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Finite Element Analysis Book · May 2014

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Official Lecture Notes for INME 6037

Finite Element Analysis Vijay K. Goyal, Ph.D. Associate Professor, Department of Mechanical Engineering, University of Puerto Rico at Mayag¨ uez, Mayag¨ uez, Puerto Rico

Vinay K. Goyal, Ph.D. Engineering Specialist, Structural Mechanics Subdivision, The Aerospace Corporation, Los Angeles, California Visiting Lecturer, Department of Mechanical and Aerospace Engineering, University of California at Los Angeles, Los Angeles, California

This material is only for those students enrolled at UPRM and UCLA. All others must request permission from the authors ([email protected]).

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c 2014, Vijay K. Goyal and Vinay K. Goyal Copyright

c Copyright January 2014 by Vijay K. Goyal and Vinay K. Goyal

The authors have used their best efforts in preparing this textbook. These efforts include the development, research, and testing of the theories and formulas to determine their effectiveness. The authors shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these formulas.

Type Set in Times Roman 12 pt using LATEX. The book is under review and consideration for publication by Pearson Education.

Dedication

To the Almighty God: Father, Son Yeshua, and the Holy Spirit; Math, Science and Engineering show the Wonders of God, who revealed himself through Yeshua; If God cannot be found then all the knowledge is meaningless. “Through Him all things were made; without Him nothing was made that has been made.” – BIBLE: John 1:3 To my wife Maricelis, my son Jeremiah, and my daughter Naarah; They are God’s blessing to my life.

Preface

Over the past three decades, the Finite Element Analysis (FEA) has become easier to use and implement, enabling engineering designers to carry out more effective simulations. The steady growth of analysis software coincides with the availability and affordability of high performance computing architectures, making FEA applications a possible reality for most engineers. Although there are some excellent books for engineering analysis using finite element techniques to solve engineering problems, in this manuscript we intend to guide the reader through the finite element method through the very basic concepts to the extent of a first-year graduate student. Intended Audience This book is intended to provide theoretical and practical foundation of the finite element method (FEM). The target audience are first-year graduate students who have had little, or no, exposure to finite element analysis. Practicing engineers will also benefit from the approach of this manuscript as they will be able to learn the theoretical aspects of the finite element methods, basically, by their own. Thus, we can assure that this book will fill up a void in the personal library of many engineers who are trying to, or planning, to use finite element analysis in their next design. The recommended background to successfully read this books is: solid mechanics, calculus, continuum mechanics or theory of elasticity and basic programming knowledge is required.

Motivation When writing this textbook, we have kept the reader in mind at all times. After years of using this manuscript, engineering graduates (from the University of Puerto Rico at Mayag¨ uez) have found the manuscript very useful in their respective jobs. In teaching and applying the subject for years, we have come to the conclusion that students and engineers too often take a ”black-box” approach when using finite element analysis (FEA) software. They usually lack fundamental knowledge of what the finite element analysis software is doing. Hence, the main goal of the book is to provide a fundamental engineering treatment associated with FEM at a level that is reasonably accessible to those being exposed to the topic for the first time. The book also attempts to provide an intuitive as well as mathematical

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insight into the subject of finite element analysis by using modern mathematical tools. After the reader has completed this book, he/she will be able to: 1. Explain what is the finite element method. 2. Use theory of elasticity to provide computational formulation of a structural problem. 3. Develop a computer code for springs, axial bars, beams, plates. 4. Conduct an engineering design project applying concepts of finite element analysis. Mathematical Level We kept the mathematics to a basic knowledge in linear algebra, differential equations, and programming. For those who have not been exposed to linear algebra, we have included an appendix that will enable the reader to self-study, or review, this topic. The authors assume that the readers have little experience with programming languages and numerical methods.

