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Design Engineer’s Handbook
Keith L. Richards
و المحتوى كما يلي :
Contents
List of Figures xv
List of Tables xxiii
Preface xxv
Chapter 1 Beams .1
1.1 Basic Theory 1
1.1.1 Introduction 1
1.1.2 Simple Elastic Bending 1
1.1.3 Shearing Force and Bending Moment 2
1.1.4 Shearing Force and Bending Moment Diagrams .5
1.1.4.1 Cantilever with a Concentrated End Load 5
1.1.4.2 Cantilever with a Uniformly Distributed Load, w, per
Unit Length .5
1.1.4.3 Simply Supported Beam with Central Concentrated Load 6
1.1.4.4 Simply Supported Beam with Uniformly Distributed
Load, w, per Unit Length 7
1.2 Stresses Induced by Bending .7
1.2.1 Pure Bending 7
1.2.2 Stress Due to Bending 8
1.2.3 The General Bending Formula .8
1.2.4 Example 9
1.2.4.1 To Determine Reaction Forces . 10
1.2.4.2 Shear Force Diagram 11
1.2.5 Best Position of Supports for Beams with Overhanging Ends . 12
1.3 Deflection in Beams 15
1.3.1 Area Moment 15
1.3.1.1 Example 17
1.3.2 Slope Deflection . 21
1.3.3 Deflection in Beams .26
1.3.3.1 Macaulay’s Method .26
1.4 Shear Deflection in Beams 29
1.4.1 Introduction 29
1.4.2 Determine the Shear Deflection in a Simply Supported Beam
with a Central Point Load 30
1.4.3 Shear Deflection of Short Beams .30
1.4.4 Short Beam with Uniformly Distributed Load over the Entire Span .30
1.4.5 Shear Deflection in a Short Beam with Uniformly Distributed
Load over Center Span . 31
1.5 Properties of a Plane Area . 32
1.5.1 Notation 32
1.5.2 General Definitions 32
1.5.2.1 Area 32
1.5.2.2 First Moment of Area . 32
1.5.2.3 Centroid: Center of Area 33vi Contents
1.5.2.4 Second Moment of Area . 33
1.5.2.5 Product of Inertia 34
1.5.2.6 Polar Moment of Inertia 34
1.5.2.7 Radius of Gyration 34
1.5.2.8 Moment of Inertia about Inclined Axes 35
1.5.2.9 Principal Axes 35
1.5.2.10 Principal Moments of Inertia 35
1.5.2.11 Mohr’s Circle for Moment of Inertia 36
1.5.3 Torsional Constant (J) 36
1.5.3.1 For Solid Sections .37
1.5.3.2 For Closed Sections 37
1.5.3.3 For Open Sections .37
1.5.4 Section Property Tables 37
1.5.5 Section Shear Centers 56
1.5.5.1 Location of Shear Center: Open Sections 57
1.5.5.2 Section Constants . 57
1.5.5.3 Shear Flow in an Element . 57
1.5.5.4 Shear Force in an Element 57
1.5.5.5 Torque/Moment Given by Element . 57
1.5.5.6 Horizontal Location of the Shear Center 58
1.5.5.7 Vertical Location of the Shear Center 58
1.5.5.8 Shear Center of a Curved Web .58
1.5.5.9 Horizontal Location 59
Further Reading . 62
Chapter 2 Torsion of Solid Sections 63
2.1 Introduction .63
2.2 Basic Theory 63
2.3 Modulus of Section 64
2.4 Angle of Twist .65
2.5 Pure Torsion of Open Sections 65
2.5.1 Thick-Walled Open Sections 65
2.5.2 Thin-Walled Open Sections . 67
2.6 Thin-Walled Closed Sections 68
2.6.1 Single Cell Sections 68
2.7 Curved Members . 71
2.7.1 Curved Torsion Members . 71
2.7.2 Circular Section 72
2.7.3 Square Section 73
2.7.4 Rectangular Sections 74
2.7.5 Springs 74
2.8 Torsional Failure of Tubes . 75
2.8.1 Modulus of Rupture: A Theoretical Approach 76
2.8.1.1 Instability Failure 76
2.8.2 Material Failure under Plastic Torsion . 78
2.9 Sand Heap Analogy for Torsional Strength . 78
2.9.1 Method (Solid Cross Section) . 78
Example 2.1 81
Example 2.2 81Contents vii
Chapter 3 Design and Analysis of Lugs and Shear Pins .83
3.1 Notation .83
3.2 Introduction .84
3.2.1 Method .85
3.2.2 Loading 85
3.2.3 Material Limitations .85
3.2.4 Geometric Limitations .85
3.2.5 Failure Modes .86
3.2.6 Notes .86
3.3 Analysis of Lugs with Axial Loading: Allowable Loads 87
3.3.1 Analysis Procedure to Determine the Ultimate Axial Load 87
3.4 Analysis of Lugs with Transverse Loading: Allowable Loads 91
3.5 Oblique Loading: Allowable Loading .94
3.5.1 Analysis Procedure 95
3.5.2 Out-of-Plane Loading .95
3.6 Bearing at Lug-to-Pin or -Bush Interface 96
3.7 Shear Pin Analysis .96
3.7.1 Shear Pin Bending in Double Shear Joint 96
3.7.1.1 Shear Pin Bending: Load Peaking between Center
Lug and Pin .98
3.7.1.2 Shear Pin Bending, Including Excess Strength of Lug 99
3.7.2 Pin Shear 100
3.8 Bush Analysis 101
3.9 Special Cases . 102
3.9.1 Oil Holes . 102
3.9.1.1 Axial Load 102
3.9.1.2 Transverse Load 102
3.9.1.3 Oblique Load 102
3.9.2 Eccentric Hole 102
