Marks’ Calculations for Machine Design

Marks’ Calculations for Machine Design
اسم المؤلف
Thomas H. Brown
التاريخ
23 يناير 2022
المشاهدات
27
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Marks’ Calculations for Machine Design
Thomas H. Brown, Jr., Ph.D., P.E.
Faculty Associate
Institute for Transportation Research and Education
NC State University
Raleigh, North Carolina
CONTENTS
Foreword xi
Preface xiii
Acknowledgments xv
Part 1 Strength of Machines
Chapter 1. Fundamental Loadings 3
1.1. Introduction / 3
1.2. Axial Loading / 4
1.3. Direct Shear / 11
1.4. Torsion / 16
1.5. Bending / 24
Chapter 2. Beams: Reactions, Shear Force and Bending Moment
Distributions, and Deflections 33
2.1. Introduction / 33
2.2. Simply-Supported Beams / 35
2.2.1. Concentrated Force at Midpoint / 36
2.2.2. Concentrated Force at Intermediate Point / 41
2.2.3. Concentrated Couple / 48
2.2.4. Uniform Load / 55
2.2.5. Triangular Load / 60
2.2.6. Twin Concentrated Forces / 67
2.2.7. Single Overhang: Concentrated Force at Free End / 73
2.2.8. Single Overhang: Uniform Load / 79
2.2.9. Double Overhang: Concentrated Forces at Free Ends / 86
2.2.10. Double Overhang: Uniform Load / 92
2.3. Cantilevered Beams / 97
2.3.1. Concentrated Force at Free End / 98
2.3.2. Concentrated Force at Intermediate Point / 104
2.3.3. Concentrated Couple / 110
2.3.4. Uniform Load / 115
2.3.5. Triangular Load / 120
Chapter 3. Advanced Loadings 127
3.1. Introduction / 127
3.2. Pressure Loadings / 127
vii
For more information about this title, click hereviii CONTENTS
3.2.1. Thin-Walled Vessels / 128
3.2.2. Thick-Walled Cylinders / 130
3.2.3. Press or Shrink Fits / 134
3.3. Contact Loading / 139
3.3.1. Spheres in Contact / 139
3.3.2. Cylinders in Contact / 143
3.4. Rotational Loading / 147
Chapter 4. Combined Loadings 153
4.1. Introduction / 153
4.2. Axial and Torsion / 156
4.3. Axial and Bending / 159
4.4. Axial and Thermal / 164
4.5. Torsion and Bending / 167
4.6. Axial and Pressure / 172
4.7. Torsion and Pressure / 175
4.8. Bending and Pressure / 184
Chapter 5. Principal Stresses and Mohr’s Circle 189
5.1. Introduction / 189
5.2. Principal Stresses / 190
5.3. Mohr’s Circle / 205
Chapter 6. Static Design and Column Buckling 233
6.1. Static Design / 233
6.1.1. Static Design for Ductile Materials / 234
6.1.2. Static Design for Brittle Materials / 246
6.1.3. Stress-Concentration Factors / 258
6.2. Column Buckling / 260
6.2.1. Euler Formula / 261
6.2.2. Parabolic Formula / 263
6.2.3. Secant Formula / 266
6.2.4. Short Columns / 270
Chapter 7. Fatigue and Dynamic Design 273
7.1. Introduction / 273
7.2. Reversed Loading / 274
7.3. Marin Equation / 279
7.4. Fluctuating Loading / 285
7.5. Combined Loading / 311
Part 2 Application to Machines
Chapter 8. Machine Assembly 321
8.1. Introduction / 321
8.2. Bolted Connections / 321CONTENTS ix
8.2.1. The Fastener Assembly / 321
8.2.2. The Members / 326
8.2.3. Bolt Strength and Preload / 331
8.2.4. The External Load / 332
8.2.5. Static Loading / 335
8.2.6. Fatigue Loading / 337
8.3. Welded Connections / 348
8.3.1. Axial and Transverse Loading / 348
8.3.2. Torsional Loading / 352
8.3.3. Bending Loading / 356
8.3.4. Fillet Welds Treated as Lines / 360
8.3.5. Fatigue Loading / 365
Chapter 9. Machine Energy 367
9.1. Introduction / 367
9.2. Helical Springs / 367
9.2.1. Loads, Stresses, and Deflection / 367
9.2.2. Spring Rate / 371
9.2.3. Work and Energy / 375
9.2.4. Series and Parallel Arrangements / 377
9.2.5. Extension Springs / 379
9.2.6. Compression Springs / 380
9.2.7. Critical Frequency / 383
9.2.8. Fatigue Loading / 385
9.3. Flywheels / 388
