The Finite Element Hlethod using 0IHRHB

CRC Press BocaRaton New York London Tokyo

TABLE OF CONTENTS

C H A P T E R 1. I N T R O D U C T I O N T O M A T L A B

1.1 Finite Element Method 1 1.2 Overview of the Book 1 1.3 About MATLAB 2 1.4 Vector and Matrix Manipulations 2 1.5 Matrix Functions 4 1.6 Da ta Analysis Functions 10 1.7 Tools for Polynomials 12 1.8 Making Comp lex Numbers 14 1.9 Nonlinear Algebraic Equations 15

1.10 Solving Differential Equations 16 1.11 Loop and Logical Statement 17 1.12 Writing Function Subroutines 20 1.13 File Manipulation 22 1.14 Basic Input-Output Functions 23 1.15 Plot t ing Tools 24

C H A P T E R 2. A P P R O X I M A T I O N T E C H N I Q U E S

2.1 Methods of Weighted Residual 31 2.2 Weak Formulation 34 2.3 Piecewise Continuous Trial Function 35 2.4 Galerkin's Finite Element Formulation 37 2.5 Variational Method 42 2.6 Rayleigh-Ritz Method 42 2.7 Rayleigh-Ritz Finite Element Method 43

Problems 46

C H A P T E R 3 . F I N I T E E L E M E N T P R O G R A M M I N G

3.1 Overall Program Structure 51 3.2 Input Da ta 52 3.3 Assembly of Element Matrices and Vectors 54 3.4 Application of Constraints 54 3.5 Example Programs 56

Problems 69

C H A P T E R 4. D I R E C T A P P R O A C H W I T H S P R I N G S Y S T E M

4.1 Linear Spring 71 4.2 Axial Member 74 4.3 Torsional Member 77

4.4 Other Systems 77 Problems 80

CHAPTER 5. LAPLACE'S A N D POISSON'S EQUATIONS

5.1 Governing Equation 83 5.2 Linear Triangulär Element 86 5.3 Bilinear Rectangular Element 90 5.4 Boundary Integral 92 5.5 Transient Analysis 96 5.6 Time Integration Technique 98 5.7 Axisymmetric Analysis 101 5.8 Three-Dimensional Analysis 103 5.9 MATLAB Application to 2-D Steady State Analysis 107

5.10 MATLAB Application to Axisymmetric Analysis 119 5.11 MATLAB Application to Transient Analysis 124 5.12 MATLAB Application to 3-D Steady State Analysis 148

Problems 153

CHAPTER 6. ISOPARAMETRIC ELEMENTS

6.1 One-Dimensional Elements 157 6.2 Quadrilateral Elements 160 6.3 Triangulär Elements 168 6.4 Gauss Quadrature 170 6.5 MATLAB Application to Gauss Quadrature 176 6.6 MATLAB Application to Laplace Equation 183

Problems 193

CHAPTER 7. TRUSS STRUCTURES

7.1 One-Dimensional Truss 197 7.2 Plane Truss 199 7.3 Space Truss 203 7.4 MATLAB Application to Static Analysis 204 7.5 MATLAB Application to Eigenvalue Analysis 215 7.6 MATLAB Application to Transient Analysis 223

Problems 231

CHAPTER 8. BEAM AND FRAME STRUCTURES

8.1 Euler-Bernoulli Beam 235 8.2 Timoshenko Beam 242 8.3 Beam Elements With Only Displacement Degrees of Freedom 245 8.4 Mixed Beam Element 249 8.5 Hybrid Beam Element 253

8.6 Composite Beams 256 8.7 Two-Dimensional Frame Element 259 8.8 Three-Dimensional Frame Element 264 8.9 MATLAB Application to Static Analysis 265

8.10 MATLAB Application to Eigenvalue Analysis 280 8.11 MATLAB Application to Transient Analysis 284 8.12 MATLAB Application to Modal Analysis of Undamped System 288 8.13 MATLAB Application to Modal Analysis of Damped System 296 8.14 MATLAB Application to Frequency Response Analysis 299

Problems 304

CHAPTER 9. ELASTICITY PROBLEM 307

9.1 Plane Stress and Plane Strain 307 9.2 Force Vector 311 9.3 Energy Method 313 9.4 Three-Dimensional Solid 315 9.5 Axisymmetric Solid 318 9.6 Dynamic Analysis 320 9.7 MATLAB Application to 2-D Stress Analysis 321 9.8 MATLAB Application to Axisymmetric Analysis 334 9.9 MATLAB Application to 3-D Stress Analysis 345

Problems 356

C H A P T E R 10. PLATE STRUCTURES 361

10.1 Classical Plate Theory 361 10.2 Classical Plate Bending Element 364 10.3 Shear Deformable Plate Element 366 10.4 Plate Element With Displacement Degrees of Freedom 368 10.5 Mixed Plate Element 373 10.6 Hybrid Plate Element 378 10.7 MATLAB Application 382

Problems 391

C H A P T E R 11. CONTROL OF FLEXIBLE STRUCTURES 393

11.1 Introduction 393 11.2 Stability Theory 394 11.3 Stability of Multiple Degrees of Freedom Systems 398 11.4 Analysis of a Second Order System 401 11.5 State Space Form Description 406 11.6 Transfer Function Analysis 417 11.7 Control Law Design for State Space Systems 427 11.8 Linear Quadratic Regulator 438 11.9 Modal Control for Second Order Systems 448

11.10 Dynamic Observer 451

11.11 Compensator Design 456 11.12 Output Feedback Design by Using Collocated Sensor/Actuator 460

Problems 475

REFERENCES 479

A P P E N D I X A: FEA MATLAB FUNCTION FILES 483

INDEX 509