Analysis of Engineering Structures and Material Behavior

Theoretical and experimental study of the mechanical behavior of structures under load

Analysis of Engineering Structures and Material Behavior is a textbook covering introductory and advanced topics in structural analysis. It begins with an introduction to the topic, before covering fundamental concepts of stress, strain and information about mechanical testing of materials. Material behaviors, yield criteria and loads imposed on the engineering elements are also discussed. The book then moves on to cover more advanced areas including relationships between stress and strain, rheological models, creep of metallic materials and fracture mechanics. Finally, the finite element method and its applications are considered.

Key features:

• Covers introductory and advanced topics in structural analysis, including load, stress, strain, creep, fatigue and finite element analysis of structural elements.
• Includes examples and considers mathematical formulations.
• A pedagogical approach to the topic.

Analysis of Engineering Structures and Material Behavior is suitable as a textbook for structural analysis and mechanics courses in structural, civil and mechanical engineering, as well as a valuable guide for practicing engineers.

Professor Josip Brnić, D. Sc., University of Rijeka - Faculty of Engineering, Croatia

Frequently Used Symbols and the Meaning of Symbols xv

Principal SI Units and the US Equivalents xxiii

SI Prefixes, Basic Units, Physical Constants, the Greek Alphabet xxv

Important Notice Before Reading the Book xxvii

Preface xxix

About the Author xxxi

Acknowledgements xxxiii

1 Introduction 1

1.1 The Task of Design and Manufacture 1

1.2 Factors that Influence the Design of Engineering Structures 1

1.3 The Importance of Optimization in the Process of Design and the Selection of Structural Materials 3

1.4 Commonly Observed Failure Modes in Engineering Practice 4

1.5 Structures and the Analysis of Structures 5

References 5

2 Stress 7

2.1 Definition of Average Stress and Stress at a Point 7

2.2 Stress Components and Equilibrium Equations 8

2.2.1 Stress Components 8

2.2.2 Equilibrium Equations 9

2.3 Stress Tensor 10

2.3.1 Mean and Deviatoric Stress Tensors 10

2.4 States of Stress 12

2.4.1 Uniaxial State of Stress 12

2.4.2 Two-dimensional State of Stress 14

2.4.3 Three-dimensional State of Stress 18

2.4.3.1 Stress on an Arbitrary Plane 20

2.4.3.2 Stress on an Octahedral Plane 21

2.4.3.3 Principal Stresses and Stress Invariants 22

2.5 Transformation of Stress Components 24

References 28

3 Strain 29

3.1 Definition of Strain 29

3.1.1 Some Properties of Materials Associated with Strain 30

3.1.1.1 Poisson’s Ratio 30

3.1.1.2 Volumetric Strain 30

3.1.1.3 Bulk Modulus 31

3.1.1.4 Modulus of Elasticity 32

3.1.1.5 Shear Modulus (Modulus of Rigidity) 32

3.2 Strain–Displacement Equations 33

3.3 Strain Tensors 35

3.3.1 Small Strain Tensor 35

3.3.2 Finite Strain Tensor 38

3.3.3 Mean and Deviatoric Strain Tensors 40

3.3.4 Principal Strains and Strain Invariants 41

3.3.4.1 Strain Tensor 41

3.3.4.2 Deviatoric Strain Tensor 42

3.4 Transformation of Strain Components 43