9.3.1. Inertial Energy of a Flywheel / 388
9.3.2. Internal Combustion Engine Flywheels / 392
9.3.3. Punch Press Flywheels / 395
9.3.4. Composite Flywheels / 401
Chapter 10. Machine Motion 409
10.1. Introduction / 409
10.2. Linkages / 410
10.2.1. Classic Designs / 410
10.2.2. Relative Motion / 412
10.2.3. Cyclic Motion / 421
10.3. Gear Trains / 424
10.3.1. Spur Gears / 425
10.3.2. Planetary Gears / 428
10.4. Wheels and Pulleys / 431
10.4.1. Rolling Wheels / 432
10.4.2. Pulley Systems / 435
Bibliography 439
Index 441
INDEX
Acceleration analysis, 416–419
Advanced loadings, 127
American Welding Society (AWS), 348
Amplitude stress, 285
Angle of twist, 19
Angular rotation, 392
Area:
tensile-stress, 324
Average stress, 196
Axial loading, 4, 156, 159, 164, 172
Axial strain, 5
Axial stress, 4
in cylinders:
thin-walled, 129, 191
thick-walled, 133–134
prismatic, 4
Beams, 33
cantilevered, 33–34, 97
double overhanging, 35
simply-supported, 33, 35
single overhanging, 35
Beam loadings:
concentrated couple, 48, 110
concentrated force at free end(s), 73, 86, 98
concentrated force at intermediate point,
41, 104
concentrated force at midpoint, 36, 159
triangular load, 60, 120
twin concentrated forces, 67
uniform load, 55, 79, 92, 115
Beam supports:
cantilever, 35
pin, 34
roller, 34
Bending, 24, 158, 167, 184
Bergstrasser factor, 370
Biaxial stress element, 148
Bolt:
length, 323
strength, 331
Bolted connections, 321
Butt welds, 348
Cantilevered beams, 33–34, 97, 98, 104, 110,
115, 120
Change in length:
prismatic bar, 8
Classic mechanism designs, 410
Coefficient of expansion, 10
Coefficient of speed fluctuation, 393, 397
Column buckling, 260
Euler formula, 261
Parabolic formula, 263
Secant formula, 266
Short columns, 270
Column end types, 262
Combined loadings:
dynamic, 311
static:
axial and bending, 159
axial and pressure, 172
axial and thermal, 164
axial and torsion, 156
bending and pressure, 184
torsion and bending, 167
torsion and pressure, 175
Complex planetary gear trains, 430
Complex pulley systems, 437
Composite flywheels, 401–403
Compression springs, 380–382
Concentrated couple loading, 48, 110
Cone angle in bolted connections, 327
Contact loading:
cylinders in contact, 143
spheres in contact, 139
INDEX
Contact pressure, maximum:
between cylinders, 144
between spheres, 140
Corrosion effect, 283
Coulomb-Mohr theory, 248
graphical representation, 248
Critical frequency, 383–384
Cycle frequency effect, 283
Cyclic loading, 276
Cyclic motion, 421–424
Cylinders:
thin-walled, 129
thick-walled, 130
Design:
dynamic, 273
static, 233
Direct shear loading, 3, 11
Disassemblable joint bolt preload,
332
Distortion-energy theory, 237
graphical representation, 237
Double overhanging beams, 35, 86, 92
Elastic limit, 6, 14
Eccentricity ratio, 267
graphical representation, 267
Effective diameter, 281
Electrolytic plating effect, 283
Elongation method for bolt preload,
332
Endurance limit, 276
Energy, 375–376
Extension springs, 379–380
hook geometry, 379
External load on bolted joints, 332–334
Euler formula, 261
Factors, Marin equation:
load type, 282
miscellaneous effects, 282–283
size, 280–281
surface finish, 280
Factors of safety:
against bolted joint separation, 335–336
dynamic design:
fluctuating loading:
Gerber theory, 288–289
Goodman theory, 288–292
Modified Goodman theory, 288–289
stress-concentration factor, 304
torsional loading, 305
static design:
brittle materials:
Coulomb-Mohr theory, 249
Maximum-normal-stress theory, 248
Modified Coulomb-Mohr theory, 250
ductile materials:
Distortion-energy theory, 238
Maximum-normal-stress theory, 235
