Modeling and Dimensioning of Structures: An Introduction

Preface xvii

Part 1. Level 1 1

Chapter 1. The Basics of Linear Elastic Behavior 3

1.1. Cohesion forces 4

1.2. The notion of stress 6

1.2.1. Definition 6

1.2.2. Graphical representation 7

1.2.3. Normal and shear stresses 8

1.3. Hooke’s law derived from a uniaxially applied force 9

1.3.1. The stretch test 9

1.3.2. Linear mechanical behavior 12

1.3.3. Elastic mechanical behavior 12

1.3.4. Interpretation of the test at a macroscopic level 13

1.3.5. Interpretation of the test at a mesoscopic level 13

1.3.6. Interpretation of the test at a microscopic level 16

1.3.7. Summary 18

1.4. Plane state of stresses 20

1.4.1. Definition 20

1.4.2. Behavior relationships for state of plane stresses 22

1.4.3. Summary 35

1.5. Particular case of straight beams 36

1.5.1. Preliminary observations 36

1.5.2. Effects linked to the resultant forces and moments 38

Chapter 2. Mechanical Behavior of Structures: An Energy Approach 51

2.1. Work and energy 51

2.1.1. Elementary work developed by a force 51

2.1.2. Elementary work developed by a moment 52

2.2. Conversion of work into energy 53

2.2.1. Potential energy of deformation 53

2.2.2. Potential energy for a spring 55

2.3. Some standard expressions for potential deformation energy 58

2.3.1. Deformation energies in a straight beam 58

2.3.2. Deformation energy under plane stresses 74

2.4. Work produced by external forces on a structure 81

2.4.1. Beam under plane bending subjected to two forces 82

2.4.2. Beam in plane bending subject to “n” forces 99

2.4.3. Generalization to any structure 103

2.4.4. Summary 112

2.5. Links of a structure with its surroundings 113

2.5.1. Example 113

2.5.2. Generalization 118

2.6. Stiffness of a structure 119

2.6.1. Preliminary note 119

2.6.2. Stiffness matrix 121

2.6.3. Examples 121

2.6.4. Influence of the positioning 125

2.6.5. Deformation energy and stiffness matrix 139

Chapter 3. Discretization of a Structure into Finite Elements 143

3.1. Preliminary observations 143

3.1.1. Problem faced 143

3.1.2. Practical obtaining of the deformation energy for a complex structure 144

3.1.3. Local and global coordinates 147

3.2. Stiffness matrix of some simple finite elements 153

3.2.1. Truss element loaded under traction (or compression) 153

3.2.3. Beam element under plane bending 168

3.2.4. Triangular element for the plane state of stresses 178

3.3. Getting the global stiffness matrix of a structure 191

3.3.1. Objective 191

3.3.2. Mechanism of the assembly of elementary matrices 191

3.3.1. Introduction 201

3.4. Resolution of the system {F}={K}??{d} 203

3.4.1. Linkage conditions 203

3.4.2. Generalization of the method 205

3.5. Different types of finite elements available in industrial software 207

Chapter 4. Applications: Discretization of Simple Structures 209

4.1. Stiffness matrix of a spring 209

4.1.1. Helical spring 209

4.1.2. Spiral spring. 211

4.2. Assembly of elements 213

4.2.1. Example 1 213

4.2.2. Example 2 217

4.2.3. Example 3 222

4.2.4. Assembly of a truss element and a beam element under simple plane bending 228

4.3. Behavior in the global coordinate system 232

4.3.1. Plane assembly of two truss elements 232

4.4. Bracket 246

4.4.1. Objectives 246

4.4.2. Modelizing 247

4.4.3. Calculation of the elementary stiffness matrix in the global system 252

4.4.4. Assembly of the global stiffness matrix [K]str 256

4.4.5. Establishing the linkage and loading conditions 259

4.4.6. Resolution of the linear system 􀁞F􀁠str = [K]str􀁹􀁞d􀁠str 260

4.4.7. Additional study of the behavior of the bracket 262

4.4.8. Using computing software 267

Part 2. Level 2 269

Chapter 5. Other Types of Finite Elements 271

5.1. Return to local and global coordinate systems 271

5.1.1. Transfer matrix 271

5.1.2. Summary 273

5.2. Complete beam element (any loading case) 274

5.2.1. Preliminary comments 274

5.2.2. Obtaining the stiffness matrix in the local coordinate system 276

5.2.3. Improvement in performances of this beam element 283

5.2.4. Summary 288

5.3. Elements for the plane state of stress 291

5.3.1. Triangular element 291

5.3.2. Quadrilateral element in plane state of stress 295

5.4. Plate element 300

5.4.1. Preliminary notes 300

5.4.2. Resultant forces and moments for cohesion forces 302

5.4.3. Plate element in bending 305

5.4.4. Complete plate element 310

5.5. Elements for complete states of stresses 315

5.5.1. Preliminary notes 315

5.5.2. Solid tetrahedric element 318

5.5.3. Solid parallelepipedic element 321

5.6. Shell elements 327

5.6.1. Preliminaries 327

5.6.2. Specific case of axisymmetric shells 328

5.6.3. Axisymmetric shell element with axisymmetric boundaries 329

Chapter 6. Introduction to Finite Elements for Structural Dynamics 331

6.1. Principles and characteristics of dynamic study 332

