Computational Statics and Dynamics: An Introduction Based on the Finite Element Method

ISBN-13: 9783031096723 / Angielski

Andreas Oechsner

Opis

This book is the 3rd edition of an introduction to modern computational mechanics based on the finite element method. This third edition is largely extended, adding many new examples to let the reader understand the principles better by performing calculations by hand, as well as numerical example to practice the finite element approach to engineering problems. The new edition comes together with a set of digital flash cards with questions and answers that improve learning success. Featuring over 100 more pages, the new edition will help students succeed in mechanics courses by showing them how to apply the fundamental knowledge they gained in the first years of their engineering education to more advanced topics. In order to deepen readers' understanding of the equations and theories discussed, each chapter also includes supplementary problems. These problems start with fundamental knowledge questions on the theory presented in the respective chapter, followed by calculation problems. In total, over 80 such calculation problems are provided, along with brief solutions for each. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app.

Spis treści

1 Introduction to the Finite Element Method. . . . . . . . . . . . . . . 1

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Rods and Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Derivation of the Governing Differential Equation . . . . . . . . . . 14

2.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Derivation of the Principal Finite Element Equation . . 21

2.3.2 Derivation of Interpolation Functions . . . . . . . . . . . . . . . 45

2.3.3 Assembly of Elements and Consideration of Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.4 Post-Computation: Determination of Strain, Stress

and further Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.5 Analogies to other Field Problems . . . . . . . . . . . . . . . . . . 62

2.3.6 Solved Rod Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4 Assembly of Elements to Plane Truss Structures . . . . . . . . . . . . 77

2.4.1 Rotational Transformation in a Plane . . . . . . . . . . . . . . . 77

2.4.2 Solved Truss Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3 Euler-Bernoulli Beams and Frames . . . . . . . . . . . . . . . . . . . . . . . 115

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2 Derivation of the Governing Differential Equation . . . . . . . . . . 118

3.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xiii

xiv Contents

3.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.3.1 Derivation of the Principal Finite Element Equation . . 133

3.3.2 Derivation of Interpolation Functions . . . . . . . . . . . . . . . 149

3.3.3 Assembly of Elements and Consideration of Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

3.3.4 Post-Computation: Determination of Strain, Stress

and further Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.3.5 Solved Beam Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

3.4 Assembly of Elements to Plane Frame Structures . . . . . . . . . . . 195

3.4.1 Rotation of a Beam Element . . . . . . . . . . . . . . . . . . . . . . . 195

3.4.2 Generalized Beam Element . . . . . . . . . . . . . . . . . . . . . . . . 198

3.4.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

3.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

4 Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4.2 Derivation of the Governing Differential Equation . . . . . . . . . . 258

