 14 140.30 15 150.02 5 N Pentagonal D s 16 150.02 A 6 N Hexagonal 17 169.29 s 18 169.29 A+D p 19 170.73 D Group Blade p 20 177.06 Movement 21 177.06 S = suction side P = pressure side Note: In the high stress location column, the first letter indicates the highest stress location, the second letter indicates the 2 highest stress location. st  n d  nd  91  Table 10: Ratio of Natural Frequencies of Full FE Model to 1:14 F E Model and Comparison to Reference [9] Mode  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Frequency (Hz) Full F E F E Model o f Turbine M o d e l 1:14 Turbine Model (1)  Ratio (2) divided by(l)  Reference [9]* on 1:14 Turbine Model Freq. (Hz)  Error o f F E Model According to Ref. [9] Results (%)  43.37  (2) 438.04  10.1  -  -  46.40  468.64  10.1  493 - 496  -4.9 to -5.5  58.42 87.79  590.04 886.68  10.1 10.1  -  -  946 - 950  -6.3 to -6.7  100.12 115.59  1011.21 1167.46  10.1 10.1  -  -  123.95  1251.90  10.1  1346- 1348  -7.0 t o - 7 . 1  140.30  1417.03  10.1  -  -  150.02  1515.20  10.1  -  -  169.29  1709.83  10.1  -  -  170.73 177.06  1724.37 1788.31  10.1 10.1  -  -  -  182.20  1840.22  10.1  -  -  188.48  1903.65  10.1  -  -  * Note: Reference [9] results were stated in ranges and therefore ranges of errors were computed. On the other hand, mode shapes of frequency higher than 1348 Hz were not identifiable, thus comparison was not made.  92  Table 11: Effect of Spin Softening and Stress Stiffening on Natural Frequencies M o d e I N o Effect Frequency (Hz)  J  Spinning plus Gravity and Pressure Freq (Hz) Changes  Freq (Hz)  Changes  43.37  43.38  (%) +0.02  43.41  (%) +0.09  |  46.40  46.53  +0.03  46.57  +0.37  I  58.42 87.79  58.43 87.93  +0.02 +0.16  58.47 87.96  +0.08 +0.19  I |  100.12 115.59  100.11 115.56  -0.01 -0.03  100.12 115.58  +0.00 -0.01  123.95  124.07  +0.01  124.09  +0.11  140.30  140.30  +0.00  140.31  +0.01  150.02  150.11  +0.06  150.13  +0.07  169.29  169.33  +0.02  169.35  +0.04  170.73 177.06  170.66 177.08  -0.04 +0.01  170.68 177.08  -0.03 +0.01  1  4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  Spinning plus Gravity  I  Table 12: Highest Stress and Its Location for Harmonic Response Analysis No.  Excitation Frequency (Hz)  M a x . Stress (MPa)  M a x . Stress Location  2 Max. Location  1  42.5 60.0 85.0 120.0 127.5 170.0 180.0  321 2172 93 168 188 7102 734  C(S) A(S) D(P) D(P) D(P) D(P)  A(S) C(S) A(S) C(S) C(S) A(P) D(P)  2  3 4 5 6 7  A(P)  (S) = Suction Side  (P) = Pressure Side  L o c a t i o n based on labeling in Figure 2  Stress = V o n Mises Stress  93  n a  Table 13: Total Strain Range and Predicted Initiation Life Loading Case  M a x . Total Strain  M i n . Total Strain  Total Strain Range  Predicted Initiation Life (cycles)  Predicted Initiation Life (Days)*  2.5% H R 5% H R  0.11338E-2 0.13244E-2  0.48242E-3 0.12224E-3  6.5138E-4 1.2022E-3  2.062E+9 2.362E+7  397.7 4.556  * Note that the number of days calculation based on 60 Hz oscillation assumption and the turbine is operating 24 hours a day, 365.25 days a year.  Table 14: Stress Intensity Factor Range of 2.5%HR and 5%HR loadings Crack Length (m)  A K i at 2 . 5 % H R (MPaVm) Plain Plain Strain Stress  A K i at 5 % H R (MPaVm) Plain Plain Strain Stress  0.00250 0.00641 0.01282 0.01924 0.02566 0.03209 0.03852 0.04496 0.05141 0.05786 0.06431 0.07078 0.07724 0.08372 0.09020 0.09669 0.10319 0.10969 0.11620 0.12260 0.12901 0.13542 0.14184 0.14827 0.15470  10.71 17.90 23.64 25.96 26.92 27.17 27.06 26.79 26.48 26.06 25.60 25.15 24.60 24.05 23.54 22.96 22.38 21.85 21.26 20.71 20.19 19.61 19.05 18.52 17.94  21.42 35.79 47.29 51.93 53.84 54.35 54^12 53.59 52.96 52.11 51.20 50.30 49! 19 48.10 47.08 45.91 44.77 43.70 42.'53 41.41 40.38 39.23 38.10 37.04 35.87  9.74 16.29 21.52 23.63 24.49 24.73 24.62 24.38 24.10 23.71 23.30 22.89 22.38 21.89 21.42 20.89 20.37 19.88 19.35 18.84 18.37 17.85 17.34 16.85 16.32  94  19.49 32.57 43.03 47.26 48.99 49.46 49.25 48.76 48.20 47.42 46.59 45.77 44.76 43.77 42.84 41.78 40.74 39.76 38.70 37.69 36.75 35.70 34.67 33.71 32.64  Table 15: Total Fatigue Life Prediction Results 2 . 