## Disc With Hole Rotating Radial Displacement

Introduction xv

Notation xvii

1 Unsymmetricai Bending 1

Summary 1

Introduction 2

1.1 Product second moment of area 3

1.2 Principal second moments of area 4

1.3 Mohr's circle of second moments of area 6

1.4 Land's circle of second moments of area 7

1.5 Rotation of axes: determination of moments of area in terms of the principal values 8

1.6 The ellipse of second moments of area 9

1.7 Momental ellipse 11

1.8 Stress determination 11

1.9 Alternative procedure for stress determination 11

1.10 Alternative procedure using the momental ellipse 13

1.11 Deflections 15 Examples 16 Problems 24

2 Struts 28

Summary 28

Introduction 30

2.1 Euler's theory 31

2.2 Equivalent strut length 35

2.3 Comparison of Euler theory with experimental results 36

2.4 Euler "validity limit" 37

2.5 Rankine or Rankine-Gordon formula 38

2.6 Perry-Robertson formula 39

2.7 British Standard procedure (BS 449) 41

2.8 Struts with initial curvature 41

2.9 Struts with eccentric load 42

2.10 Laterally loaded struts 46

2.11 Alternative procedure for any strut-loading condition 48

2.12 Struts with unsymmetrical cross-section 49

Examples 50

Problems 56

3 Strains Beyond the Elastic Limit 61

Summary 61

Introduction 62

3.1 Plastic bending of rectangular-sectioned beams 64

3.2 Shape factor - symmetrical sections 65

3.3 Application to I-section beams 67

3.4 Partially plastic bending of unsymmetrical sections 67

3.5 Shape factor - unsymmetrical sections 69

3.6 Deflections of partially plastic beams 69

3.7 Length of yielded area in beams 69

3.8 Collapse loads - plastic limit design 71

3.9 Residual stresses after yielding: elastic-perfectly plastic material 73

3.10 Torsion of shafts beyond the elastic limit - plastic torsion 75

3.11 Angles of twist of shafts strained beyond the elastic limit 11

3.12 Plastic torsion of hollow tubes 11

3.13 Plastic torsion of case-hardened shafts 79

3.14 Residual stresses after yield in torsion 79

3.15 Plastic bending and torsion of strain-hardening materials 80

(a) Inelastic bending 80

(b) Inelastic torsion 83

3.16 Residual stresses - strain-hardening materials 84

3.17 Influence of residual stresses on bending and torsional strengths 84

3.18 Plastic yielding in the eccentric loading of rectangular sections 85

3.19 Plastic yielding and residual stresses under axial loading with stress concentrations 86

3.20 Plastic yielding of axially symmetric components 87

(a) Thick cylinders - collapse pressure 87

(b) Thick cylinders - "auto-frettage" 89

(c) Rotating discs 94 Examples 96 Problems 109

4 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients 117

Summary 117

4.1 Thin rotating ring or cylinder 118

4.2 Rotating solid disc 119

4.3 Rotating disc with a central hole 122

4.4 Rotating thick cylinders or solid shafts 124

4.5 Rotating disc of uniform strength 125

4.6 Combined rotational and thermal stresses in uniform discs and thick cylinders 126

Examples 129

Problems 136

5 Torsion of Non-Circular and Thin-Walled Sections 141

Summary 141

5.1 Rectangular sections 142

5.2 Narrow rectangular sections 143

5.3 Thin-walled open sections 143

5.4 Thin-walled split tube 145

5.5 Other solid (non-tubular) shafts 145

5.6 Thin-walled closed tubes of non-circular section (Bredt-Batho theory) 147

5.7 Use of "equivalent J " for torsion of non-circular sections 149

5.8 Thin-walled cellular sections 150

5.9 Torsion of thin-walled stiffened sections 151

5.10 Membrane analogy 152

5.11 Effect of warping of open sections 153 Examples 154 Problems 160

6 Experimental Stress Analysis 166

Introduction 166

6.1 Brittle lacquers 167

6.2 Strain gauges 171

6.3 Unbalanced bridge circuit 173

6.4 Null balance or balanced bridge circuit 173

6.5 Gauge construction 173

6.6 Gauge selection 175

6.7 Temperature compensation 175

6.8 Installation procedure 176

6.9 Basic measurement systems 177

6.11 Other types of strain gauge 180

6.12 Photoelasticity 181

6.13 Plane-polarised light - basic polariscope arrangements 182

6.14 Temporary birefringence 183

6.15 Production of fringe patterns 184

6.16 Interpretation of fringe patterns 185

6.17 Calibration 186

6.18 Fractional fringe order determination - compensation techniques 187

6.19 lsoclinics-circular polarisation 188

6.20 Stress separation procedures 190

6.21 Three-dimensional photoelasticity 190

6.22 Reflective coating technique 190

6.23 Other methods of strain measurement 192 Bibliography 192

7 Circular Plates and Diaphragms 193

Summary 193

A. CIRCULAR PLATES 195

7.1 Stresses 195

7.2 Bending moments 197

7.3 General equation for slope and deflection 198

7.4 General case of a circular plate or diaphragm subjected to combined uniformly distributed load q (pressure) and central concentrated load F 199