3.9.3 Multiple Shear Connection . 102
3.10 Stresses Due to Interference-Fit Pins and Bushes . 104
3.10.1 Solid Circular Interference-Fit Shear Pins . 104
3.10.2 Interference-Fit Bushes 105
3.11 Stress Concentration Factor at Lug-to-Pin Interface . 106
3.12 Examples . 106
Further Reading . 111
Chapter 4 Mechanical Fasteners . 113
4.1 Threaded Fasteners 113
4.2 Basic Types of Threaded Fasteners: 113
4.3 Thread Standards . 113
4.4 Thread Profiles 113
4.5 Thread Series . 113
4.6 Thread Designations 115
4.6.1 Metric Series 115
4.6.2 Imperial (Inch Series) . 115
4.7 Material and Strength Designations 117
4.8 Tensile and Shear Stress Areas 118
4.8.1 Tensile Stress Area . 118viii Contents
4.8.2 Shear Area of External Threads . 119
4.8.3 Shear Area of Internal Threads 120
4.9 Length of Engagement .120
4.9.1 Length of Engagement Using Equal Strength Materials .120
4.9.2 Length of Engagement Using Dissimilar Strength Materials 121
4.10 Fastener and Nut Design Philosophies 121
4.11 Pitching of Fasteners . 122
4.11.1 Pressure Cone . 122
4.12 Tension Connections 122
4.13 Torque-Tension Relationship . 123
4.14 Proof Load and Proof Stress .124
4.14.1 Fastener Preload .124
4.15 Introduction to Pretension .124
4.15.1 Why Preload? .126
4.16 Joint Diagrams .126
4.16.1 Joint Diagrams with an External Load Applied 127
4.16.2 Effects of a Large Increase in the External Load 128
4.16.3 The Effect of a Compressive External Load 129
4.16.4 Nomenclature . 130
4.16.5 Notes . 130
4.17 Fastener Stiffness 131
4.18 Joint Stiffness 133
4.18.1 Calculation of Load Distribution Using Fastener/Joint Stiffness . 133
4.19 Thermal Loading . 134
4.19.1 Initial Tension in Bolt . 135
4.20 Fasteners Subject to Combined Shear and Tension . 136
4.20.1 Interaction Curves: Load Ratios and Factors of Safety . 136
4.20.2 Interaction Curve 136
4.20.3 Interaction Equation . 137
4.21 Eccentric Loads . 137
Example 4.1 137
4.21.1 Permissible Shear Stress . 139
4.22 Prying Forces . 139
Example 4.2 139
4.23 Fasteners Subject to Alternating External Force . 141
4.23.1 Factor of Safety (FoS) with No Preload . 142
4.23.2 The Minimum Preload to Prevent Any Loss of Compression . 143
4.23.3 Calculate the FoS for the Bolt with a Preload of 22,000 N 143
4.23.4 The Minimum Force in the Part when the Preload is 22,000 N 145
Chapter 5 Limits and Fits 147
5.1 Introduction . 147
5.2 Tolerance Grade Numbers . 147
5.2.1 Tolerance 147
5.2.2 International Tolerance Grade Numbers 147
5.3 Fundamental Deviations 148
5.3.1 Preamble . 148
5.3.2 Fundamental Deviation 149
5.3.3 Fundamental Deviations for Shafts 150
5.3.4 Fundamental Deviations for Holes . 150Contents ix
5.3.5 Upper and Lower Deviations 151
5.3.5.1 Shaft Letter Codes c, d, f, g, and h . 151
5.3.5.2 Shaft Letter Codes k, n, p, s, and u . 151
5.3.5.3 Hole Letter Code H . 151
5.4 Preferred Fits Using the Basic Hole System 152
5.4.1 Loose Running Fit (Example) 152
5.4.2 Location Clearance Fit (Example) . 153
5.5 Surface Finish 154
Chapter 6 Thick Cylinders 155
6.1 Introduction . 155
6.2 A Thick-Walled Cylinder Subject to Internal and External Pressures 155
6.3 General Equations for a Thick-Walled Cylinder Subject to an Internal
Pressure . 157
6.4 The General Equation for a Thick-Walled Cylinder Subject to Internal
and External Pressures 158
Example 6.1: Interference Fit . 160
Example 6.2: Radial Distribution of Stress 163
Chapter 7 Compound Cylinders . 165
7.1 Introduction . 165
7.2 Shrinkage Allowance 165
Example 7.1 167
Example 7.2 170
Further Reading . 174
Chapter 8 The Design and Analysis of Helical Compression Springs Manufactured from
Round Wire 175
8.1 Elastic Stresses and Deflections of Helical Compression Springs
Manufactured from Round Wire . 175
8.1.1 Introduction 175
8.1.2 Notation 176
8.1.3 Notes . 176
8.1.4 Compression Spring Characteristics 177
8.1.4.1 Material Specifications . 177
8.1.4.2 Wire Diameter 177
8.1.4.3 Mean Diameter . 177
8.1.4.4 Spring Index c . 177
8.1.4.5 Spring Rate . 177
8.1.4.6 Number of Active Coils 177
8.1.4.7 Total Number of Coils 177
8.1.4.8 Solid Length . 178
8.1.4.9 Initial (Free) Length . 178
8.1.4.10 Clearance at Maximum Load . 178
8.1.4.11 Direction of Wind . 178
8.1.4.12 Allowable Stresses 178
8.1.4.13 Finish 178x Contents
8.1.5 Static Shear Stress 178