5 % H R Loading 5 % H R Loading Plain Strain Plain Stress Plain Strain Plain Stress Initiation Life (cycles) Propagation Life (cycles) Total Life (cycles) Total Life (Days)  2.06E+09  2.06E+09  2.36E+07  2.36E+07  1.16E+06  1.56E+06  1.26E+05  1.70E+05  2.06E+09  2.06E+09  2.37E+07  2.38E+07  397.99  398.06  4.58  4.59  Table 16: Effect of Installation of 40-70 Blade Stiffeners Mode Number  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  Frequency (Hz) % Experimental change Result  F E Model No Stiffeners  F E Model with Stiffeners  43.37  43.74  +0.9  46.40  49.01  58.42 87.79  Experimental with Stiffeners  % change  -  -  -  +5.6  43.58  48.88  +12.2  58.72 98.56  +0.52 +12.3  -  -  -  90.69  104.55  +15.4  100.12 115.59  104.42 121.44  +4.3 +5.1  -  -  -  123.95  144.47  +16.6  127.60  149.78  +17.4  140.30  149.49  +6.6  -  -  -  150.02  177.51  +18.3  152.93  180.56  +18.1  95  T a b l e 17: Effect of Thickness Modifications Percentage change o f Band Thickness* (%) +40 +20 -26.3 -55.7  -20 -40 -60  Percentage Change o f Crown Thickness (%)  +40 +60 +20 +40 +60  * Percentage Change of Band thickness refers to areas indicated in Figure 50.  96  Percentage Change o f Rotational Natural Frequency (%) -3.6 -1.8 +2.4 +5.3 +7.42 +9.80 +5.9 +11.2 +15.5  Appendix A: Conversion Method from Model Test Data to Prototype Data D = Diameter o f runner N = M o d e l Ratio = D / D p  CO = =  Angular Velocityf = frequency F == Centrifugal Force  (rad/s) (Hz) (N)  (m) (RPM) (m/s)  W = Weight a == Stress P == Specific weight  (N) (MPa) (kg/m )  m  H = Head Pressure n == Rotational Speed U = Peripheral Speed Suffix  (m)  p: prototype  3  m: model  Dimensional Analysis: (1) Peripheral speed o f runner  (Al) (2) Revolution  U -D p  m  (A2)  U -D m  N yH  p  n  (3) Centrifugal Force Centrifugal Force due to runner's weight  F p F„  W p W Centrifugal Force o f Waterni  D p D m  F„  W-D  •H. Ym  H., m m  \ „,J D  1 Hn  r  W... • D„CO.,  =N  \ »>J n  " H,.  (A3)  (A4)  (4) Stress Stress due to centrifugal force o f runner  a,  F D p n. F -D]  r -H  1  P  p  y -H  m  m  H  (A5)  Stress due to centrifugal force of water  H.,  (A6)  (5) Frequency o f Oscillating Stress  fp_ f Jm  H. N^H„ 1  97  (A7)  Appendix B: Theory of Modal Analysis Assuming free vibration and non-damped structure, the force equation may be expressed as: \M\ii}  + [K\u}  = {0}  <» B  where [M\ = mass matrix [K\ = stiffness matrix {u} = acceleration vector {u} = displacement vector For a linear system, free vibrations w i l l be harmonic in nature. Thus, the displacement vector can be written as: = {0}.  M  where  {O}  =  C O S ( ^ . -t)  ( > B2  eigenvector representing the mode shape of the i' frequency  h  natural  (JO . = the i' natural frequency (radian per unit time) h  t  = time  Substituting into equation ( B l ) , we get (-cof[M}  +  {K]){<S>},={0}  (  B  3  )  For non-trivial solution, the determinant of ([K] - co [M]) is zero. Thus, 2  \[K]-co [M]\ 2  = 0  (  B  4  )  This is a typical eigenvalue problem with n values o f co and n eigenvectors {O} that can satisfy equation (B4) where n is the total number o f D O F for the structure. A N S Y S provides a number o f eigenvalue extraction methods. Details are discussed i n the following.  