6.1.1. Example 1 332

6.1.2. Example 2 338

6.2. Mass properties of beams 346

6.2.1. Finite beam element in dynamic bending plane 346

6.2.2. Discretization of a beam for dynamic bending 350

6.2.3. Other types of dynamic behaviors of a beam 357

6.3. Generalization 363

6.4. Summary 364

Chapter 7. Criteria for Dimensioning 365

7.1. Designing and dimensioning 365

7.2. Dimensioning in statics 370

7.2.1. The two types of criteria 370

7.2.2. Elasticity limit criterion 373

7.2.3. Non-rupture criterion 382

7.3. Dimensioning in fatigue 393

7.3.1. Fatigue phenomenon 393

7.3.2. Fatigue test 394

7.3.3. Modeling of the fatigue 397

7.3.4. Estimation of fatigue strength 399

Chapter 8. Practical Aspects of Finite Element Modeling 407

8.1. Use of finite element software 407

8.1.1. Introduction 407

8.1.2. Summary tables of the properties of elements 408

8.1.3. Connection between elements of different types 415

8.1.4. Other practical aspects 420

8.2. Example 1: machine-tool shaft 432

8.2.1. Simulation exercise 432

8.2.2. Data 433

8.2.3. Successive steps of modeling 434

8.3. Example 2: thin-walled structures 440

8.3.1. Model based on beam elements 441

8.3.2. Model in plate elements 448

8.3.3. Model in beam and plate elements 449

8.4. Example 3: modeling of a massive structure 450

8.4.1. Problem 450

8.4.2. Steps of modeling 451

8.4.3. Comments on the validity of the model 456

8.5. Summary of the successive modeling steps 457

8.5.1. Preliminary analysis 457

8.5.2. Model verification and validation 458

8.5.3. Corresponding use of the software 460

Part 3. Supplements 463

Chapter 9. Behavior of Straight Beams 465

9.1. The “straight beam” model 466

9.1.1. Definition 466

9.1.2. Main or “principal” axis of a cross-section 466

9.1.3. Applied loadings 468

9.1.4. Cohesion force and moment on a current cross-section 469

9.1.5. Hypothesis of the beam theory 474

9.1.6. Microscopic equilibrium 479

9.2. Mesoscopic equilibrium or equilibrium extended to a whole cross-section 482

9.3. Behavior relations and stresses 486

9.3.1. Normal resultant 486

9.3.2. Torsional loading 490

9.3.3. Pure bending 514

9.3.4. Plane bending with shear resultant 530

9.3.5. Any loading 549

9.4. Application: example of detailed calculation of the resultant forces and moments of cohesive forces
551

9.4.1. Preliminary static analysis 551

9.4.2. Resultant force and moment on every cross-section 553

Chapter 10. Additional Elements of Elasticity 563

10.1. Reverting to the plane state of stresses 563

10.1.1. Influence of the coordinate system 563

10.1.2. Principal directions and stresses 566

10.1.3. Mohr graphical representation 568

10.1.4. Summary 575

10.1.5. Some remarkable plane states of stresses with their Mohr representation 576

10.1.6. Experimental evaluation of deformations to define stresses 588

10.1.7. Deformation energy in principal axes 593

10.2. Complete state of stresses 593

10.2.1. Principal directions and stresses 593

10.2.2. Stresses in any x , y , z axes 594

10.2.3. Deformations 597

10.2.4. Behavior relations 599

10.2.5. Strain potential energy 604

10.2.6. Summary 609

10.2.7. Components of the strain potential energy 612

Chapter 11. Structural Joints 619

11.1. General information on connections by means of cylindrical fasteners 620

11.1.1. Contact pressure 620

11.1.2. General information on riveting 622

11.1.3. General information on bolted joints 624

11.1.4. Deterioration of riveted and bolted joints 626

11.2. Bolted joint 631

11.2.1. Simplified case where the tightening is neglected 631

11.2.2. Case of pre-tightening 654

11.3. Riveted joint 666

11.3.1. Hypotheses 666

11.3.2. Characteristics of the modeled joining interface 666

11.3.3. Forces on each attachment 667

11.3.4. Graphic representation of the shear stresses 668

11.3.5. Summary 670

11.4. Welded joints 671

11.4.1. Preliminary observations and hypotheses 671

11.4.2. Determination of the stresses in the weld bead cross-section 673

11.4.3. Summary 686

11.4.4. Example 688

Chapter 12. Mathematical Prerequisites 691

12.1. Matrix calculus 691

12.1.1. General information 691

12.1.2. Matrix operations 692

12.1.3. Quadratic form 696

12.1.4. Eigenvalues and eigenvectors of a matrix 697

12.2. Change in orthonormal coordinate system 698

12.2.1. Case of coplanar coordinate systems 698

12.2.2. Cases of any general coordinate systems 699

Appendix A. Modeling of Common Mechanical Joints 703

A.1. Definition 703

A.1.1. Monolithic unit 703

A.1.2. Joints 703

A.1.3. Perfect joints 704

A.2. Common standardized mechanical joints (ISO 3952) 704

Appendix B. Mechanical Properties of Materials 711

B.1. Mechanical properties of some materials used for structures 711

B.1.1. Steels and casting 711

B.1.2. Non-ferrous metals 712

Appendix C. List of Summaries 713

Bibliography 717