8.1.5.1 Basic Formulas . 178
8.1.5.2 Maximum Shear Stress . 179
8.1.5.3 Useful Relations 179
8.1.5.4 Relationships . 180
8.1.6 Spring and Deflection Characteristics 185
8.1.7 Solid Length Characteristics 186
8.1.8 Buckling of Compression Springs 186
8.1.9 Transverse Loading 187
8.1.10 Increase of Spring Diameter under Compression 189
8.1.10.1 Ends Free to Rotate . 189
8.1.10.2 Ends Restrained against Rotation . 189
8.1.11 Helix Warping in Compression Springs . 189
8.1.12 Natural Frequency 189
8.1.12.1 First Natural Frequency 190
8.1.13 Example 1 . 190
8.1.13.1 Design of a Helical Compression Spring 190
8.1.13.1 End Coil Formulation . 191
8.1.13.2 Working Stress 191
8.1.13.3 Determination of D, d, and c 191
8.1.13.4 Determining Spring Rate, Deflection Characteristics,
and Number of Coils . 194
8.1.13.5 Conformity with Limitations of Section 8.3 . 195
8.1.13.6 Closed Length Characteristics 195
8.1.13.7 Buckling Characteristics . 196
8.1.13.8 Determination of Coil Growth under Compression . 197
8.1.13.9 Natural Frequency of the Spring 197
8.1.13.10 Stress Increase Due to Helix Warping 198
8.2 Allowable Stresses for Helical Compression Springs Manufactured
from Round Wire . 198
8.2.1 Static Strength Data 198
8.2.2 Fatigue Data . 198
8.2.2.1 Estimation of an S-N Curve . 199
8.2.3 Factors Affecting Spring Life 199
8.2.3.1 Spring Geometry 199
8.2.3.2 Corrosion 201
8.2.3.3 Surface Finish . 201
8.2.3.4 Elevated Temperatures 201
8.2.4 Treatments for Improving the Fatigue Life of Springs 201
8.2.4.1 Prestressing . 201
8.2.4.2 Shot Peening . 201
8.2.4.3 Abrasive Cleaning 201
8.3 Notes on the Design of Helical Compression Springs Made
from Round Wire .205
8.3.1 General Notes .205
8.3.2 Prestressing 205
8.3.3 Choice of Material 206
8.3.3.1 Operation Reliability 206
8.3.3.2 Corrosion and Protection 206
8.3.3.3 Working Temperature .206
8.3.3.4 Special Requirements .206Contents xi
8.3.4 Loading 206
8.3.4.1 Cyclic Loading and Fatigue Properties 208
8.3.4.2 Transverse Loading 208
8.3.4.3 Impact .208
8.3.4.4 Eccentric Loading .208
8.3.4.5 Buckling 208
8.3.5 Design Features 209
8.3.5.1 End Forms .209
8.3.5.2 Free Length . 210
8.3.5.3 Solid Length . 210
8.3.5.4 Tolerances . 210
8.3.5.5 Surface Finish . 211
8.3.5.6 Surface Treatment . 211
8.3.6 Design Procedures 211
8.3.6.1 Basic Design . 211
8.3.7 Manufacturing Requirements . 211
8.4 Nested Helical Compression Springs 214
8.4.1 General Notes to Section 8.4 214
8.4.2 Example 2 .220
8.4.3 Nested Springs in Series .223
8.4.4 Example 3 .224
Further Reading .227
Chapter 9 Introduction to Analytical Stress Analysis and the Use of the Mohr Circle 229
9.1 Introduction .229
9.2 Notation .229
9.3 Two-Dimensional Stress Analysis .230
9.4 Principal Stresses and Principal Planes 231
9.4.1 Maximum Shear Stress 232
9.4.2 Geometric Interpretation 233
9.5 Construction of the Mohr Circle .234
9.5.1 Conclusions and Deductions 235
9.6 Relationship between Direct and Shear Stress 236
9.7 The Pole of the Mohr Circle 237
9.7.1 A Few Special Cases 237
9.8 Examples . 239
9.9 The Analysis of Strain .245
9.9.1 Sign Conventions for Strains 245
9.10 Comparison of Stress and Strain Equations 246
9.10.1 The Strain Rosette 247
9.10.2 Construction .247
9.10.3 Conclusion 251
9.11 Theories of Elastic Failure 251
9.11.1 Steady Load Failure Theories 251
9.11.2 Maximum Principal Stress (Rankin’s) Theory 251
9.11.3 Maximum Principal Strain (St. Venant’s) Theory . 252
9.11.4 Maximum Shear Stress (Guest’s or Tresca’s) Theory 253
9.11.5 Distortion Energy Theory 253xii Contents
9.11.5.1 Strain Energy (Haigh’s) Theory 253
9.11.5.2 Shear Strain Energy (Von Mises’s) Theory 255
9.11.6 Conclusions 255
9.12 Interaction Curves, Stress Ratio’s Margins of Safety, and Factors of Safety .256
9.12.1 Interaction: Stress Ratio .256
9.12.2 Interactive Curve 256
9.12.3 Interaction, Stress Ratios, Yield Conditions 257
9.12.4 Interaction Equations: Yield Conditions 258
9.12.5 Interaction Equations: Failure Conditions 259
9.12.5.1 Compact Structures: No Bending .259
9.12.6 Compact Structures: Bending 259
9.12.7 General Interaction Relationships (Figures 9.33–9.36) 260
9.12.8 Determination of Safety Factors 263
Chapter 10 Introduction to Experimental Stress Analysis 265