3.4.1 Mohr’s Circle 44

3.5 Strain Measurement 44

References 48

4 Mechanical Testing of Materials 51

4.1 Material Properties 51

4.2 Types of Material Testing 52

4.3 Test Methods Related to Mechanical Properties 52

4.4 Testing Machines and Specimens 52

4.4.1 Static Tensile Testing Machine and Specimens 52

4.4.2 Impact Testing Machine and Specimens 54

4.4.3 Hardness Testing Machine 54

4.4.4 Fatigue Testing Machines 56

4.5 Test Results 56

4.5.1 Static Tensile Test Results 56

4.5.1.1 Engineering Stress–Strain Diagram 56

4.5.1.2 Creep Diagram/Curve 62

4.5.1.3 Relaxation Diagram/Curve 62

4.5.2 Dynamic Test Results 63

4.5.2.1 Tensile, Flexural and Torsional Test Results 63

4.5.2.2 Toughness Test Results 64

4.5.2.3 Fracture Toughness Test Results 64

References 64

5 Material Behavior and Yield Criteria 67

5.1 Elastic and Inelastic Responses of a Solid 67

5.2 Yield Criteria 67

5.2.1 Ductile Materials 71

5.2.1.1 Maximum Shear Stress Criterion (Tresca Criterion) 71

5.2.1.2 Distortional Energy Density Criterion (von Mises Criterion) 74

5.2.2 Brittle Materials 76

5.2.2.1 Maximum Normal Stress Criterion 76

5.2.2.2 Maximum Normal Strain Criterion 76

References 78

6 Loads Imposed on Engineering Elements 79

6.1 Axial Loading 79

6.1.1 Normal Stress 81

6.1.2 The Principal Stress 82

6.2 Torsion 85

6.2.1 Elastic Torsion – Shear Stress and Strain Analysis 86

6.2.1.1 Prismatic Bars: Circular Cross-section 86

6.2.1.2 Prismatic Bars: Noncircular Cross-section 95

6.2.1.3 Thin-walled Structures 96

6.2.2 Warping (Distortion) of a Cross-section 101

6.2.3 Inelastic Torsion and Residual Stress 103

6.2.3.1 Residual Stress 105

6.3 Bending 109

6.3.1 Beam Supports, Types of Beams, Types of Loads 109

6.3.2 Internal Forces – Bending Moments (Mf), Shear Force (Q), Distributed Load (q) 111

6.3.3 Principal Moments of Inertia of an Area (I1, I2) and Extreme Values of Product of Inertia (Ixy) of an Area 112

6.3.3.1 Axes Parallel to the Centroidal Axes 114

6.3.3.2 Rotation of the Coordinate Axes at the Observed Point (Rotated Axes) 115

6.3.4 Symmetrical Bending 116

6.3.4.1 Pure Bending 116

6.3.4.2 Nonuniform Bending 122

6.3.5 Nonsymmetrical Bending 126

6.3.6 Loading of Thin-walled Engineering Elements; Shear Center 133

6.3.6.1 Shear Center 134

6.3.7 Beam Deflections 136

6.3.8 Bending of Curved Elements 140

6.4 Stability of Columns 149

6.4.1 Critical Buckling Force in the Elastic Range 150

6.4.1.1 Pin-ended Columns 150

6.4.1.2 Columns with Other End Conditions 153

6.4.2 Critical Buckling Stress in the Elastic Range 155

6.4.3 Buckling – Plastic Range 156

6.4.3.1 Local Buckling of the Column 157

6.5 Eccentric Axial Loads 159

6.5.1 Eccentric Axial Load Acting in a Plane of Symmetry 159

6.5.2 General Case of an Eccentric Axial Load 161

References 164

7 Relationships Between Stress and Strain 167

7.1 Fundamental Considerations 167

7.2 Anisotropic Materials 169

7.3 Isotropic Materials 171

7.3.1 Determination of Hooke’s Law – Method of Superposition 175

7.3.2 Engineering Constants of Elasticity 178

7.4 Orthotropic Materials 180

7.5 Linear Stress–Strain–Temperature Relations for Isotropic Materials 184

