رسالة ماجستير بعنوان The “45 Degree Rule” and Its Impact on Strength and Stiffness of a Shaft Subjected to a Torsional Load

رسالة ماجستير بعنوان
The “45 Degree Rule” and Its Impact on Strength and Stiffness of a Shaft Subjected to a Torsional Load
Thesis Submitted to
The School of Engineering of the
University of Dayton
In Partial Fulfillment of the Requirements for
The Degree of Master of Science in Mechanical Engineering
By
Cory Alfred Nation
Dayton, Ohio
Table of Contents
Abstract Iv
Dedication Vii
Acknowledgements Viii
Table of Contents Ix
List of Figures Xi
List of Tables Xv
List of Symbols and Abbreviations Xvi
Chapter 1: Introduction 1
1 1 Research Area 1
1 2 Problem Motivating Work 3
1 3 Claims Motivating Research 4
1 4 Application and General Principle 4
1 5 Common Definitions and Terminology 5
1 6 Limitations and Assumptions 8
1 7 Summary of Research Methodology 9
CHAPTER 2: LITERATURE REVIEW 13
CHAPTER 3: DESIGN EQUATIONS 16
3 1 Summary of Loading 16
3 2 Strength Based Design Equations 16
3 3 Stiffness (Deflection) Based Design Equations 17
3 4 Stress Concentration Factor Effect 18
3 5 Impact to Critical Speed 19
CHAPTER 4: FINITE ELEMENT ANALYSIS 21
4 1 Analysis Technique 21
4 2 Mesh Sensitivity Analysis 24
4 3 Stress Flow Line Simulation 33x
4 4 Part Geometry, Loading, and Boundary Condition Summary 39
4 5 Analysis & Results 39
CHAPTER 5: ADDITIONAL CONSIDERATIONS 43
5 1 Additional Geometry Analysis 43
5 2 Addition of Defects in (Removed) 45 Degree Region 44
CHAPTER 6: DISCUSSION OF RESULTS 50
6 1 Overall Results Discussion 50
6 2 Standard Deviation Results 51
6 3 Complete Stress Concentration Charts 55
6 4 Additional Geometry Results Discussion 59
6 5 Analysis with Simulated Defects Discussion 63
CHAPTER 7: CONCLUSION & FUTURE WORK 64
7 1 Overall Results Discussion 64
7 2 Future Work 65
BIBLIOGRAPHY 67
APPENDIX A: FINITE ELEMENT ANALYSIS RESULTS 70
A 1 r/d Versus Kt Data 70
A 2 D/d Versus Kt Data 74
APPENDIX B: NATION FEA RESULTS COMPARED TO PETERSON’S Kt CHARTS 83xi
LIST OF FIGURES
Figure 1: Method for Modeling Abrupt Change in Diameter 2
Figure 2: Description of Fillet Radii 6
Figure 3: Description of Transverse and Uniaxial Holes 7
Figure 4: Stepped Shaft Geometry Outline 10
Figure 5: Sloped Shaft Geometry Outline 10
Figure 6: Meshed Part 23
Figure 7: Meshed Part in the Area of Interest 23
Figure 8: Part Fixture and Loading Summary 24
Figure 9: Sample Geometry Using Full Shaft Shoulder 26
Figure 10: Sample Geometry Using 45° Sloped Shaft Shoulder 26
Figure 11: Stress Concentration Factors Determined From Constant Geometry Model,
Varying Mesh Size 27
Figure 12: Three Areas of Interest for Error Study 30
Figure 13: FEA Reported Error Level for Various Mesh Sizes 31
Figure 14: Overall Stress Pattern Seen for 0 03125″ Mesh Size 33
Figure 15: Force Flow Lines Using Full Shaft Shoulder (σѵ, max=2,401 psi) 34
Figure 16: Force Flow Lines Using 45° Sloped Shaft Shoulder (σѵ, max =2,244 psi) 34
Figure 17: Displacement Flow Lines Using Full Shaft Shoulder (UY=3 599 x 10-4 in ) 36xii
Figure 18: Displacement Flow Lines Using 45° Sloped Shaft Shoulder
(UY=3 722 x 10-4 in ) 37
Figure 19: Representation of Converting Point Displacement to Angular Displacement 38
Figure 20: FEA Sample Result for 90° Step Shaft 41
Figure 21: FEA Sample Result for 45° Step Shaft 42
Figure 22: Sample Geometry Using 30° Shaft Shoulder 43
Figure 23: Sample Geometry Using 60° Shaft Shoulder 44
Figure 24: Simulated Defect – 0 25” Diameter by 0 75” Deep Hole 45
Figure 25: Simulated Defect – 0 5” Diameter by 0 5” Deep Hole 46
Figure 26: Simulated Defect – 1 0” by 1 0” Cut, 0 50” Wide 46
Figure 27: Simulated Defect – 0 5” by 0 5” Cut, 0 25” Wide 47
Figure 28: Sketch Illustrating Defects Contained Within 45° Region 47
Figure 29: Simulated Defect – 0 5” diameter by 1 5” deep holes 48
Figure 30: Simulated Defect – 1 0” by 2 0” Cut, 0 25” Wide 48