Eigenvalue and Eigenvector Extraction Method A N S Y S provides two different types o f procedures: (1) full extraction from reduced matrix, and (2) partial extraction from full matrix. Since no presumption can be made on the vibration behavior of the runner, it is not easy to reduce the matrix before performing the analysis. Focus is placed, therefore, on the second type o f procedure. There are 4 types o f extraction methods available: (1) subspace iteration, (2) block Lanczos, (3) unsymmetric eigehsolver, and (4) damped eigensolver. Since the turbine runner does not have unsymmetric matrix, the unsymmetric eigensolver would be omitted. O n the other hand, damping o f the turbine runner would be mainly cause by water acted as added mass on the turbine. Thus, damped eigensolver is not appropriate for eigenvalue extraction. Damping using added mass effect would be discussed i n a separate section 2.3.  Subspace Iteration Method The subspace iteration method is an effective method widely used for eigenvalue extraction and is described in detail by Bathe [11, pp.954-978]. Some reasons o f its popularity are that the theory is relatively easy to understand and the method is easy to  98  program. The procedure is particularly suited for the calculation o f a few eigenvalues and eigenvectors o f a large finite element system. The solution procedure was named the subspace iteration method because the iteration is equivalent to iterating with a q-dimensional subspace. A total o f 6 hours C P U time was required for extracting 25 modes from the turbine F E model using the described P C system. 10  Block Lanczos Eigenvalue Extraction Method The block Lanczos eigenvalue extraction method is available for large symmetric eigenvalue problems. The eigensolver used in A N S Y S is based on theory found by Grimes et al [12]. This new variation o f algorithm performs recursive operation using a block o f vectors as opposed to the classical method that only uses one single vector. This solver has a higher convergence rate compared to the subspace iteration method. Extracting same number o f modes from the same model using the block Lanczos method required only 1.5 hours (Compared to 6 hours for subspace iteration method). Since subspace and block Lanczos method extract exactly the same natural frequencies and mode shapes, block Lanczos method was used for the analysis o f this work.  The solution procedure consists of the following three steps: 1. Establish q starting iteration vectors, q > p, where p is the number of eigenvalues and vectors to be calculated. 2. Use simultaneous inverse iteration o the q vectors and Ritz analysis to extract the "best" eigenvalue and eigenvector approximations from the q iteration vectors. 3. After iteration convergence, use the Sturm sequence check to verify that the required eigenvalues and corresponding eigenvectors have been calculated.  99  Appendix C: Derivation of Surface Panel Method for Added Mass Effect (Reference [16]) Notation [A] = Diagonal matrix o f panel areas [C] = Damping matrix {F/} = F l u i d force vector {F } = Structural and body force vector [H] = Coefficient matrix relating source strength to velocity potential [K] = Structural stiffness matrix [L] = Coefficient matrix relating source strength to normal velocity at panel control points [M\ = Structural mass matrix [MA] = A d d e d fluid mass matrix [7] = Transformation matrix for normal to global direction  g = Acceleration due to gravity G = Green's function n = Normal p — Pressure t = time  s  z = Height o f free surface f = Structural surface boundary  {«} = Nodal displacement Vector p = fluid density <j= Source strength (assumed constant on a panel) tj) = Velocity potential function  { V } = Vector o f control point normal velocities A structural dynamic response analysis by finite element method requires solution o f the structural equilibrium equations in the form: N  [M]u + [C]u + [K]u = F (t) + F (t) f  ( ) C1  s  To obtain the fluid forces on the structure, the relationship between the fluid pressure field and the interface accelerations must be defined. Using potential flow theory: 0 -»0  as d£  x -> oo  _du  (C2) (C3)  dn dt where T is the fluid/structure interface and the substantive derivative is used on the right hand side o f (C3). Linearized free surface boundary condition i