MATLAB Guide to Finite Elements (eBook)

Preface to the Second Edition 6
Preface to the First Edition 8
Table of Contents 10
1 Introduction 12
1.1 Steps of the Finite Element Method 12
1.2 MATLAB Functions for Finite Element Analysis 13
1.3 MATLAB Tutorial 15
2 The Spring Element 22
2.1 Basic Equations 22
2.2 MATLAB Functions Used 23
3 The Linear Bar Element 38
3.1 Basic Equations 38
3.2 MATLAB Functions Used 39
4 The Quadratic Bar Element 55
4.1 Basic Equations 55
4.2 MATLAB Functions Used 56
5 The Plane Truss Element 71
5.1 Basic Equations 71
5.2 MATLAB Functions Used 72
6 The Space Truss Element 100
6.1 Basic Equations 100
6.2 MATLAB Functions Used 101
7 The Beam Element 117
7.1 Basic Equations 117
7.2 MATLAB Functions Used 118
8 The Plane Frame Element 144
8.1 Basic Equations 144
8.2 MATLAB Functions Used 146
9 The Grid Element 182
9.1 Basic Equations 182
9.2 MATLAB Functions Used 183
10 The Space Frame Element 199
10.1 Basic Equations 199
10.2 MATLAB Functions Used 201
11 The Linear Triangular Element 223
11.1 Basic Equations 223
11.2 MATLAB Functions Used 225
12 The Quadratic Triangular Element 254
12.1 Basic Equations 254
12.2 MATLAB Functions Used 256
13 The Bilinear Quadrilateral Element 279
13.1 Basic Equations 279
13.2 MATLAB Functions Used 282
14 The Quadratic Quadrilateral Element 315
14.1 Basic Equations 315
14.2 MATLAB Functions Used 318
15 The Linear Tetrahedral (Solid) Element 341
15.1 Basic Equations 341
15.2 MATLAB Functions Used 344
16 The Linear Brick (Solid) Element 370
16.1 Basic Equations 370
16.2 MATLAB Functions Used 374
17 Other Elements 400
17.1 Applications of Finite Elements in Other Areas 400
17.2 Basic Equations of the Fluid Flow 1D Element 401
17.3 MATLAB Functions Used in the Fluid Flow 1D Element 403
References 405
Answers to Problems 407
Contents of the Accompanying CD-ROM 427
Index 428

"1 Introduction (p. 1-5)

This short introductory chapter is divided into two parts. In the ?rst part there is a summary of the steps of the ?nite element method. The second part includes a short tutorial on MATLAB.

1.1 Steps of the Finite Element Method

There are many excellent textbooks available on ?nite element analysis like those in [1–18]. Therefore this book will not present any theoretical formulations or derivations of ?nite element equations. Only the main equations are summarized for each chapter followed by examples. In addition only problems from linear elastic structural mechanics are used throughout the book. The ?nite element method is a numerical procedure for solving engineering problems. Linear elastic behavior is assumed throughout this book. The problems in this book are taken from structural engineering but the method can be applied to other ?elds of engineering as well. In this book six steps are used to solve each problem using ?nite elements. The six steps of ?nite element analysis are summarized as follows:

1. Discretizing the domain – this step involves subdividing the domain into elements and nodes. For discrete systems like trusses and frames the system is already discretized and this step is unnecessary. In this case the answers obtained are exact. However, for continuous systems like plates and shells this step becomes very important and the answers obtained are only approximate. In this case, the accuracy of the solution depends on the discretization used. In this book this step will be performed manually (for continuous systems).

2. Writing the element stiffness matrices – the element stiffness equations need to be written for each element in the domain. In this book this step will be performed using MATLAB.

3. Assembling the global stiffness matrix – this will be done using the direct stiffness approach. In this book this step will be performed using MATLAB.

4. Applying the boundary conditions – like supports and applied loads and displacements. In this book this step will be performed manually.

5. Solving the equations – this will be done by partitioning the global stiffness matrix and then solving the resulting equations using Gaussian elimination. In this book the partitioning process will be performed manually while the solution part will be performed using MATLAB with Gaussian elimination.

6. Post-processing – to obtain additional information like the reactions and element forces and stresses. In this book this step will be performed using MATLAB. It is seen from the above steps that the solution process involves using a combination of MATLAB and some limited manual operations. The manual operations employed are very simple dealing only with discretization (step 1), applying boundary conditions (step 4) and partitioning the global stiffness matrix (part of step 5)."