References 186

8 Rheological Models 189

8.1 Introduction 189

8.2 Time-independent Behavior Modeling 190

8.2.1 Elastic Deformation Modeling 190

8.2.1.1 Hooke’s Element (H Model) 190

8.2.2 Deformation Modeling after the Elastic Limit 192

8.2.2.1 Saint Venant Element (SV Model) 192

8.2.2.2 Saint Venant Element–Spring/(SV–Spring) 192

8.2.2.3 Saint Venant Element | Spring−Spring/(SV | Spring−Spring) 192

8.3 Time-dependent Behavior Modeling 194

8.3.1 Newton Element (N Model): Linear Viscous Dashpot Element 195

8.3.2 Maxwell Model (M = H−N) 195

8.3.2.1 Generalized Maxwell Model 197

8.3.3 Voigt-Kelvin Model (K = H | N) 198

8.3.3.1 Generalized Voigt–Kelvin Model 199

8.3.4 Standard Linear Solid Model (SLS) 200

8.3.5 Voigt–Kelvin−Hooke’s Model (K−H) 201

8.3.6 Burgers’ Model 202

8.4 Differential Form of Constitutive Equations 205

References 207

9 Creep in Metallic Materials 209

9.1 Introduction 209

9.2 Plastic Deformation – General 211

9.2.1 Slip 211

9.2.2 Cleavage 212

9.2.3 Twinning 213

9.2.4 Grain Boundary Sliding 213

9.2.5 Void Coalescence 214

9.3 The Creep Phenomenon and Its Geometrical Representation 214

9.3.1 Creep Deformation Maps and Fracture Mechanism Maps 216

9.3.1.1 Creep Deformation Mechanisms 216

9.3.1.2 Fracture Micromechanisms and Macromechanisms 220

9.3.1.3 Creep Fracture Mechanisms 221

9.3.2 Short-time Uniaxial Creep Tests, Creep Modeling and Microstructure Analysis 223

9.3.2.1 Short-time Uniaxial Creep Tests 223

9.3.2.2 Creep Modeling 225

9.3.2.3 Microstructure Analysis – Basic 227

9.3.3 Long-term Creep Behavior Prediction Based on the Short-time Creep Process 228

9.3.3.1 Extrapolation Methods 230

9.3.3.2 Time–Temperature Parameters 231

9.3.4 Multiaxial Creep 232

9.4 Relaxation Phenomenon and Modeling 234

References 236

10 Fracture Mechanics 239

10.1 Introduction 239

10.2 Fracture Classification 240

10.3 Fatigue Phenomenon 242

10.3.1 Known Starting Points 242

10.3.2 Stress versus Life Curves (σ–N/S–N), Endurance Limit 242

10.4 Linear Elastic Fracture Mechanics (LEFM) 248

10.4.1 Basic Consideration 248

10.4.2 Crack Opening Modes 251

10.4.2.1 Stress Intensity Factor (K/SIF) 252

10.4.2.2 Plastic Zone Size around the Crack Tip 260

10.4.2.3 Plastic Zone Shape around the Crack Tip 263

10.5 Elastic–Plastic Fracture Mechanics (EPFM) 266

10.5.1 The J Integral 267

10.6 Experimental Determination of Fracture Toughness 270

10.6.1 Test Specimens: Shapes, Dimensions, Orientations and Pre-cracking 271

10.6.1.1 Shapes and Dimensions of the Specimens 271

10.6.1.2 Orientation of a Specimen Made from Base Material 272

10.6.1.3 Fatigue Pre-cracking 274

10.6.2 Fracture Toughness, KIc and the K–R Curve 274

10.6.2.1 R-curve (K–R Curve) 274

10.6.2.2 Plane Strain Fracture Toughness (KIc) Testing 277

10.6.3 Fracture Toughness JIc and the J–R Curve 279

10.6.3.1 R-curve (J–R Curve) 279

10.6.3.2 Fracture Toughness ( JIc) Determination/Testing 280

10.7 Charpy Impact Energy Testing 284

10.8 Crack Propagation 288

10.8.1 Introduction 288

10.8.2 Fatigue Crack Growth 289

10.8.2.1 The Paris Equation 294

10.8.2.2 The Walker Equation 296

10.8.2.3 The Forman Equation 297