Chapter Organization and Topical Coverage The format of this book is student-friendly since each chapter begins with instructional objectives and ends with a chapter summary highlighting the most important aspects of the chapter with an outline of ongoing research within the topics presented in the chapter. It includes a variety of examples, specifically worked with a pedagogical approach, using a step-by-step procedure which is easy to apply to a wide range of engineering problems. At the end of each chapter one can find a variety of problems that have been carefully worked-out in an accompanying solution manual to the textbook, available online to the instructors. Emphasis was placed on applications to keep the reader interested in the subject. The contents of this book are intended for a single term course. In Chapter 1 we provide a background of the finite element method, including how continuum mechanics fits within the FEA. We have included several complementary appendices for those who are not familiar with linear elasticity. Chapter 2 is an introduction of the fundamental philosophy used in FEA. It explains the reader the advantage of converting a differential equation into a strong and weak forms, as well as providing its functional, whenever it exists. Chapter 3 focuses entirely on spring analysis and shows the reader how each stage of the finite element analysis works. We begin with simple spring analysis to help the

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student grasp concepts regarding stiffness matrices, force and displacement vectors, their assembly, and the system solution using MATLABr programming. Appendix B provides a review of linear algebra using MATLABr , which is essential for this chapter. The goal is to explain how the preprocessor, processor and postprocessor work and how these are implemented into a computer program. Chapter 4 extends the fundamental concepts in finite element analysis presented in Chapter 3 to solve second order ordinary differential equations. The chapter uses the energy and work principles combined with a vectorial notation to derive the elemental equations. Chapter 5 develops the finite element formulation for the one-dimensional fourth-order ordinary differential equations. As a special case we will apply these concepts to beam theory. The primary difference between the finite elements we will derive in this chapter and the finite elements of the previous chapter is the order of the governing differential equation. Chapter 6 discusses how to analyze axial bar and beam structures when they are rotated about the z-axis and are part of a whole system. The procedure is identical to that covered in previous chapters with the major difference being that we need to use a transformation matrix for taking into the rotation. Chapter 7 discusses how to solve second order partial differential equations defined over a two dimensional domain. Chapter 8 is basically an extension of the one- and two-dimensional elements. Axisymmetric elements are used to model structural components that are rotationally symmetric about an axis. In short, this unique book will help the reader, whether a student or a practicing engineer, to independently learn the topics through carefully worked out examples and apply them to real engineering problems. Any comments and suggestions can be sent to [email protected]. Vijay K. Goyal 1/2014

Acknowledgments

There are many people who have made this work possible. First and foremost, I am mostly thankful to Yeshua, for giving me the opportunity to live in this time. All my success I give to him for He has been my strength and inspiration at all times. Secondly, I express my special appreciation to my wife Maricelis, my son Jeremiah, and my daughter Naarah for their and inspiration behind this effort. I could not have completed this task without their prayers, love, understanding, encouragement, and . Thirdly, I would like to thank all the graduate students who collaborated to complete this book: Juan Rein´es and Angel Quintero. In addition, many thanks to the invaluable inputs from the undergraduate students who used the manuscript form of this book during the 2004–2014 period at the University of Puerto Rico at Mayag¨ uez. Lastly, I cannot leave behind all the people who have given their suggestions to this work, such as Dr. Paul A. Sundaram, whom I consider my mentor. Special thanks to all the friends who encouraged and helped me achieve this goal. God bless and thank you all,

Vijay K. Goyal

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Table of Contents

List of Contents

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List of Figures

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List of Tables

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Chapter 1. Introduction to Finite Element Method 1.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 What is FEM? . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Background of the FEA . . . . . . . . . . . . . . . . . . . 1.1.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of Continuum Mechanics . . . . . . . . . . . . . . . . . 1.2.1 Computational Mechanics . . . . . . . . . . . . . . . . . . 1.2.2 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Typical Elasticity Field for a Continuous Deformable Body 1.2.4 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . 1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. Approximations to Differential Equations 2.1 Differential Equations . . . . . . . . . . . . . . . . . . . 2.1.1 What Are Differential Equations? . . . . . . . . 2.1.2 Exact Solution . . . . . . . . . . . . . . . . . . 2.1.3 Approximate Solution . . . . . . . . . . . . . . 2.1.4 Integral Approximation via Strong Formulation 2.2 Strong Form: Methods of Weighted Residual . . . . . . 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . 2.2.2 Trial Functions . . . . . . . . . . . . . . . . . . 2.2.3 Procedure . . . . . . . . . . . . . . . . . . . . . 2.2.4 Four Categories . . . . . . . . . . . . . . . . . . 2.2.5 Algebraic Representation . . . . . . . . . . . . . 2.3 Weak Formulation and Quadratic Functionals . . . . . 2.3.1 Definition . . . . . . . . . . . . . . . . . . . . .

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