## A First Course in the Finite Element Method, Sixth Edition, SI Version

By Daryl L. Logan

**Contents:**

Preface to the SI Edition ix

Preface x

Digital Resource xii

Acknowledgments xv

Notation xvi

1 Introduction 1

Chapter Objectives 1

Prologue 1

1.1 Brief History 3

1.2 Introduction to Matrix Notation 4

1.3 Role of the Computer 6

1.4 General Steps of the Finite Element Method 7

1.5 Applications of the Finite Element Method 15

1.6 Advantages of the Finite Element Method 21

1.7 Computer Programs for the Finite Element Method 25

References 27

Problems 30

2 Introduction to the Stiffness (Displacement) Method 31

Chapter Objectives 31

Introduction 31

2.1 Definition of the Stiffness Matrix 32

2.2 Derivation of the Stiffness Matrix

for a Spring Element 32

2.3 Example of a Spring Assemblage 36

2.4 Assembling the Total Stiffness Matrix by Superposition

(Direct Stiffness Method) 38

2.5 Boundary Conditions 40

2.6 Potential Energy Approach to Derive Spring Element Equations 55

Summary Equations 65

References 66

Problems 66

3 Development of Truss Equations 72

Chapter Objectives 72

Introduction 72

3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 73

3.2 Selecting a Displacement Function in Step 2 of the Derivation

of Stiffness Matrix for the One-Dimensional Bar Element 78

3.3 Transformation of Vectors in Two Dimensions 82

3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 84

3.5 Computation of Stress for a Bar in the x – y Plane 89

3.6 Solution of a Plane Truss 91

3.7 Transformation Matrix and Stiffness Matrix for a Bar

in Three-Dimensional Space 100

3.8 Use of Symmetry in Structures 109

3.9 Inclined, or Skewed, Supports 112

3.10 Potential Energy Approach to Derive Bar Element Equations 121

3.11 Comparison of Finite Element Solution to Exact Solution for Bar 132

3.12 Galerkin’s Residual Method and Its Use to Derive

the One-Dimensional Bar Element Equations 136

3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 139

3.14 Flowchart for Solution of Three-Dimensional Truss Problems 143

3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 144

Summary Equations 146

References 147

Problems 147

4 Development of Beam Equations 169

Chapter Objectives 169

Introduction 169

4.1 Beam Stiffness 170

4.2 Example of Assemblage of Beam Stiffness Matrices 180

4.3 Examples of Beam Analysis Using the Direct Stiffness Method 182

4.4 Distributed Loading 195

4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 208

4.6 Beam Element with Nodal Hinge 214

4.7 Potential Energy Approach to Derive Beam Element Equations 222

4.8 Galerkin’s Method for Deriving Beam Element Equations 225

Summary Equations 227

References 228

Problems 229

5 Frame and Grid Equations 239

Chapter Objectives 239

Introduction 239

5.1 Two-Dimensional Arbitrarily Oriented Beam Element 239

5.2 Rigid Plane Frame Examples 243

5.3 Inclined or Skewed Supports—Frame Element 261

5.4 Grid Equations 262

5.5 Beam Element Arbitrarily Oriented in Space 280

5.6 Concept of Substructure Analysis 295

Summary Equations 300

References 302

Problems 303

6 Development of the Plane Stress and Plane Strain

Stiffness Equations 337

Chapter Objectives 337

Introduction 337

6.1 Basic Concepts of Plane Stress and Plane Strain 338

6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 342

6.3 Treatment of Body and Surface Forces 357

6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 362

6.5 Finite Element Solution of a Plane Stress Problem 363

6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) 374

Summary Equations 379

References 384

Problems 384

7 Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 391

Chapter Objectives 391

Introduction 391

7.1 Finite Element Modeling 392

7.2 Equilibrium and Compatibility of Finite Element Results 405

7.3 Convergence of Solution and Mesh Refinement 408

7.4 Interpretation of Stresses 411

7.5 Flowchart for the Solution of Plane Stress/Strain Problems 413

7.6 Computer Program–Assisted Step-by-Step Solution, Other Models, and Results