7.5 Uniformly loaded circular plate with edges clamped 200

7.6 Uniformly loaded circular plate with edges freely supported 202

7.7 Circular plate with central concentrated load F and edges clamped 203

7.8 Circular plate with central concentrated load F and edges freely supported 205

7.9 Circular plate subjected to a load F distributed round a circle 206

7.10 Application to the loading of annular rings 208

7.11 Summary of end conditions 208

7.12 Stress distributions in circular plates and diaphragms subjected to lateral pressures 209

7.13 Discussion of results - limitations of theory 211

7.14 Other loading cases of practical importance 212

B. BENDING OF RECTANGULAR PLATES 213

7.15 Rectangular plates with simply supported edges carrying uniformly distributed loads 213

7.16 Rectangular plates with clamped edges carrying uniformly distributed loads 214

Examples 215

Problems 218

8 Introduction to Advanced Elasticity Theory 220

8.1 Types of stress 220

8.2 The cartesian stress components: notation and sign convention 220 8.2.1 Sign conventions 221

8.3 The state of stress at a point 221

8.4 Direct, shear and resultant stresses on an oblique plane 224

8.4.1 Line of action of resultant stress 226

8.4.2 Line of action of normal stress 227

8.4.3 Line of action of shear stress 221

8.4.4 Shear stress in any other direction on the plane 227

8.5 Principal stresses and strains in three dimensions - Mohr's circle representation 228

8.6 Graphical determination of the direction of the shear stress zn on an inclined plane in a three-dimensional principal stress system 229

8.7 The combined Mohr diagram for three-dimensional stress and strain systems 230

8.8 Application of the combined circle to two-dimensional stress systems 232

8.9 Graphical construction for the state of stress at a point 234

8.10 Construction for the state of strain on a general strain plane 235

8.11 State of stress-tensor notation 235

8.12 The stress equations of equilibrium 236

8.13 Principal stresses in a three-dimensional cartesian stress system 242 8.13.1 Solution of cubic equations 242

8.14 Stress invariants - Eigen values and Eigen vectors 243

8.15 Stress invariants 244

8.16 Reduced stresses 246

8.17 Strain invariants 247

8.18 Alternative procedure for determination of principal stresses 247 8.18.1 Evaluation of direction cosines for principal stresses 248

8.19 Octahedral planes and stresses 249

8.20 Deviatoric stresses 251

8.21 Deviatoric strains 253

8.22 Plane stress and plane strain 254

8.22.1 Plane stress 255

8.22.2 Plane strain 255

8.23 The stress-strain relations 256

8.24 The strain-displacement relationships 257

8.25 The strain equations of transformation 259

8.26 Compatibility 261

8.27 The stress function concept 263

8.27.1 Forms of Airy stress function in Cartesian coordinates 265

8.27.2 Case I - Bending of a simply supported beam by a uniformly distributed loading 267

8.27.3 The use of polar coordinates in two dimensions 271

8.27.4 Forms of stress function in polar coordinates 272

8.27.5 Case 2 - Axi-symmetric case: solid shaft and thick cylinder radially loaded with uniform pressure 273

8.27.6 Case 3 - The pure bending of a rectangular section curved beam 273

8.27.7 Case 4 - Asymmetric case n = 1. Shear loading of a circular arc cantilever beam 274

8.27.8 Case 5 - The asymmetric cases n > 2 -stress concentration at a circular hole in a tension field 276

8.27.9 Other useful solutions of the biharmonic equation 279

Examples 283

Problems 290

9 Introduction to the Finite Element Method 300

Introduction 300

9.1 Basis of the finite element method 300

9.2 Applicability of the finite element method 302

9.3 Formulation of the finite element method 303

9.4 General procedure of the finite element method 303

9.4.1 Identification of the appropriateness of analysis by the finite element method 303

9.4.2 Identification of the type of analysis 305

9.4.3 Idealisation 305

9.4.4 Discretisation of the solution region 305

9.4.5 Creation of the material model 312

9.4.6 Node and element ordering 312

9.4.7 Application of boundary conditions 316

9.4.8 Creation of a data file 317

9.4.9 Computer, processing, steps 318

9.4.10 Interpretation and validation of results 318

9.4.11 Modification and re-run 319

9.5 Fundamental arguments 319

9.5.1 Equilibrium 319

9.5.2 Compatibility 321

9.5.3 Stress-strain law 322

9.5.4 Force/displacement relation 322

9.6 The principle of virtual work 323

9.7 A rod element 324

9.7.1 Formulation of a rod element using fundamental equations 324

9.7.2 Formulation of a rod element using the principle of virtual work equation 328

9.8 A simple beam element 334

9.8.1 Formulation of a simple beam element using fundamental equations 334

9.8.2 Formulation of a simple beam element using the principle of virtual work equation 339

9.9 A simple triangular plane membrane element 343 9.9.1 Formulation of a simple triangular plane membrane element using the principle of virtual work equation 344