f a free surface present: r  d#  d<f> ( ) 0 + g. dt " dz To solve the potential function using surface panel method, f is defined by a distribution o f singularities o f constant strength on panels on the interface surface. In case o f using a simple source distribution, the field potential at x=(x,y,z) is written in the form: 2  C4  2  <p(x)=[ G(x, x' )CT(X' )dT(x')  ( ) C5  where x ' represents the coordinate vector o f a point on the panel surface and a is the source strength. In discrete form (C5) becomes: w r (C6) «K*) = £ ^ [ G ( x , V ) d T ( x / ) Where the constant panel source strength has been removed from the panel integral. The kernel function G in the current formulation is the free space Green's function for a source o f strength o f 4%.  100  ^  rt  1  1  r  ( C 7 )  \x-xA  The gradient o f the velocity potential function is then forced to satisfy the boundary condition o f (C3) at a number o f control points on the body. The gradient is given by  on  ~T  i  on  r X  The integrals i n equation (C6) and (C8) can be evaluated exactly, and these analytical solution are used when the panel-to-control point separation is relatively small. For a particular set o f control points, the evaluation o f the integrals in equation (C6) and (C8) leads to the system  {fi = [H]{*}  (C9)  {V„} = [L]{a}  ( °) C1  with [H] a coefficient matrix relating source strength to control point potential and [L] a coefficient matrix relating source strength to control point fluid normal velocities. Relation (C9) and (CIO) define completely the fluid flow field given the distribution o f source strengths a. The fluid velocities normal to the panels at the control points can be related v i a equation (C3) to the local nodal point velocities i n the finite element model.  (c»>  m  lK]=[T  at  Rearranging (CIO) and ( C l 1), the panel source strength becomes  w)-m-'irm  (C12)  at  Substituting equation (C12) into (C9), the velocity potential at a control point is  m  m=lH]lLriT  >  (ci3  at  In case o f using dipoles o f constant strength on a panel, the integral equation is  •*  on  *  on  Similar to the derivation in source formulation:  [HW } = [Lm  (C15)  n  Solving for the control point potential and substituting for the surface velocity:  w=m-[/rrf  (C16)  at  The pressure field i n the fluid o f density p can be determined from the velocity potential via the Bernoulli equation  (C17)  d$ P  =  ~ ~dt P  Using equation (C13), therefore:  d iu\ 2  dt  101  ( ) C18  B y introducing the panel area matrix [A] and transform the local coordinates to the global coordinate v i a [T] and [ T ] , the fluid force vector can be expressed as T  d iu\ [A][H][Lf[T]-±!-  ( )  2  {F } = -p[T]' f  C19  B y neglecting rigid body motion and surface displacement derivatives and nonlinear terms from time derivatives, equation (C19) reduces to {F } = -[M ]{ii} f  (C20)  A  Therefore added mass matrix by source formulation is [M ] = p[T] [A][H Added mass matrix by dipole formulation is  ][!]-' [T]  T  A  (C21)  [M ] = -p[T] [A][L]- [H][T] T  (C22)  ]  A  The modified structural equilibrium equation is [M + M ] {ii} + [C] {u} + [K] {u} = {F (0 } A  x  102  (C23)  Appendix D: Summary of Element Properties Shell Element  Elastic Shell Element Assumptions and Restrictions Zero area elements are not allowed. This occurs most often whenever the elements are not numbered properly. Zero thickness elements or elements tapering down to a zero thickness at any corner are not allowed. The applied transverse thermal gradient is assumed to vary linearly through the thickness and vary bilinearly over the shell surface. A n assemblage o f flat shell elements can produce a good approximation to a curved shell surface provided that each flat element does not extend over more than a 15 degrees arc. If an elastic foundation stiffness is input, one-fourth o f the total is applied at each