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# Note for Finite Element Method - FEM By Abhishek Apoorv

• Finite Element Methods - FEM
• Note
• Mechanical Engineering
• 11 Topics
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Abhishek Apoorv
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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: Web: Yijun.Liu@uc.edu 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 i

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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 ii

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Lecture Notes: Introduction to the Finite Element Method Table of Contents Copyright Notice ....................................................................................................... ii 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 iii

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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 iv