10.1 Photoelasticity .265
10.1.1 The Principles of Photoelasticity 265
10.1.2 Principles 266
10.1.3 Isoclinics and Isochromatics 266
10.1.4 Plane Polariscope . 267
10.1.5 Circular Polariscope . 267
10.1.6 Two-Dimensional and Three-Dimensional Photoelasticity . 267
10.1.7 Further Development 268
10.2 Photoelastic Coatings 268
10.2.1 Preparation of the Coating .269
10.2.2 Analysis of the Coating 269
10.2.3 Coating Materials . 270
10.2.4 Full Field Interpretation of Strain Distribution 270
10.3 Introduction to Brittle Lacquer Coatings 270
10.3.1 Introduction 270
10.3.2 Loading and Testing Techniques 271
10.3.3 Effects of Change in Relative Humidity and Temperature 271
10.3.4 Measuring Strain under Static Loading . 272
10.4 Introduction to Strain Gauges . 272
10.4.1 Vibrating Wire Strain Gauges 272
10.4.2 Electrical Resistance Strain Gauges . 273
10.4.3 Unbalanced Bridge Circuit . 275
10.4.4 Null Balance or Balanced Bridge Circuit . 275
10.4.5 Installation Procedures . 275
10.5 Extensometers 277
10.5.1 Contact Extensometers .277
10.5.2 Noncontact Extensometers . 278
10.5.2.1 General Notes . 278
10.5.3 Applications 278
Chapter 11 Introduction to Fatigue and Fracture 279
11.1 Introduction and Background to the History of Fatigue . 279
11.1.1 Later Developments 281Contents xiii
11.1.2 Recent Developments .282
11.1.3 Basic Definitions 284
11.2 The Fatigue Process .285
11.3 Initiation of Fatigue Cracks .287
11.4 Factors Affecting Fatigue Life 288
11.4.1 Stress Amplitude 289
11.4.2 Mean Stress 289
11.5 Stress Concentrations 291
11.5.1 The Elastic Stress Concentration Factor 292
11.5.2 The Fatigue Stress Concentration Factor .293
11.6 Structural Life Estimations .294
11.7 Introduction to Linear Elastic Fracture Mechanics .294
11.7.1 Preamble .295
11.7.2 Comparison of Fatigue and Fracture Mechanics .295
11.7.3 The Difference between Classical Fatigue Analysis and
Fracture Mechanics 295
11.7.4 Stress Intensity .296
11.7.4.1 General Stress Intensity Solution 297
11.7.5 Fracture Toughness and Crack Growth 298
11.8 Fatigue Design Philosophy 300
11.8.1 Fail-Safe .300
11.8.2 Safe-Life .300
11.9 Cycle Counting Methods . 301
11.9.1 Introduction to Spectrum Cycle Counting . 301
Chapter 12 Introduction to Geared Systems .305
12.1 Introduction .305
12.2 Types of Gears .305
12.2.1 Spur Gears 305
12.2.2 Internal Spur Gears 306
12.2.3 Rack and Pinion .306
12.2.4 Helical Gears 307
12.2.5 Double Helical Gears .308
12.2.6 Spiral Bevel Gears 308
12.2.7 Bevel Gears 308
12.2.8 Spiral Gears 309
12.2.9 Worm and Worm Wheels .309
12.3 Form of Tooth 310
12.4 Layout of Involute Curves . 311
12.5 Involute Functions . 313
12.6 Basic Gear Transmission Theory 316
12.6.1 Torque and Efficiency . 317
12.7 Types of Gear Trains . 317
12.7.1 Simple Gear Train 317
12.7.2 Compound Gears 319
12.8 Power Transmission in a Gear Train . 319
12.9 Referred Moment of Inertia, (Ireferred) 322
12.10 Gear Train Applications 323xiv Contents
Chapter 13 Introduction to Cams and Followers 329
13.1 Introduction . 329
13.2 Background 329
13.3 Requirements of a Cam Mechanism . 330
13.4 Terminology 330
13.4.1 Plate Cams 330
13.4.2 Cylindrical Cams . 331
13.4.3 Typical Cam Follower Arrangements for Plate-Type Cams 332
13.5 The Timing Diagram . 332
13.6 Cam Laws 333
13.6.1 Constant Velocity of the Follower 335
13.6.2 Parabolic Motion 336
13.6.3 Simple Harmonic Motion . 336
13.6.4 Cycloidal Motion 339
13.7 Pressure Angle . 341
13.8 Design Procedure 342
13.9 Graphical Construction of a Cam Profile 342xv
List of Figures
FIGURE 1.1 Stresses in a cantilever beam. 2
FIGURE 1.2 Stresses generated in a beam between supports. .2
FIGURE 1.3 Shearing forces and bending moments acting on a beam in (a)–(c). .4
FIGURE 1.4 Simply supported beam. 4
FIGURE 1.5 Conventions used in shearing and bending 4
FIGURE 1.6 Cantilever with a concentrated load .5
FIGURE 1.7 Cantilever with a uniformly distributed load .6
FIGURE 1.8 Simply supported beam with a central load .6
FIGURE 1.9 Simply supported beam with a uniformly distributed load. 7
FIGURE 1.10 Surfaces in a section due to a bending moment .7
FIGURE 1.11 (a) Represents a cross-section of a beam, to which (b) a bending moment M
has been applied .8
FIGURE 1.12 Stresses induced in a beam due to bending. 9
