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Table of Contents
About the Authors xiii Preface xv
1 Forces and Moments 1
1.1 Introduction 1
1.2 Units 1
1.3 Forces in Mechanics of Materials 3
1.4 Concentrated Forces 4
1.5 Moment of a Concentrated Force 9
1.6 Distributed Forces—Force and Moment Resultants 19
1.7 Internal Forces and Stresses—Stress Resultants 27
1.8 Restraint Forces and Restraint Force Resultants 32
1.9 Summary and Conclusions 33
2 Static Equilibrium 35
2.1 Introduction 35
2.2 Free Body Diagrams 35
2.3 Equilibrium—Concentrated Forces 38
2.3.1 Two Force Members and Pin Jointed Trusses 38
2.3.2 Slender Rigid Bars 44
2.3.3 Pulleys and Cables 49
2.3.4 Springs 52
2.4 Equilibrium—Distributed Forces 55
2.5 Equilibrium in Three Dimensions 59
2.6 Equilibrium—Internal Forces and Stresses 62
2.6.1 Equilibrium of Internal Forces in Three Dimensions 65
2.6.2 Equilibrium in Two Dimensions—Plane Stress 69
2.6.3 Equilibrium in One Dimension—Uniaxial Stress 70
2.7 Summary and Conclusions 70
3 Displacement, Strain, and Material Properties 71
3.1 Introduction 71
3.2 Displacement and Strain 71
3.2.1 Displacement 72
3.2.2 Strain 72
3.3 Compatibility 76
3.4 Linear Material Properties 77
3.4.1 Hooke’s Law in One Dimension—Tension 77
3.4.2 Poisson’s Ratio 81
3.4.3 Hooke’s Law in One Dimension—Shear in Isotropic Materials 82
3.4.4 Hooke’s Law in Two Dimensions for Isotropic Materials 83
3.4.5 Generalized Hooke’s Law for Isotropic Materials 84
3.5 Some Simple Solutions for Stress, Strain, and Displacement 85
3.6 Thermal Strain 89
3.7 Engineering Materials 90
3.8 Fiber Reinforced Composite Laminates 90
3.8.1 Hooke’s Law in Two Dimensions for a FRP Lamina 91
3.8.2 Properties of Unidirectional Lamina 94
3.9 Plan for the Following Chapters 96
3.10 Summary and Conclusions 98
4 Classical Analysis of the Axially Loaded Slender Bar 99
4.1 Introduction 99
4.2 Solutions from the Theory of Elasticity 99
4.3 Derivation and Solution of the Governing Equations 109
4.4 The Statically Determinate Case 116
4.5 The Statically Indeterminate Case 129
4.6 Variable Cross Sections 136
4.7 Thermal Stress and Strain in an Axially Loaded Bar 142
4.8 Shearing Stress in an Axially Loaded Bar 143
4.9 Design of Axially Loaded Bars 145
4.10 Analysis and Design of Pin Jointed Trusses 149
4.11 Work and Energy—Castigliano’s Second Theorem 153
4.12 Summary and Conclusions 162
5 A General Method for the Axially Loaded Slender Bar 165
5.1 Introduction 165
5.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 165
5.3 The Assembled Global Equations and Their Solution 169
5.4 A General Method—Distributed Applied Loads 182
5.5 Variable Cross Sections 196
5.6 Analysis and Design of Pin-jointed Trusses 202
5.7 Summary and Conclusions 211
6 Torsion 213
6.1 Introduction 213
6.2 Torsional Displacement, Strain, and Stress 213
6.3 Derivation and Solution of the Governing Equations 216
6.4 Solutions from the Theory of Elasticity 225
6.5 Torsional Stress in Thin Walled Cross Sections 229
6.6 Work and Energy—Torsional Stiffness in a Thin Walled Tube 231
6.7 Torsional Stress and Stiffness in Multicell Sections 239
6.8 Torsional Stress and Displacement in Thin Walled Open Sections 242
6.9 A General (Finite Element) Method 245
6.10 Continuously Variable Cross Sections 254
6.11 Summary and Conclusions 255
7 Classical Analysis of the Bending of Beams 257
7.1 Introduction 257
7.2 Area Properties—Sign Conventions 257
7.2.1 Area Properties 257
7.2.2 Sign Conventions 259
7.3 Derivation and Solution of the Governing Equations 260
7.4 The Statically Determinate Case 271
7.5 Work and Energy—Castigliano’s Second Theorem 278
7.6 The Statically Indeterminate Case 281
7.7 Solutions from the Theory of Elasticity 290
7.8 Variable Cross Sections 300
7.9 Shear Stress in Non Rectangular Cross Sections—Thin Walled Cross Sections 302
7.10 Design of Beams 309
7.11 Large Displacements 313
7.12 Summary and Conclusions 314
8 A General Method (FEM) for the Bending of Beams 315
8.1 Introduction 315
8.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 315
8.3 The Global Equations and their Solution 320
8.4 Distributed Loads in FEM 327
8.5 Variable Cross Sections 341
8.6 Summary and Conclusions 345
9 More about Stress and Strain, and Material Properties 347
9.1 Introduction 347
9.2 Transformation of Stress in Two Dimensions 347
9.3 Principal Axes and Principal Stresses in Two Dimensions 350
9.4 Transformation of Strain in Two Dimensions 354
9.5 Strain Rosettes 356
9.6 Stress Transformation and Principal Stresses in Three Dimensions 358
9.7 Allowable and Ultimate Stress, and Factors of Safety 361
9.8 Fatigue 363
9.9 Creep 364
9.10 Orthotropic Materials—Composites 365