10.8.2.4 The Forman–Newman–de Koning Equation 297

10.8.3 Creep Crack Growth 297

10.8.4 Life Assessment of Engineering Components 298

10.8.4.1 Constant Amplitude Loading 298

10.8.4.2 Variable Amplitude Loading 298

10.8.5 Crack Closure 299

10.8.5.1 Elber Crack Closure Phenomenon 299

10.8.6 A Brief Review of Testing of Unnotched, Axially Loaded Specimens 301

References 309

11 The Finite Element Method and Applications 313

11.1 The Finite Element Method (FEM) in the Analysis of Engineering Problems 313

11.1.1 Applications of FEM 313

11.1.2 The Advantages of Using the FEM 314

11.1.3 A Brief Overview of the Historical Development of the FEM 314

11.2 Linear Analysis of Structural Behavior 315

11.2.1 Formulations of Equilibrium Equations 316

11.2.1.1 Variational Formulation of the Finite Element (Equilibrium) Equation 318

11.2.2 Structures 334

11.2.3 Finite Elements 334

11.2.4 Shape Functions – Cartesian and Natural (Dimensionless) Coordinate Systems 334

11.2.4.1 Cartesian Coordinate System 335

11.2.4.2 Natural (Dimensionless) Coordinate System 341

11.2.5 One-dimensional Finite Elements 347

11.2.5.1 Basic 1-D Finite Elements 347

11.2.5.2 Finite Elements of Higher Order 359

11.2.6 Two-dimensional Finite Elements 363

11.2.6.1 Basic 2-D Finite Elements 367

11.2.6.2 Finite Elements of Higher Order 376

11.2.6.3 Transformation Procedure for the Finite Element Equation 378

11.2.7 Three-dimensional Finite Elements 379

11.2.7.1 Basic 3-D Finite Elements 381

11.2.7.2 Finite Elements of Higher Order 388

11.2.8 Isoparametric Finite Elements 393

11.2.8.1 Introduction 393

11.2.8.2 Isoparametric Representation 395

11.2.9 Bending of Elastic Flat Plates 398

11.2.9.1 Deformation Theories for Elastic Plates 398

11.2.9.2 Finite Elements Based on Kirchhoff Plate Theory 407

11.2.10 Basics of Dynamic Behavior of Elastic Structures 410

11.2.10.1 Mass Matrix of the Finite Element 413

11.2.10.2 Free, Undamped Vibrations of Constructions – Eigenvalues 414

11.3 A Brief Introduction to Nonlinear Analysis of Structural Behavior 421

11.4 Metal-forming Processes – Brief Overview 422

11.4.1 Introduction 422

11.4.2 Classification, Variables and Characteristics of Metal-forming Processes 423

11.4.2.1 Comparison of Hot and Cold Working Processes in Terms of Working Temperature, Shaping Force and Achieved Material Properties 428

11.4.3 Basic Settings Related to the Theory of Metal-forming Processes 429

11.4.3.1 Strain-rate Tensor and Data Relating to Yield Criteria 430

11.4.3.2 Virtual Work-rate Principle 433

11.4.3.3 The Prandtl–Reuss Equations 433

11.4.3.4 The Governing Equations of Plastic Deformation 437

11.4.3.5 Shape Functions 437

11.4.3.6 Strain-rate Matrix 438

11.5 The Application of the Finite Element Method in Structural Analysis 438

11.5.1 One-dimensional Finite Elements: Finite Element Analysis of Truss Structure Deformation 439

11.5.2 Two-dimensional Finite Elements: J Integral Calculation 443

11.5.3 Special Two-dimensional Finite Elements in Shear Stress Analysis 447

11.5.3.1 Introduction 447

11.5.3.2 Application of General Quadrilateral Finite Elements 450

References 451

Index 453

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