FIGURE 1.13 Uniformly distributed and multiple loads on a beam. .9
FIGURE 1.14 Shear force diagram . 11
FIGURE 1.15 Bending moment diagram 12
FIGURE 1.16 Beams with overhanging ends. 13
FIGURE 1.17 Uniformly distributed load acting on a beam with overhanging ends. (a)
Beam with a uniformly distributed load with symmetrical overhanging
supports. (b) Bending moment diagram for the beam in (a). . 14
FIGURE 1.18 Analysis of a cantilever using the area-moment method . 15
FIGURE 1.19 Beam in Theorem 2. 17
FIGURE 1.20 Load and bending moment diagram for Example 1.3.1.1. . 18
FIGURE 1.21 Slope–deflection symbols. . 21
FIGURE 1.22 Example of a continuous beam 22
FIGURE 1.23 Loading, deflection, and bending moment diagram for Example 1.2 25
FIGURE 1.24 Simply supported beam with a single offset center concentrated load 26
FIGURE 1.25 Simply supported beam with a central load .30
FIGURE 1.26 Short beam with a distributed load over the entire span 30
FIGURE 1.27 Graph for Figure 1.26 with l/d < 3.0. Uniform load over entire length 31
FIGURE 1.28 Uniform load over central portion of beam. 31xvi List of Figures
FIGURE 1.29 Graph of Figure 1.28 with l/d < 3.0 31
FIGURE 1.30 Moment of inertia–general definitions . 33
FIGURE 1.31 Moments of inertia about an inclined plane. . 35
FIGURE 1.32 Features of the Mohr’s circle. 36
FIGURE 1.33 Torsional constant for a solid section .37
FIGURE 1.34 Torsional constant for a closed section 37
FIGURE 1.35 Torsional constant for an open section .37
FIGURE 1.36 Shear flows around various sections 56
FIGURE 1.37 Location of shear center–open section. . 57
FIGURE 1.38 Shear center for a curved web 58
FIGURE 1.39 Shear flow around a semicircle 60
FIGURE 2.1 Cross section of a circular shaft subject to pure torsion. .64
FIGURE 2.2 Angle of twist .65
FIGURE 2.3 (a) L-shaped section. Concave curve. c = point of maximum stress; r =
radius of fillet; θ = angle between elements (radians); d = diameter of largest
inscribed circle; A = area of cross section. (b) Convex curve (i.e., at bulb) . 67
FIGURE 2.4 Torsion in a thin-walled section .70
FIGURE 2.5 Torsion in a closed section. A = area enclosed by median contour; b = total
length of median contour; ds = length of differential element along median
contour; t = local wall thickness 70
FIGURE 2.6 Torsion in a circular section .73
FIGURE 2.7 Torsion in a square section .73
FIGURE 2.8 Torsion in a rectangular section . 74
FIGURE 2.9 Ratio of R to b for the aspect ratio of the section. . 74
FIGURE 2.10 Spring forms. (a) Circular section; (b) square section. 75
FIGURE 2.11 Notation 75
FIGURE 2.12 Elastic and plastic stress distribution . 76
FIGURE 2.13 Sand heap analysis–example section. 78
FIGURE 2.14 Angle of repose. .80
FIGURE 2.15 Section for Example 2.2 . 81
FIGURE 3.1 Typical lug assembly. .84
FIGURE 3.2 Lug loading. .85
FIGURE 3.3 Types of lugs. .86
FIGURE 3.4 Lug failure modes. .86
FIGURE 3.5 Detail of lug with bushing. 87List of Figures xvii
FIGURE 3.6 Effect of grain direction subject to axial load. L, T, N indicates the grain
direction “F” in sketch; L = Longitudinal; T = Long transverse; N = Short
transverse (normal). .88
FIGURE 3.7 Shear efficiency factor Kbr 90
FIGURE 3.8 Factors for calculating yield axial loads attributable to shear bearing of lug
and pin 90
FIGURE 3.9 Transverse areas calculations . 91
FIGURE 3.10 Substantiated lug shapes. . 91
FIGURE 3.11 Efficiency factor for transverse load Ktru. .93
FIGURE 3.12 Lug as a cantilever carrying load .93
FIGURE 3.13 Oblique loading 94
FIGURE 3.14 Out-of-plane loading, load direction, and bending section 95
FIGURE 3.15 Pin moment arm .97
FIGURE 3.16 Pin moment arm with clearance. .98
FIGURE 3.17 Peaking factors for pin bending. (Dashed lines indicate where the
theoretical curves are not substantiated by test data.) 99
FIGURE 3.18 Bending Modulus of Rupture .99
FIGURE 3.19 Load peaking. 100
FIGURE 3.20 Allowable shear stress factor for pin in double shear. . 101
FIGURE 3.21 Modification of allowable load for the presence of a lubrication hole . 102
FIGURE 3.22 Equivalent lug for a lug with an eccentric hole 103
FIGURE 3.23 Multiple shear connections. . 103
FIGURE 3.24 Points of high stress in bore . 104
FIGURE 3.25 Bush shape factor. 105
