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Book SynopsisAnalysis of Engineering Structures and Material Behavior Professor Josip Brniae, D. Sc.
Table of ContentsFrequently 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
