Analysis of Structures: An Introduction Including Numerical Methods / Edition 1

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Analysis of Structures offers a new way of introducing engineering students to the subject of stress and deformation analysis of solid objects and to become more familiar with how numerical methods such as the finite element method are currently being used in industry. The authors seek to secure a thorough understanding of the basic numerical skills and insight of the kinds of results these methods can generate, providing the classical solutions along with the numerical solutions. Throughout the text, analytical development provides the student with the understanding that is necessary to interpret and use the solutions that are obtained using contemporary software based on the popular finite element method. These initial topics are then extended to analyze other 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 housing exercises and examples that use modern software which generates color contour plots of deformation and internal stress.

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Product Details

  • ISBN-13: 9780470977620
  • Publisher: Wiley
  • Publication date: 10/11/2011
  • Series: Wiley Series in Computational Mechanics Series , #33
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 638
  • Product dimensions: 6.90 (w) x 9.90 (h) x 1.40 (d)

Meet the Author

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

Anthony Waas is Professor of Aerospace Engineering andProfessor of Mechanical Engineering, and Director, CompositeStructures Laboratory at the University of Michigan. His currentresearch interests are damage tolerance analysis of compositematerials and components made of composite materials,nanocomposites, structural engineering, biomaterials andbioengineering, and structures and mechanical components operatingunder "hot" conditions. A recipient of many awards for teaching andresearch excellence, Professor Waas is a Fellow of ASME and theAAM, and an Associate Fellow of AIAA and has served as an AssociateEditor of the AIAA Journal (1995-02) and on the EditorialAdvisory Board of the AIAA Journal of Aircraft (1995-00). Heis currently on the editorial board of the Journal Composites: Band serves as an Associate Editor of the RAeS AeronauticalJournal, IJ of Engineering Science and Journal ofApplied Mechanics, and is on the Editorial Board of ComputerModeling in Engineering and Sciences, and the Journal of theMechanical Behavior of Materials. He was the Technical Chair ofthe 49th AIAA SDM conference.

Joe G Eisley is Professor Emeritus - AerospaceEngineering in the College of Engineering at the University ofMichigan. He is author of Mechanics of ElasticStructures.

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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 Dimensions65

2.6.2 Equilibrium in Two Dimensions—Plane Stress69

2.6.3 Equilibrium in One Dimension—Uniaxial Stress70

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—Tension77

3.4.2 Poisson’s Ratio 81

3.4.3 Hooke’s Law in One Dimension—Shear inIsotropic Materials 82

3.4.4 Hooke’s Law in Two Dimensions for IsotropicMaterials 83

3.4.5 Generalized Hooke’s Law for IsotropicMaterials 84

3.5 Some Simple Solutions for Stress, Strain, and Displacement85

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 FRPLamina 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 Bar99

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 Theorem153

4.12 Summary and Conclusions 162

5 A General Method for the Axially Loaded Slender Bar165

5.1 Introduction 165

5.2 Nodes, Elements, Shape Functions, and the Element StiffnessMatrix 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 WalledTube 231

6.7 Torsional Stress and Stiffness in Multicell Sections 239

6.8 Torsional Stress and Displacement in Thin Walled OpenSections 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 Theorem278

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—ThinWalled 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 StiffnessMatrix 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 Properties347

9.1 Introduction 347

9.2 Transformation of Stress in Two Dimensions 347

9.3 Principal Axes and Principal Stresses in Two Dimensions350

9.4 Transformation of Strain in Two Dimensions 354

9.5 Strain Rosettes 356

9.6 Stress Transformation and Principal Stresses in ThreeDimensions 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 CrossSections 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 OpenSections—Shear Center 393

10.7 Bending and Torsion in Thin Walled ClosedSections—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 Behavior430

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 Dimensions439

12.1 Introduction 439

12.2 The Governing Equations in Two Dimensions—PlaneStress 440

12.3 Finite Elements and the Stiffness Matrix for Plane Stress445

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 FEMAnalysis 470

12.9 Summary and Conclusions 477

13 Analysis of Thin Laminated Composite Material Structures479

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 AxialLoading 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 Equation548

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 Structures577

15.7 Addition of Damping 577

15.8 Summary and Conclusions 582

16 Evolution in the (Intelligent) Design and Analysis ofStructural 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 FEM585

16.2.3 Step 3. Rectangular Bar with Pin Holes—PlaneStress Analysis 586

16.2.4 Step 4. Rectangular Bar with Pin Holes—SolidBody Analysis 587

16.2.5 Step 5. Add Material Around the Hole—SolidElement Analysis 588

16.2.6 Step 6. Bosses Added—Solid Element Analysis590

16.2.7 Step 7. Reducing the Weight—Solid ElementAnalysis 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 Mathematica611

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

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