9.11 Summary and Conclusions 366
10 Combined Loadings on Slender Bars—ThinWalled Cross Sections 367
10.1 Introduction 367
10.2 Review and Summary of Slender Bar Equations 367
10.2.1 Axial Loading 367
10.2.2 Torsional Loading 369
10.2.3 Bending in One Plane 370
10.3 Axial and Torsional Loads 372
10.4 Axial and Bending Loads—2D Frames 375
10.5 Bending in Two Planes 384
10.5.1 When Iyz is Equal to Zero 384
10.5.2 When Iyz is Not Equal to Zero 386
10.6 Bending and Torsion in Thin Walled Open Sections—Shear Center 393
10.7 Bending and Torsion in Thin Walled Closed Sections—Shear Center 399
10.8 Stiffened Thin Walled Beams 405
10.9 Summary and Conclusions 416
11 Work and Energy Methods—Virtual Work 417
11.1 Introduction 417
11.2 Introduction to the Principle of Virtual Work 417
11.3 Static Analysis of Slender Bars by Virtual Work 421
11.3.1 Axially Loading 421
11.3.2 Torsional Loading 426
11.3.3 Beams in Bending 427
11.3.4 Combined Axial, Torsional, and Bending Behavior 430
11.4 Static Analysis of 3D and 2D Solids by Virtual Work 430
11.5 The Element Stiffness Matrix for Plane Stress 433
11.6 The Element Stiffness Matrix for 3D Solids 436
11.7 Summary and Conclusions 437
12 Structural Analysis in Two and Three Dimensions 439
12.1 Introduction 439
12.2 The Governing Equations in Two Dimensions—Plane Stress 440
12.3 Finite Elements and the Stiffness Matrix for Plane Stress 445
12.4 Thin Flat Plates—Classical Analysis 452
12.5 Thin Flat Plates—FEM Analysis 455
12.6 Shell Structures 459
12.7 Stiffened Shell Structures 466
12.8 Three Dimensional Structures—Classical and FEM Analysis 470
12.9 Summary and Conclusions 477
13 Analysis of Thin Laminated Composite Material Structures 479
13.1 Introduction to Classical Lamination Theory 479
13.2 Strain Displacement Equations for Laminates 480
13.3 Stress-Strain Relations for a Single Lamina 482
13.4 Stress Resultants for Laminates 486
13.5 CLT Constitutive Description 489
13.6 Determining Laminae Stress/Strains 492
13.7 Laminated Plates Subject to Transverse Loads 493
13.8 Summary and Conclusion 498
14 Buckling 499
14.1 Introduction 499
14.2 The Equations for a Beam with Combined Lateral and Axial Loading 499
14.3 Buckling of a Column 504
14.4 The Beam Column 512
14.5 The Finite Element Method for Bending and Buckling 515
14.6 Buckling of Frames 524
14.7 Buckling of Thin Plates and Other Structures 524
14.8 Summary and Conclusions 527
15 Structural Dynamics 529
15.1 Introduction 529
15.2 Dynamics of Mass/Spring Systems 529
15.2.1 Free Motion 529
15.2.2 Forced Motion—Resonance 540
15.2.3 Forced Motion—Response 547
15.3 Axial Vibration of a Slender Bar 548
15.3.1 Solutions Based on the Differential Equation 548
15.3.2 Solutions Based on FEM 560
15.4 Torsional Vibration 567
15.4.1 Torsional Mass/Spring Systems 567
15.4.2 Distributed Torsional Systems 568
15.5 Vibration of Beams in Bending 569
15.5.1 Solutions of the Differential Equation 569
15.5.2 Solutions Based on FEM 574
15.6 The Finite Element Method for all Elastic Structures 577
15.7 Addition of Damping 577
15.8 Summary and Conclusions 582
16 Evolution in the (Intelligent) Design and Analysis of Structural Members 583
16.1 Introduction 583
16.2 Evolution of a Truss Member 584
16.2.1 Step 1. Slender Bar Analysis 584
16.2.2 Step 2. Rectangular Bar—Plane Stress FEM 585
16.2.3 Step 3. Rectangular Bar with Pin Holes—Plane Stress Analysis 586
16.2.4 Step 4. Rectangular Bar with Pin Holes—Solid Body Analysis 587
16.2.5 Step 5. Add Material Around the Hole—Solid Element Analysis 588
16.2.6 Step 6. Bosses Added—Solid Element Analysis 590
16.2.7 Step 7. Reducing the Weight—Solid Element Analysis 591
16.2.8 Step 8. Buckling Analysis 592
16.3 Evolution of a Plate with a Hole—Plane Stress 592
16.4 Materials in Design 594
16.5 Summary and Conclusions 594
A Matrix Definitions and Operations 595
A.1 Introduction 595
A.2 Matrix Definitions 595
A.3 Matrix Algebra 597
A.4 Partitioned Matrices 598
A.5 Differentiating and Integrating a Matrix 598
A.6 Summary of Useful Matrix Relations 599
B Area Properties of Cross Sections 601
B.1 Introduction 601
B.2 Centroids of Cross Sections 601
B.3 Area Moments and Product of Inertia 603
B.4 Properties of Common Cross Sections 609
C Solving Sets of Linear Algebraic Equations with Mathematica 611
C.1 Introduction 611
C.2 Systems of Linear Algebraic Equations 611
C.3 Solving Numerical Equations in Mathematica 611
C.4 Solving Symbolic Equations in Mathematica 612
C.5 Matrix Multiplication 613
D Orthogonality of Normal Modes 615
D.1 Introduction 615
D.2 Proof of Orthogonality for Discrete Systems 615
D.3 Proof of Orthogonality for Continuous Systems 616
References 617
Index 619