Figure 31: Simulated Defect Deviation from Baseline Run 49
Figure 32: Deviation between the 45° and the 90° Shaft Models Versions for D/d=2 5 51
Figure 33: Deviation between the 45° and the 90° Shaft Models for D/d=2 0 52
Figure 34: Deviation between the 45° and the 90° Shaft Models for D/d=1 666 52
Figure 35: Deviation between the 45° and the 90° Shaft Models for D/d=1 25 53
Figure 36: Average Standard Deviation for an analyzed r/d for All D/d analysis Points 54
Figure 37: Fixed D/d versus Varying r/d for Both Configurations 57
Figure 38: Fixed r/d versus Varying D/d for Both Shaft Configurations 58
Figure 39: Additional Geometry Stress for 30°, 45°, 60° and 0° (or 90°) Shoulder Angles 59xiii
Figure 40: Displacement Flow Lines Using 60° Sloped Shaft Shoulder
(UY=3 840 x 10-4 in ) 60
Figure 41: Displacement Flow Lines Using 30° Sloped Shaft Shoulder
(UY=3 666 x 10-4 in ) 61
Figure 42: Additional Geometry Stiffness Tested at 30°, 45°, 60° and 0° (or 90°) 62
Figure 43: r/d FEA Data – D/d=2 5 70
Figure 44: r/d FEA Chart – D/d=2 5 70
Figure 45: r/d FEA Data – D/d=2 0 71
Figure 46: r/d FEA Chart – D/d=2 0 71
Figure 47: r/d FEA Data – D/d=1 429 72
Figure 48: r/d FEA Chart – D/d=1 429 72
Figure 49: r/d FEA Data – D/d=1 25 73
Figure 50: r/d FEA Chart – D/d=1 25 73
Figure 51: D/d FEA Data – r/d=0 01 74
Figure 52: D/d FEA Chart – r/d=0 01 74
Figure 53: D/d FEA Data – r/d=0 02 75
Figure 54: D/d FEA Chart – r/d=0 02 75
Figure 55: D/d FEA Data – r/d=0 03 76
Figure 56: D/d FEA Chart – r/d=0 03 76
Figure 57: D/d FEA Data – r/d=0 05 77
Figure 58: D/d FEA Chart – r/d=0 05 77
Figure 59: D/d FEA Data – r/d=0 07 78
Figure 60: D/d FEA Chart – r/d=0 07 78
Figure 61: D/d FEA Data – r/d=0 10 79xiv
Figure 62: D/d FEA Chart – r/d=0 10 79
Figure 63: D/d FEA Data – r/d=0 15 80
Figure 64: D/d FEA Chart – r/d=0 15 80
Figure 65: D/d FEA Data – r/d=0 20 81
Figure 66: D/d FEA Chart – r/d=0 20 81
Figure 67: D/d FEA Data – r/d=0 30 82
Figure 68: D/d FEA Chart – r/d=0 30 82
Figure 69: Stress Concentration Factors, Kt, With Data from Nation and Pilkey 84
Figure 70: D/d=1 111 Kt Data – Pilkey Compared to Nation 85
Figure 71: D/d=1 25 Kt Data – Pilkey Compared to Nation 85
Figure 72: D/d=1 666 Kt Data – Pilkey Compared to Nation 86
Figure 73: D/d=2 0 Kt Data – Pilkey Compared to Nation 86
Figure 74: D/d=2 5 Kt Data – Pilkey Compared to Nation 87xv
LIST OF TABLES
Table 1: Von Mises Stress for Various Mesh Sizes 32
Table 2: Energy Norm Error Percent for Various Mesh Size 32
Table 3: Mechanical Properties of Analyzed Shaft 41
Table 4: (D-d)/r Recommended Practice 53xvi
LIST OF SYMBOLS AND ABBREVIATIONS
Designator Definition
A = Area
API = American Petroleum Institute
D = Larger Diameter
d = Lesser Diameter
d
p = Angular Displacement (Point to Point)
Dia, Ø = Diameter
G = Modulus of Rigidity
In = Inch
I = Moment of Inertia
J = Polar Moment of Inertia
k = Stiffness
Kt = Stress Concentration Factor for Normal Stress
Kt = Theoretical Stress Concentration Factor for Shear Stress
Kts = Theoretical Stress Concentration Factor for Normal
Stress
Kts = Stress Concentration Factor for Shear Stress
Kϴ = Deflection Concentration Factor
L, l = Length
Lbf = Pound Force
M = Moment
m = Mass
R, r = Fillet Radius
N = First Critical speed
T = Torque
U = Strain Energy
UX, UY, UZ = Deflection in Rectangular Coordinates
x, y, z = Rectangular Coordinates
ρ = Density
ϴ = Angle Between Lesser Diameter and Shaft Shoulder
θmax = Maximum Angular Displacement
θnom = Nominal Angular Displacementxvii
σFail = Predicted Failure – Principle Stress
σmax = Maximum Normal Stress
σnom = Nominal Normal Stress
σx-a = Alternating and Mean Bending Stress
τFail = Predicted Failure – Shear
τmax = Maximum Shear Stress
τnom = Nominal Shear Stress
τx-a = Alternating and Mean Torsional Shear Stress
φ = Angle of twist
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