Lecture Notes: Introduction to the Finite Element Method
Lecture Notes: Introduction to the
Finite Element Method
Yijun Liu
CAE Research Laboratory
Mechanical Engineering Department
University of Cincinnati
Cincinnati, OH 45221-0072, U.S.A.
E-mail: Yijun.Liu@uc.edu
Web: http://urbana.mie.uc.edu/yliu
This document is downloaded from the course website: http://urbana.mie.uc.edu/yliu/FEM-525/FEM-525.htm (Last Updated: May 21, 2003)
© 1997-2003 by Yijun Liu, University of Cincinnati.
© 1997-2003 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to the Finite Element Method
Copyright Notice
© 1997-2003 by Yijun Liu, University of Cincinnati.
All rights reserved. Permissions are granted for personal and educational uses only. Any other uses of these lecture notes (such as for classroom lectures outside the University of Cincinnati, trainings elsewhere, and those of a commercial nature) are not permitted, unless such uses have been granted in writing by the author.
© 1997-2003 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to the Finite Element Method
Table of Contents |
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Copyright Notice ....................................................................................................... |
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Table of Contents ..................................................................................................... |
iii |
Preface ....................................................................................................................... |
v |
Chapter 1. Introduction ............................................................................................ |
1 |
I. Basic Concepts ................................................................................................... |
1 |
II. Review of Matrix Algebra ................................................................................. |
7 |
III. Spring Element .............................................................................................. |
14 |
Chapter 2. Bar and Beam Elements....................................................................... |
25 |
I. Linear Static Analysis ....................................................................................... |
25 |
II. Bar Element ..................................................................................................... |
26 |
III. Beam Element ................................................................................................. |
53 |
Chapter 3. Two-Dimensional Problems ................................................................ |
75 |
I. Review of the Basic Theory ............................................................................... |
75 |
II. Finite Elements for 2-D Problems.................................................................. |
82 |
Chapter 4. Finite Element Modeling and Solution Techniques........................... |
105 |
I. Symmetry ........................................................................................................ |
105 |
II. Substructures (Superelements) ..................................................................... |
107 |
III. Equation Solving......................................................................................... |
109 |
IV. Nature of Finite Element Solutions ............................................................. |
112 |
V. Numerical Error............................................................................................ |
114 |
© 1997-2003 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to the Finite Element Method
VI. Convergence of FE Solutions ...................................................................... |
116 |
VII. Adaptivity (h-, p-, and hp-Methods) ........................................................... |
117 |
Chapter 5. Plate and Shell Elements ................................................................... |
119 |
I. Plate Theory ................................................................................................... |
119 |
II. Plate Elements ............................................................................................. |
129 |
III. Shells and Shell Elements........................................................................... |
133 |
Chapter 6. Solid Elements for 3-D Problems ..................................................... |
138 |
I. 3-D Elasticity Theory ..................................................................................... |
138 |
II. Finite Element Formulation.......................................................................... |
142 |
III. Typical 3-D Solid Elements ......................................................................... |
144 |
Chapter 7. Structural Vibration and Dynamics................................................... |
157 |
I. Basic Equations.............................................................................................. |
157 |
II. Free Vibration............................................................................................... |
163 |
III. Damping ...................................................................................................... |
167 |
IV. Modal Equations.......................................................................................... |
168 |
V. Frequency Response Analysis....................................................................... |
171 |
VI. Transient Response Analysis ....................................................................... |
172 |
Chapter 8. Thermal Analysis ............................................................................... |
177 |
I. Temperature Field.......................................................................................... |
177 |
II. Thermal Stress Analysis................................................................................ |
180 |
Further Reading.................................................................................................... |
183 |
© 1997-2003 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to the Finite Element Method
Preface
These online lecture notes (in the form of an e-book) are intended to serve as an introduction to the finite element method (FEM) for undergraduate students or other readers who have no previous experience with this computational method. The notes cover the basic concepts in the FEM using the simplest mechanics problems as examples, and lead to the discussions and applications of the 1-D bar and beam, 2-D plane and 3-D solid elements in the analyses of structural stresses, vibrations and dynamics. The proper usage of the FEM, as a popular numerical tool in engineering, is emphasized throughout the notes.
This online document is based on the lecture notes developed by the author since 1997 for the undergraduate course on the FEM in the mechanical engineering department at the University of Cincinnati. Since this is an e-book, the author suggests that the readers keep it that way and view it either online or offline on his/her computer. The contents and styles of these notes will definitely change from time to time, and therefore hard copies may become obsolete immediately after they are printed. Readers are welcome to contact the author for any suggestions on improving this e-book and to report any mistakes in the presentations of the subjects or typographical errors. The ultimate goal of this e- book on the FEM is to make it readily available for students, researchers and engineers, worldwide, to help them learn subjects in the FEM and eventually solve their own design and analysis problems using the FEM.
The author thanks his former undergraduate and graduate students for their suggestions on the earlier versions of these lecture notes and for their contributions to many of the examples used in the current version of the notes.
Yijun Liu
Cincinnati, Ohio, USA
December 2002
© 1997-2003 Yijun Liu, University of Cincinnati |
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