Maximum-shear-stress theory, 237
yielding of a bolted connection, 339
Fastener assembly types, 321
Fatigue, 273
Fatigue loading of:
bolted connections, 337–340
helical spring, 385–386
welded connections, 365–366
Fillet weld geometry for:
bending, 357
torsion, 353
Fillet welds, 348, 350–358
Fillet welds treated as lines, 360–363
Finite life, 276, 277
First moment of area:
definition, 27
formula for the maximum value:
circular cross section, 31
rectangular cross section, 28
Fit standards, 137
Fluctuating design criteria, 287
Flywheels, 388
Four-bar linkage, 410
Four-stroke engine, 392
Fracture point, 6, 14
Free Body Diagram (FBD) of a helical
spring, 368
Frettage effect, 283
Frustum in bolted connections, 326–327
Fundamental loadings, 3
summary table of formulas, 153
Gaskets in bolted connections, 326
Gear trains, 424
Geometry of:
fillet welds as lines, 361
helical springs, 368
slider-crank linkage, 414
Gerber theory, 288–289
Goodman Diagram for:
bolted connections, 338
fluctuating torsional loading, 386
welded connections, 366
Goodman theory, 288–289INDEX 443
Grip, of a bolted connection, 323
Groove welds, 348
Helical spring deflection, 371
Helical springs, 367
Hole punching, 15
Hooke’s law:
axial, 6, 9, 10
shear, 14
Hoop stress in cylinders, 129, 191
Infinite life, 276
Instantaneous contact point, 432
Interface pressure, 135
Internal combustion engines, 392–394
Joint contact, 334
Lap joint, 350
Lateral strain, 7
Linkages, 410
Load factor on bolted connections, 335
Load type factor for Marin equation,
282
Loadings:
advanced, 127
axial, 4, 156, 159, 164
bending, 24, 159, 167, 184
combined, 153
contact, 139
direct shear, 11
fluctuating, 285
fundamental, 3
pressure, 127, 172, 175, 184
reversed, 274
rotational, 147
summary table of formulas, 153, 154
thermal, 10, 164
torsion, 16, 156, 167, 175
triangular, 60, 120
uniform, 55, 79, 92, 115
Machine:
assembly, 321
energy, 367
motion, 409
Marin equation, 279
Maximum-normal-stress theory:
brittle materials, 247
graphical representation, 247
ductile materials, 234
graphical representation, 247
Maximum-shear-stress theory, 235
graphical representation, 236
Maximum shear stress, 196
Mean stress, 285
Mechanical advantage, 436
Members, in a bolted connection,
326
Minimum shear stress, 196
Miscellaneous effects factors for Marin
equation, 282–283
Modified Coulomb-Mohr theory, 249
graphical representation, 249
Modified Goodman theory, 288–289
Modulus of elasticity:
axial, 6, 15
shear, 14, 15
Moment of inertia:
circular beam, 31
flywheel, 389
rectangular beam, 25
Mohr’s Circle, 205
graphical process, 208
triaxial stress, 231
Neutral axis, 24
Notch sensitivity, 260, 283
Number of active coils in a helical spring,
372
Parabolic formula, 263
Parallel arrangement of springs, 377–378
Permanent joint bolt preload, 332
Pin supports, 34
Pitch of a helical spring, 380
Plane stress element, 153–154, 189
Planetary gears, 428–431
Poisson’s ratio, 7, 15
Polar moment of inertia:
hollow shaft, 17
solid shaft, 17
thin-walled rectangular tube, 22
welded connection, 354
Potential energy of a spring, 376
Preload, bolt, 331–332
techniques to verify, 332
Press fits, 134, 175
Pressure:
contact:
between cylinders, 144
between spheres, 140
interface, 135
internal, 128444 INDEX
Pressure loadings, 127, 172, 175, 184
summary table of formulas, 154
Pressure vessels:
thin-walled:
cylindrical, 129, 191
spherical, 128
Pressurized tank, 184, 190
Principal stresses, 190, 195
Prismatic bar, 4, 156, 258, 322
Proof strength, 332
Proportional limit, 6, 14
Pulley systems, 435–437
Punch press:
cycle, 396
flywheels, 395–398