4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

4.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

4.2.3 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

4.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

4.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

4.3.1 Derivation of the Principal Finite Element Equation . . 269

4.3.2 Linear Interpolation Functions for the Displacement

and Rotational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

4.3.3 Higher-Order Interpolation Functions for the Beam

with Shear Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 303

4.3.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

4.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

5 Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

5.2 Derivation of the Governing Differential Equation . . . . . . . . . . 328

5.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

5.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

5.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

5.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

5.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

5.3.1 Derivation of the Principal Finite Element Equation . . 337

Contents xv

5.3.2 Four-Node Planar Element . . . . . . . . . . . . . . . . . . . . . . . . 341

5.3.3 Solved Plane Elasticity Problems . . . . . . . . . . . . . . . . . . . 354

5.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

6 Classical Plate Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

6.2 Derivation of the Governing Differential Equation . . . . . . . . . . 387

6.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

6.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

6.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

6.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

6.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

6.3.1 Derivation of the Principal Finite Element Equation . . 398

6.3.2 Rectangular Four-Node Plate Element . . . . . . . . . . . . . . 401

6.3.3 Distorted Four-Node Plate Element . . . . . . . . . . . . . . . . . 418

6.3.4 Solved Classical Plate Element Problems . . . . . . . . . . . . 422

6.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

7 Shear Deformable Plate Elements . . . . . . . . . . . . . . . . . . . . . . . . 437

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

7.2 Derivation of the Governing Differential Equation . . . . . . . . . . 438

7.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

7.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

7.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

7.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

7.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

7.3.1 Derivation of the Principal Finite Element Equation . . 450

7.3.2 Rectangular Four-Node Plate Element . . . . . . . . . . . . . . 457

7.3.3 Solved Thick Plate Element Problems . . . . . . . . . . . . . . . 463

7.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

8 Three-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

8.1 Derivation of the Governing Differential Equation . . . . . . . . . . 473

8.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

8.1.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

8.1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

8.1.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

8.2 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

8.2.1 Derivation of the Principal Finite Element Equation . . 479

8.2.2 Hexahedron Solid Elements . . . . . . . . . . . . . . . . . . . . . . . . 484

xvi Contents

8.2.3 Solved Three-Dimensional Element Problems . . . . . . . . 498

8.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

9 Principles of Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

9.1 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

9.2 Relationship Between Displacement, Velocity and Acceleration510

9.3 Solved Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

9.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

10 Integration Methods for Transient Problems . . . . . . . . . . . . . . 521

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

10.2 Derivation of the Governing Differential Equation . . . . . . . . . . 522

10.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

10.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

10.2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

10.2.4 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

10.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

10.3.1 Derivation of the Principal Finite Element Equation . . 524

10.3.2 Consideration of Damping . . . . . . . . . . . . . . . . . . . . . . . . . 527

10.3.3 Transient Solution Schemes . . . . . . . . . . . . . . . . . . . . . . . . 532

10.3.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

10.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

A Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

A.1 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

A.2 Frequently Used Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

A.3 Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

A.4 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

A.5 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

A.6 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

A.7 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

A.8 Integration and Coordinate Transformation . . . . . . . . . . . . . . . . 554

A.9 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

A.9.1 Simpson’s Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

A.9.2 Gauss-Legendre Quadrature . . . . . . . . . . . . . . . . . . . . . . . 563

A.10 Taylor’s Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

A.11 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

A.11.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

A.11.2 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

A.11.3 Dyadic Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

Contents xvii

A.11.4 Inverse of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

A.12 Solution of Linear Systems of Equations . . . . . . . . . . . . . . . . . . . 578

A.12.1 Elimination of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

A.12.2 Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

A.13 Elementary Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

A.14 Analytical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

A.14.1 Straight-Line Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

A.14.2 Sign of Second Derivative of a Curve . . . . . . . . . . . . . . . . 581

A.14.3 Area of a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

B Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

B.1 Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

B.2 Second Moment of Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

B.3 Parallel-Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

C Units and Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

C.1 SI Base Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

C.2 Coherent SI derived Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

C.3 Consistent Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

C.4 Conversion of Important English Units to the Metric System . 589

D Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

D.1 Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

D.2 Classical Plate Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

E Summary of Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

E.1 One-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

E.2 Two-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

E.3 Three-Dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

F Extrapolation from Integration Points to Nodes . . . . . . . . . . 641

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

Answers to Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . 651

F.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

F.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

F.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

F.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

6.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743

6.6 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754

6.7 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766

6.8 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

6.9 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

xviii Contents

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

O autorze

Andreas Öchsner is a Full Professor of Lightweight Design and Structural Simulation at Esslingen University of Applied Sciences, Germany. After completing his Dipl.-Ing. degree in Aeronautical Engineering at the University of Stuttgart (1997), he served as a research and teaching assistant at the University of Erlangen-Nuremberg from 1997 to 2003 while pursuing his Doctor of Engineering Sciences (Dr.-Ing.) degree. From 2003 to 2006, he was an Assistant Professor at the Department of Mechanical Engineering and Head of the Cellular Metals Group affiliated with the University of Aveiro, Portugal. He spent seven years (2007–2013) as a Full Professor at the Department of Applied Mechanics, Technical University of Malaysia, where he was also Head of the Advanced Materials and Structure Lab. From 2014 to 2017 he was a Full Professor at the School of Engineering, Griffith University, Australia and Leader of the Mechanical Engineering Program (Head of Discipline and Program Director).

Blurb

This book is the 3rd edition of an introduction to modern computational mechanics based on the finite element method. This third edition is largely extended, adding many new examples to let the reader understand the principles better by performing calculations by hand, as well as numerical example to practice the finite element approach to engineering problems. The new edition comes together with a set of digital flash cards with questions and answers that improve learning success. Featuring over 100 more pages, the new edition will help students succeed in mechanics courses by showing them how to apply the fundamental knowledge they gained in the first years of their engineering education to more advanced topics.

In order to deepen readers’ understanding of the equations and theories discussed, each chapter also includes supplementary problems. These problems start with fundamental knowledge questions on the theory presented in the respective chapter, followed by calculation problems. In total, over 80 such calculation problems are provided, along with brief solutions for each.