FIGURE 3.26 Stress concentration factor for round lugs. 106
FIGURE 3.27 K0.2 stress concentration factor for square lugs 107
FIGURE 3.28 K
(e < 0.1) stress concentration factor for square lugs . 107
FIGURE 3.29 Correction factor for pin to hole clearance. . 108
FIGURE 3.30 Effect of lug thickness. 108
FIGURE 3.31 Detail of fitting . 109
FIGURE 4.1 Bolt . 114
FIGURE 4.2 Screw 114
FIGURE 4.3 Types of heads 114
FIGURE 4.4 Typical thread profile . 115
FIGURE 4.5 Tensile stress area. . 118xviii List of Figures
FIGURE 4.6 Shear area of external thread . 119
FIGURE 4.7 Shear area of internal thread 121
FIGURE 4.8 Effects of flange thickness on fastener pitch 122
FIGURE 4.9 A typical bolted joint. 123
FIGURE 4.10 Bolt preload 124
FIGURE 4.11 Bolt stiffness. . 125
FIGURE 4.12 Abutment stiffness. 125
FIGURE 4.13 Tension diagram .125
FIGURE 4.14 Description of preload 126
FIGURE 4.15 Basic joint diagram. . 127
FIGURE 4.16 Basic pre-tension diagram. 127
FIGURE 4.17 Pre-tension diagram with preload 128
FIGURE 4.18 Effects of fastener stiffness 128
FIGURE 4.19 Effect of increase in the external load. 129
FIGURE 4.20 Effect of a compressive load. . 130
FIGURE 4.21 Bolted assembly with a preload and external load applied 131
FIGURE 4.22 Bolt loading diagram with preload and external load applied . 131
FIGURE 4.23 Bolt loading diagram with external load matching preload . 132
FIGURE 4.24 Stiffness characteristics of a bolt. 133
FIGURE 4.25 Multiple clamping surfaces 133
FIGURE 4.26 Interaction curve. . 137
FIGURE 4.27 Bracket subject to shear loading. . 138
FIGURE 4.28 Centroid of bolt group 138
FIGURE 4.29 Angle of forces for Example 4.1. . 139
FIGURE 4.30 Bracket without preload. 140
FIGURE 4.31 Bracket with preload. . 142
FIGURE 4.32 Modified Goodman diagram for Example 4.3 . 143
FIGURE 4.33 Effect of preload on the magnitude of the alternating force 145
FIGURE 5.1 Nomenclature used in limits and fits. 148
FIGURE 6.1 Elements of a thick-walled cylinder . 156
FIGURE 6.2 Stress relationships . 156
FIGURE 6.3 Thick cylinder subject to an internal pressure . 158
FIGURE 6.4 Interference fit 160
FIGURE 6.5 Components of shaft and hub. . 160List of Figures xix
FIGURE 7.1 Compound cylinder 166
FIGURE 7.2 Algebraic combination of stresses. (a) Stress distribution in a homogeneous
cylinder. (b) Stress distribution in a compound cylinder. 166
FIGURE 7.3 Variation in the circumferential stress across the cylinder 174
FIGURE 8.1 Compression spring with ends closed and ground flat under 4 stages of
loading 186
FIGURE 8.2 End function “H” for various end conditions. . 187
FIGURE 8.3 Critical values of HD/L
o. . 188
FIGURE 8.4 Compression spring under combined axial and transverse loading. . 188
FIGURE 8.5 Stress increase due to helix warping 190
FIGURE 8.6 Natural frequency curve. . 191
FIGURE 8.7 Stresses in spring subject to various cyclic loading . 192
FIGURE 8.8 Damping coefficient . 192
FIGURE 8.9 Compression spring diagram. 193
FIGURE 8.10 Shear stress in outer fibers of spring wire during prestressing 199
FIGURE 8.11 Shear stress distribution before, during, and after prestressing superimposed
on wire diameter. . 199
FIGURE 8.12 Presentation of fatigue data. 200
FIGURE 8.13 Estimated S-N curve for BS 1408 Range 3 at a constant mean stress:
500MN/m2. 200
FIGURE 8.14 Spring end forms 209
FIGURE 8.15 Nest of three springs. . 214
FIGURE 8.16 Nested springs in series. 223
FIGURE 9.1 A unit thickness element subject to a biaxial stress field .230
FIGURE 9.2 Section of element in Figure 9.1 . 231
FIGURE 9.3 Principal stresses and planes: complex stress system 232
FIGURE 9.4 Principal stresses and planes: maximum shear stresses. . 233
FIGURE 9.5 Sign conventions. . 235
FIGURE 9.6 Mohr circle diagram. . 235
FIGURE 9.7 Stresses acting on the element. 235
FIGURE 9.8 Section through a prismatic bar. 236
FIGURE 9.9 Distribution of stress from Figure 9.8 237
FIGURE 9.10 The pole of the Mohr circle (a). .238
FIGURE 9.11 The pole of the Mohr circle (b) 238
FIGURE 9.12 Types of loading that is treated using the Mohr circle . 239xx List of Figures
FIGURE 9.13 Numerical example for treatment by Mohr circle 240
FIGURE 9.14 Example. 241
FIGURE 9.15 Example–loading on a shaft. .242