Punching time, 397
Pure motions:
rotation, 412
translation, 412
Pure shear element, 156
Quick-return linkage, 411
Radial interference, 135
Radial stress, 131–132, 148
Radius of gyration: 260
circular cross section, 265
rectangular cross section, 261
Rated torque, 395
Recovery time, punch presses, 397
Relative motion, 412–415, 416–419
Riveted joint, 11, 30
Roller supports, 34
Rolling wheels, 432–434
Rotated plane stress element, 190, 205
Rotating disk, 148
R. R. Moore rotating-beam machine, 275
Secant formula, 266
Section modulus, 26
Series arrangement of springs, 377–378
Shear:
direct, 11
strain, 13, 19
stress, 12, 16, 22, 27
maximum, 196
minimum, 196
Shear-stress correction factor, 370
Shrink fits, 134, 175
Simply-supported beams, 33, 35, 36, 41, 48, 55,
60, 67, 73, 79, 86, 92
Single overhanging beams, 35, 73, 79
Size factor for Marin equation, 280–281
Slenderness ratio, 260, 261, 263, 266, 271
Slider-crank linkage, 411
S-N Diagram, 275–276
Solid disk flywheel, 388–389
Spheres:
thin-walled, 128
Spring deflection, 371
Spring index, 369
Spring rate:
bolt, 322–323
capscrew, 322–323
frustum of a cone, 327
helical springs, 371–372
members in a bolted connection, 326–327
Spur gears, 425–427
Static design:
coordinate system, 234
brittle materials, 246
comparison with experimental data, 250
recommendations, 251–252
ductile materials, 234
comparison with experimental data, 238
recommendations, 238–239
theories:
Coulomb-Mohr, 248
Distortion-energy, 237
Maximum-normal-stress, 234, 247
Maximum-shear-stress, 235
Modified Coulomb-Mohr, 249
Stability of helical springs, 381–382
Static loading of bolted connections, 335–336
Stiffness:
bolt, 322–323
capscrew, 322–323
frustum of a cone, 327
members in a bolted connection, 326–327
Strain:
axial, 5
lateral, 7
shear, 13, 19
thermal, 10
Strength, proof, 332
Stress:
alternating, 285
average, 196
axial, 4, 129, 133–134
bending, 24
contact, 140, 144
critical, 261, 264, 266, 271INDEX 445
direct shear, 12
hoop, 129, 191
mean, 285
normal, in spheres, 128
principal, 190
radial, 131–132, 148
shear, 16, 22, 27
tangential, 131, 148
thermal, 10
triaxial, 230
Stress-concentration factors, 258, 283,
304
Stress elements:
biaxial, 148, 219, 223
maximum, 195
plane, 153, 205
pure shear, 156, 157, 219, 227
uniaxial, 155, 219
Stress-strain diagrams:
axial loading:
brittle materials, 7
ductile materials, 6
high-strength bolt or capscrew, 332
shear loading:
brittle materials, 14
ductile materials, 14
Supports:
cantilever, 33–34
pin, 34
roller, 34
Surface finish factor for Marin equation, 280
Synchronous angular velocity, 395
Tangential stress, 130, 148
Tee joint, 351
Temperature factor in Marin equation, 282
Tensile-stress area, 324, 331
Thermal:
loading, 164
strain, 10
stress, 10
Thick-walled cylinders, 130
Thin rotating disks, 148
Thin-walled:
tubes, 22
vessels, 128
cylindrical, 129
spherical, 128
Thread length, 323–324
Torque as a function of:
angular velocity, 395
rotation angle, 392
Torque wrench method for bolt preload,
332
Torsion, 16, 156, 167, 175
Torsional loading:
fluctuating, 304–305
welded connections, 352–354
Transformation equations, 190
Transverse joint, 351, 352
Triangular loading, 60, 120
Triaxial stress, 230
Turn-of-the-nut method for bolt preload,
332
Two-stroke engine, 392
Ultimate strength, 6, 14
Uniaxial stress element, 155
Uniform loading, 55, 79, 92, 115
Velocity analysis, 412–415
Vessels:
thin-walled:
cylinders, 129
spheres, 128
Wahl factor, 370
Welded connections, 348
Wheels and pulleys, 431
Work and energy, 375–376
Yield point, 6, 14

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