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# Analysis of Structures. An Introduction Including Numerical Methods

• Published: September 2011
• Region: Global
• 638 Pages
• John Wiley and Sons Ltd

Analysis of Structures offers an original way of introducing engineering students to the subject of stress and deformation analysis of solid objects, and helps them become more familiar with how numerical methods such as the finite element method are used in industry.

Eisley and Waas secure for the reader a thorough understanding of the basic numerical skills and insight into interpreting the results these methods can generate.

Throughout the text, they include analytical development alongside the computational equivalent, providing the student with the understanding that is necessary to interpret and use the solutions that are obtained using software based on the finite element method. They then extend these methods to the analysis of solid and structural components that are used in modern aerospace, mechanical and civil engineering applications.

Analysis of Structures is accompanied by a book companion website company website housing exercises and examples that use modern software which generates color contour plots of deformation and internal stress.It offers invaluable guidance and understanding to senior level and graduate students studying courses in stress

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

10.1 Introduction 367

10.2 Review and Summary of Slender Bar Equations 367

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

11.3.3 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

15.8 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

Anthony M. Waas and Joe G. Eisley, University of Michigan, USA

Anthony Waas is Professor of Aerospace Engineering and Professor of Mechanical Engineering, and Director, Composite Structures Laboratory at the University of Michigan. His current research interests are damage tolerance analysis of composite materials and components made of composite materials, nanocomposites, structural engineering, biomaterials and bioengineering, and structures and mechanical components operating under "hot" conditions. A recipient of many awards for teaching and research excellence, Professor Waas is a Fellow of ASME and the AAM, and an Associate Fellow of AIAA and has served as an Associate Editor of the AIAA Journal (1995-02) and on the Editorial Advisory Board of the AIAA Journal of Aircraft (1995-00). He is currently on the editorial board of the Journal Composites: B and serves as an Associate Editor of the RAeS Aeronautical Journal, IJ of Engineering Science and Journal of Applied Mechanics, and is on the Editorial Board of Computer Modeling in Engineering and Sciences, and the Journal of the Mechanical Behavior of Materials. He was the Technical Chair of the 49th AIAA SDM conference.

Joe G Eisley is Professor Emeritus - Aerospace Engineering in the College of Engineering at the University of Michigan. He is author of Mechanics of Elastic Structures.

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