FIGURE 9.16 Example–universal beam part 1. .244
FIGURE 9.17 Example–universal beam part 2. .245
FIGURE 9.18 Sign conventions for strain .246
FIGURE 9.19 The strain rosette. 247
FIGURE 9.20 Mohr strain circle .248
FIGURE 9.21 Rosette strains 248
FIGURE 9.22 Rosette axes. 249
FIGURE 9.23 Rearrangement of rosette axes .249
FIGURE 9.24 Strain plot .250
FIGURE 9.25 Construction of Mohr strain circle .250
FIGURE 9.26 Construction of Mohr strain circle (continued). 250
FIGURE 9.27 Principle orthogonal principal stresses 251
FIGURE 9.28 Maximum principal stress (Rankin’s) theory. . 252
FIGURE 9.29 Maximum principal strain (St. Venant’s) theory 253
FIGURE 9.30 Maximum shear stress (Guest or Tresca’s) theory. Note: In quadrants I and
III the maximum principal stress theory and the maximum shear stress
theory are the same for the biaxial case. .254
FIGURE 9.31 Strain energy (Haigh’s) theory. 254
FIGURE 9.32 Interaction curve. .256
FIGURE 9.33 General interactive relationships 260
FIGURE 9.34 Direct compression and bending 260
FIGURE 9.35 Offset compression and bending 260
FIGURE 9.36 Amplification factor “k” for bending moment . 261
FIGURE 9.37 Truss diagram. .262
FIGURE 9.38 Interaction curve for truss beam example 263
FIGURE 10.1 Schematic of the working components for a plane polariscope. . 267
FIGURE 10.2 Schematic of the working components for a circular polariscope .268
FIGURE 10.3 Schematic representation of a reflection polariscope. 269
FIGURE 10.4 Schematic diagram of a vibrating wire strain gauge. 273
FIGURE 10.5 Elements of a linear electrical resistance strain gauge. . 273
FIGURE 10.6 Wheatstone bridge circuit 274List of Figures xxi
FIGURE 10.7 Three lead-wire system for half-bridge (dummy-active) setup. . 275
FIGURE 10.8 Three lead-wire system for quarter-bridge (dummy-active) set-up with a
single self-temperature compensated gauge . 275
FIGURE 10.9 Huston’s extensometer 277
FIGURE 10.10 Extensometer for plastic test coupons 277
FIGURE 10.11 Extensometer for metal test coupons. . 278
FIGURE 11.1 Wöhler’s original railway axle .280
FIGURE 11.2 Representation of a rotating cantilever test rig. .280
FIGURE 11.3 The S-N curve as described by Wöhler. (1 centners per zoll2 = 0.75 Mpa.) 280
FIGURE 11.4 The stress and strain curve for cyclic loading 281
FIGURE 11.5 Fatigue stress–static stress diagram . 281
FIGURE 11.6 Various modes of failure 283
FIGURE 11.7 Crack-growth rate curve .284
FIGURE 11.8 Stress versus time curve .284
FIGURE 11.9 Schematic arrangement of a Wöhler fatigue test. 285
FIGURE 11.10 Typical fatigue curves. .286
FIGURE 11.11 Example of a typical flight cycle 286
FIGURE 11.12 Electrohydraulic fatigue test machine 287
FIGURE 11.13 Wood’s model for fatigue crack initiation 288
FIGURE 11.14 Stress cycles with different mean stresses and “R”-ratios. 289
FIGURE 11.15 Amplitudes of the stress cycle. 290
FIGURE 11.16 Comparison of Gerber, Soderberg, and Goodman laws viewed against stress
amplitude and mean tensile stress. 290
FIGURE 11.17 Typical master diagram 290
FIGURE 11.18 Effect of a stress concentration 292
FIGURE 11.19 Experimental S-N curves for a notched and un-notched specimen .293
FIGURE 11.20 Mode I crack under a biaxial stress field. 297
FIGURE 11.21 Fatigue crack as a function of life measured in cycles of stress. .299
FIGURE 11.22 Fatigue crack growth rate as a function of stress intensity function. 299
FIGURE 11.23 Schematic stress history of a detail subject to random variable amplitude
loading 301
FIGURE 11.24 Stress–strain response for strain history shown in Figure 11.25. 302
FIGURE 11.25 Rainflow diagram. .303
FIGURE 12.1 Spur gear. .306
FIGURE 12.2 Internal spur gear. 306xxii List of Figures
FIGURE 12.3 Rack and pinion. 307
FIGURE 12.4 Helical gears. .307
FIGURE 12.5 Double helical gears .308
FIGURE 12.6 Spiral bevel gear .308
FIGURE 12.7 Bevel gears .309
FIGURE 12.8 Spiral gears. .309
FIGURE 12.9 Worm and worm wheel 310
FIGURE 12.10 Development of the involute curve. . 311
FIGURE 12.11 Description of the involute tooth. 312
FIGURE 12.12 Layout of the involute tooth. 313
FIGURE 12.13 Involute function. . 314
FIGURE 12.14 Spur gear dimensional terms. 315
FIGURE 12.15 Simple gear drive. 316
FIGURE 12.16 Simple gear train 318
