**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**