for Plane Stress/Strain Problems 414

References 420

Problems 421

8 Development of the Linear-Strain Triangle Equations 437

Chapter Objectives 437

Introduction 437

8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 437

8.2 Example LST Stiffness Determination 442

8.3 Comparison of Elements 444

Summary Equations 447

References 448

Problems 448

9 Axisymmetric Elements 451

Chapter Objectives 451

Introduction 451

9.1 Derivation of the Stiffness Matrix 451

9.2 Solution of an Axisymmetric Pressure Vessel 462

9.3 Applications of Axisymmetric Elements 468

Summary Equations 473

References 475

Problems 475

10 Isoparametric Formulation 486

Chapter Objectives 486

Introduction 486

10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 487

10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix 492

10.3 Newton-Cotes and Gaussian Quadrature 503

10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 509

10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) 515

Summary Equations 526

References 530

Problems 530

11 Three-Dimensional Stress Analysis 536

Chapter Objectives 536

Introduction 536

11.1 Three-Dimensional Stress and Strain 537

11.2 Tetrahedral Element 539

11.3 Isoparametric Formulation and Hexahedral Element 547

Summary Equations 555

References 558

Problems 558

12 Plate Bending Element 572

Chapter Objectives 572

Introduction 572

12.1 Basic Concepts of Plate Bending 572

12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations 577

12.3 Some Plate Element Numerical Comparisons 582

12.4 Computer Solutions for Plate Bending Problems 584

Summary Equations 588

References 590

Problems 591

13 Heat Transfer and Mass Transport 599

Chapter Objectives 599

Introduction 599

13.1 Derivation of the Basic Differential Equation 601

13.2 Heat Transfer with Convection 604

13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h 605

13.4 One-Dimensional Finite Element Formulation Using a Variational Method 607

13.5 Two-Dimensional Finite Element Formulation 626

13.6 Line or Point Sources 636

13.7 Three-Dimensional Heat Transfer by the Finite Element Method 639

13.8 One-Dimensional Heat Transfer with Mass Transport 641

13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 642

13.10 Flowchart and Examples of a Heat Transfer Program 646

Summary Equations 651

References 654

Problems 655

14 Fluid Flow in Porous Media and through

Hydraulic Networks; and Electrical Networks and Electrostatics 673

Chapter Objectives 673

Introduction 673

14.1 Derivation of the Basic Differential Equations 674

14.2 One-Dimensional Finite Element Formulation 678

14.3 Two-Dimensional Finite Element Formulation 691

14.4 Flowchart and Example of a Fluid-Flow Program 696

14.5 Electrical Networks 697

14.6 Electrostatics 701

Summary Equations 715

References 719

Problems 720

15 Thermal Stress 727

Chapter Objectives 727

Introduction 727

15.1 Formulation of the Thermal Stress Problem and Examples 727

Summary Equations 752

Reference 753

Problems 754

16 Structural Dynamics and Time-Dependent

Heat Transfer 761

Chapter Objectives 761

Introduction 761

16.1 Dynamics of a Spring-Mass System 762

16.2 Direct Derivation of the Bar Element Equations 764

16.3 Numerical Integration in Time 768

16.4 Natural Frequencies of a One-Dimensional Bar 780

16.5 Time-Dependent One-Dimensional Bar Analysis 784

16.6 Beam Element Mass Matrices and Natural Frequencies 789

16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices 796

16.8 Time-Dependent Heat Transfer 801

16.9 Computer Program Example Solutions for Structural Dynamics 808

Summary Equations 817

References 821

Problems 822

Appendix A Matrix Algebra 827

Appendix B Methods for Solution of Simultaneous

Linear Equations 843

Appendix C Equations from Elasticity Theory 865

Appendix D Equivalent Nodal Forces 873

Appendix E Principle of Virtual Work 876

Appendix F Geometric Properties of Structural Steel Wide-Flange

Sections (W Shapes) 880

Answers to Selected Problems 908

Index 938