FIGURE 12.17 A compound gear set. 319
FIGURE 12.18 Diagram of motor/hoist gear drive .320
FIGURE 12.19 Referred inertia for a simple gear. . 322
FIGURE 12.20 Diagram of hoist. . 323
FIGURE 12.21 Torque required by hoist. . 325
FIGURE 12.22 Diagram of hoist mechanism for Example 12.3. . 326
FIGURE 13.1 Cam driving an offset translating follower. . 330
FIGURE 13.2 Cam driving a swinging arm follower. 331
FIGURE 13.3 Cylindrical cam and follower . 331
FIGURE 13.4 Typical cam follower designs . 332
FIGURE 13.5 Typical timing diagram 333
FIGURE 13.6 Timing diagram. 334
FIGURE 13.7 Constant velocity curve 335
FIGURE 13.8 Parabolic motion. . 337
FIGURE 13.9 Simple harmonic curve. . 339
FIGURE 13.10 Cycloidal motion curve 340
FIGURE 13.11 Pressure angle. . 341
FIGURE 13.12 Constant velocity of follower–graphical construction. 343
FIGURE 13.13 Swinging link follower having a uniform angular velocity–graphical
construction 345xxiii
List of Tables
TABLE 1.1 Standard Bending Cases 3
TABLE 1.2 Section Properties 10
TABLE 1.3 Bending Moment Calculations 11
TABLE 1.4 Sectional Properties . 12
TABLE 1.5 Form Factor for Shear Deflection 29
TABLE 1.6 Section Property Tables for Various Sections .38
TABLE 1.7 Shear Center Positions for Various Sections . 61
TABLE 2.1 Torsional Properties of Solid Sections 66
TABLE 2.2 Torsional Constants for Various Sections (Part 1) .68
TABLE 2.3 Torsional Constants for Various Sections (Part 2) 69
TABLE 2.4 Torsional Constants for Thin-Walled Open Sections 71
TABLE 2.5 Torsional Constants for Thin-Walled Closed Sections 72
TABLE 3.1 Pin Shear and Moment Arms in Multiple Shear Lugs 103
TABLE 4.1 National and International Standards 114
TABLE 4.2 ISO Metric Course Thread 116
TABLE 4.3 UNC Threads 116
TABLE 4.4 Metric Grades of Steels for Fasteners . 117
TABLE 4.5 SAE Grades of Steels for Fasteners . 117
TABLE 4.6 ASTM Standard for Fastener Steels 117
TABLE 5.1 Tolerance Grades . 149
TABLE 5.2 Fundamental Deviation (Shaft) . 150
TABLE 5.3 Fundamental Deviation (Hole) 150
TABLE 5.4 Preferred Fits Using the Basic Hole 152
TABLE 5.5 Hole and Shaft Sizes for Loose Running Fit (H11/c11) . 152
TABLE 5.6 Hole and Shaft Sizes for Location Clearance Fit (H7/h6) . 153
TABLE 5.7 Surface Roughness Values Obtainable by Standard Manufacturing Processes 154
TABLE 6.1 Nomenclature . 156
TABLE 6.2 Spreadsheet Results for Example 6.2 163
TABLE 8.1 Allowances for End Coil Forms 176
TABLE 8.2 Material Specifications 177xxiv List of Tables
TABLE 8.3 Values of Design Functions C1, C2, C3, C4, and C5
(to be used for imperial springs) 180
TABLE 8.4 Values of Design Functions C1, C2, C3, C4, and C5
(to be used for metric springs (SI Units)) . 183
TABLE 8.5 Number of Ineffective Coils for Various End Coil Forms 185
TABLE 8.6 Effect of End Coil Forms on the Solid Length of the Spring 187
TABLE 8.7 Spring Wire Data: Imperial Sizes and Allowables .202
TABLE 8.8 Spring Wire Data: Metric Sizes and Allowable Stresses 203
TABLE 8.9 Spring Wire Data: Variation of Material Strength with Temperature 204
TABLE 8.10 Spring Wire Data: Maximum Allowable Temperatures and Variations
of Bending and Torsional Modulus with Temperature 204
TABLE 8.11 Alloy Steel Springs 207
TABLE 8.12 Nonferrous Materials .207
TABLE 8.13 Nominal Solid Length for End Coil Formations . 210
TABLE 8.14 Example of Spring Manufacturing Requirements 212
TABLE 8.15 Information Required by the Manufacturer 213
TABLE 8.16 Spring Specification Data Not Relevant to the Manufacturer . 213
TABLE 8.17 Recommended Values of Clearance Factor x 215
TABLE 8.18 To Reduce the Outside Diameter . 216
TABLE 8.19 To Reduce Stress . 217
TABLE 8.20 To Reduce Length 219
TABLE 8.21 Summary of Nest Details for Example 2 222
TABLE 8.22 Summary of Nest Details for Example 2 222
TABLE 9.1 Interaction Equations – Yield Conditions .258
TABLE 10.1 Isochromatic Fringe Characteristics 271
TABLE 11.1 Similarities and Dissimilarities between Fatigue and Fracture Mechanics 296
TABLE 12.1 Spur Gear Design Formulae 315
TABLE 13.1 Constant Velocity Values . 336
TABLE 13.2 Parabolic Motion Values . 338
TABLE 13.3 Simple Harmonic Motion Values 338
TABLE 13.4 Cycloidal Motion Values . 341xxv
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