9.10 Formation of assembled stiffness matrix by use of a dof.

correspondence table 347

9.11 Application of boundary conditions and partitioning 349

9.12 Solution for displacements and reactions 349

Bibliography 350

Examples 350

Problems 375

10 Contact Stress, Residual Stress and Stress Concentrations 381

Summary 381

10.1 Contact stresses 382 Introduction 382

10.1.1 General case of contact between two curved surfaces 385

10.1.2 Special case 1 - Contact of parallel cylinders 386

10.1.3 Combined normal and tangential loading 388

10.1.4 Special case 2 - Contacting spheres 389

10.1.5 Design considerations 390

10.1.6 Contact loading of gear teeth 391

10.1.7 Contact stresses in spur and helical gearing 392

10.1.8 Bearing failures 393

10.2 Residual stresses 394 Introduction 394

10.2.1 Reasons for residual stresses 395

(a) Mechanical processes 395

(b) Chemical treatment 397

(c) Heat treatment 398

(e) Castings 401

10.2.2 The influence of residual stress on failure 402

10.2.3 Measurement of residual stresses 402 The hole-drilling technique 404 X-ray diffraction 407

10.2.4 Summary of the principal effects of residual stress 408

10.3 Stress concentrations 408 Introduction 408

10.3.1 Evaluation of stress concentration factors 413

10.3.2 St. Venant's principle 420

10.3.3 Theoretical considerations of stress concentrations due to concentrated loads 422

(a) Concentrated load on the edge of an infinite plate 422

(b) Concentrated load on the edge of a beam in bending 423

10.3.4 Fatigue stress concentration factor 423

10.3.5 Notch sensitivity 424

10.3.6 Strain concentration - Neuber's rule 425

10.3.7 Designing to reduce stress concentrations 426

(a) Fillet radius 427

(b) Key ways or splines 427

(c) Grooves and notches 429

(d) Gear teeth 430

(f) Oil holes 431

(g) Screw threads 431

(h) Press or shrink fit members 433

10.3.8 Use of stress concentration factors with yield criteria 434

10.3.9 Design procedure 434 References 435 Examples 437 Problems 442

11 Fatigue, Creep and Fracture 443

Summary 443

11.1 Fatigue 446 Introduction 446

11.1.1 The S/N curve 446

11.1.3 Effect of mean stress 451

11.1.4 Effect of stress concentration 453

11.1.5 Cumulative damage 454

11.1.6 Cyclic stress-strain 455

11.1.7 Combating fatigue 458

11.1.8 Slip bands and fatigue 460

11.2 Creep 462 Introduction 462

11.2.1 The creep test 462

11.2.2 Presentation of creep data 465

11.2.3 The stress-rupture test 466

11.2.4 Parameter methods 467

11.2.5 Stress relaxation 470

11.2.6 Creep-resistant alloys 471

11.3 Fracture mechanics 472 Introduction 472

11.3.1 Energy variation in cracked bodies 473

(a) Constant displacement 474

(b) Constant loading 474

11.3.2 Linear elastic fracture mechanics (L.E.F.M.) 475

(a) Griffith's criterion for fracture 475

(b) Stress intensity factor 477

11.3.3 Elastic-plastic fracture mechanics (E.P.F.M.) 481

11.3.4 Fracture toughness 483

11.3.5 Plane strain and plane stress fracture modes 484

11.3.6 General yielding fracture mechanics 484

11.3.7 Fatigue crack growth 486

11.3.8 Crack tip plasticity under fatigue loading 488

11.3.9 Measurement of fatigue crack growth 489

References 490

Examples 491

Problems 503

12 Miscellaneous topics 509

12.1 Bending of beams with initial curvature 509

12.2 Bending of wide beams 515

12.3 General expression for stresses in thin-walled shells subjected to pressure or self-weight 517

12.4 Bending stresses at discontinuities in thin shells 518

12.5 Viscoelasticity 521 References 527 Examples 527 Problems 527

Appendix 1. Typical mechanical and physical properties for engineering metals 534

Appendix 2. Typical mechanical properties of non-metals 535

Appendix 3. Other properties of non-metals 536

Index

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