node. Shear deflection is not included in this thin-shell element. A triangular element may be formed by defining duplicate K and L node numbers. The extra shapes are automatically deleted for triangular elements so that the membrane stiffness reduces to a constant strain formulation. For large deflection analyses, the element must be triangular. The four nodes defining the element should lie in an exact flat plane; however, a small out-of-plane tolerance is permitted so that the element may have a slighty warped shape. A moderately warped element w i l l produce a warning message in the printout. If the warpage is too severe, a fatal message results and a triangular element should be used. If the lumped mass matrix formulation is specified, the effect o f the implied offsets on the mass matrix is ignored for warped elements.  103  Plastic Shell Element Assumptions and Restrictions Zero area elements are not allowed. This occurs most often whenever the elements are not numbered properly. Zero thickness elements or elements tapering down to a zero thickness at any corner are not allowed. Under bending loads and tapered elements produce inferior stress results and refined meshes may be required. Use o f this element in triangular form produces results o f inferior quality compared to the quadrilateral form. However, under thermal loads, when the element is doubly curved (warped), triangular elastic shell element produce more accurate stress result than do quadrilateral shaped element. Quadrilateral elastic shell elements may produce inaccurate stresses under thermal loads for doubly curved or warped domains. The applied transverse thermal gradient is assumed to vary linearly through the thickness. The out-of-plane (normal) stress for this element varies linearly through the thickness. The transverse shear stresses are assumed to be constant through the thickness. Shear deflections are included. Elastic rectangular elements without membrane loads give constant curvature results, i.e., node stresses are the same as the centroidal stresses. For linearly varying results use elastic shell elements. Triangular elements are not geometrically invariant and the element produces a constant curvature solution. Only the lumped mass matrix is available. M,N,O,P,U,V,W,X  3D 20-Node Solid Element Assumptions and Restrictions The element must not have a zero volume. Also, the element may not be twisted such that the element has two separate volumes. This occurs most frequently when the element is not numbered properly. A n edge with removed midside node implies that the displacement varies linearly, rather than parabolically, along that edge. Degeneration to the form o f pyramid should be used with caution. The element sizes, when degenerated, should be small i n order to minimize the stress gradients. Pyramid elements are best used as filler elements or in meshing transition zones.  104  Appendix E: Computation of Initiation Life in MathCAD. BC Hydro and Power Authority GMS Turbine Cracking Investigation Fatigue Initiation Life Prediction Strain Life Approach Date: April 14, 1999 By Brian Sze Bun Chan Assuming ASTM A27 similar to AISI 1020 properties 3  ksi := 10 psi E := 27000-ksi a = 123.2-ksi b = -0.12 := 0.44 f  c := -0.51  Loading Condition: 9 5 % Static + 5% Harmonic Response Analysis s £  m  a  =  x  min  A s  N  = 0.12224-10"  ~ max ~ = 10000 e  f  0.13244-10  2  3  min  8  Given  N  f  = 2.362-10  7  Days =  (60-3600-24) Days = 4.556 Years =  365.2! Years = 0.012  105  Loading Condition: 97.5% Static + 2.5% Harmonic Response Analysis s  = 0.11338 10"  2  m  s  a  = 0.48242-10"  3  m  m  "  A s  N  x  f  8  max ~ min 8  = 10000  Given ^ = N  f  E  f  ( 2 . N ) % ^ N f  := F i n d ( N  X  b  f  f  N = 2.062-10  9  f  Days := — (60-3600-24) Days = 397.681 Days Years = ——-— 365.25 Years = 1.089  106  

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