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This comprehensive guide explores the versatility and affordability of Finite Element Analysis (FEA) as a powerful tool for solving engineering problems. The book provides a practical introduction to FEA, covering applications in mechanical, civil, electrical engineering, and physics. It presents a balanced blend of theory and applications, catering to both beginners and those seeking to enhance their FEA skills.
The journey begins with mathematical preliminaries and an introduction to the Finite Element Method. It progresses through various applications such as axially loaded members, trusses, beams, stress analysis, thermal analysis, fluid flow analysis, dynamic analysis, and engineering electromagnetics analysis. Each chapter includes comparisons of analytical methods, FEA hand calculations, and software-based solutions, helping readers understand the strengths and limitations of each approach.
Understanding these concepts is crucial for effective problem-solving in engineering. This book transitions readers from fundamental principles to advanced engineering applications, blending theoretical knowledge with practical skills. Companion files with executable models and animations enhance the learning experience, making this guide an essential resource for mastering FEA techniques.
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FINITE ELEMENTANALYSIS
Second Edition
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FINITE ELEMENTANALYSIS
A Primer
Second Edition
SARHAN M. MUSA(Prairie View A & M University)
MERCURY LEARNING AND INFORMATIONBoston, Massachusetts
Copyright ©2024 by MERCURY LEARNING AND INFORMATION. An Imprint of DeGruyter Inc. All rights reserved.Reprinted and revised with permission.
Original title and copyright: A Primer on Finite Element Analysis.Copyright © 2011 by Laxmi Publications Pvt. Ltd. All rights reserved. ISBN : 978-93-81159-10-1
This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means, media, electronic display or mechanical display, including, but not limited to, photocopy, recording, Internet postings, or scanning, without prior permission in writing from the publisher.
Publisher: David PallaiMERCURY LEARNING AND INFORMATION121 High Street, 3rd FloorBoston, MA [email protected]
S. M. Musa. FINITE ELEMENT ANALYSIS: A Primer. Second Edition.ISBN: 978-1-68392-415-9
The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others.
Images of ANSYS menus, dialog boxes and plots are copyright of ANSYS Incorporation, United States of America and have been used with prior consent. Commercial software name, company name, other product trademarks, registered trademark logos are the properties of the ANSYS Incorporation, U.S.A.
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Dedicated to my late father, Mahmoud, my late mother, Fatmeh, and my wife, Lama.
CONTENTS
Preface
CHAPTER 1: MATHEMATICAL PRELIMINARIES
1.1 Introduction
1.2 Matrix Definition
1.3 Types of Matrices
1.4 Addition or Subtraction of Matrices
1.5 Multiplication of a Matrix by Scalar
1.6 Multiplication of a Matrix by Another Matrix
1.7 Rules of Matrix Multiplications
1.8 Transpose of a Matrix Multiplication
1.9 Trace of a Matrix
1.10 Differentiation of a Matrix
1.11 Integration of a Matrix
1.12 Equality of Matrices
1.13 Determinant of a Matrix
1.14 Direct Methods for Linear Systems
1.15 Gaussian Elimination Method
1.16 Cramer’s Rule
1.17 Inverse of a Matrix
1.18 Vector Analysis
1.19 Eigenvalues and Eigenvectors
1.20 Using MATLAB
Exercises
References
CHAPTER 2: INTRODUCTION TO THE FINITE ELEMENT METHOD
2.1 Introduction
2.2 Methods of Solving Engineering Problems
2.2.1 Experimental Method
2.2.2 Analytical Method
2.2.3 Numerical Method
2.3 Procedure of Finite Element Analysis (Related to Structural Problems)
2.4 Methods of Prescribing Boundary Conditions
2.4.1 Elimination Method
2.4.2 Penalty Method
2.4.3 Multipoint Constrains Method
2.5 Practical Applications of Finite Element Analysis
2.6 Finite Element Analysis Software Package
2.7 Finite Element Analysis for Structure
2.8 Types of Elements
2.9 Direct Method for Linear Spring
Exercises
References
CHAPTER 3: FINITE ELEMENT ANALYSIS OF AXIALLY LOADED MEMBERS
3.1 Introduction
3.1.1 Two-Node Bar Element
3.1.2 Three-Node Bar Element
3.2 Bars of Constant Cross-Section Area
3.3 Bars of Varying Cross-Section Area
3.4 Stepped Bar
Exercises
References
CHAPTER 4: FINITE ELEMENT ANALYSIS TRUSSES
4.1 Introduction
4.2 Truss
Exercises
References
CHAPTER 5: FINITE ELEMENT ANALYSIS OF BEAMS
5.1 Introduction
5.2 Simply Supported Beams
5.3 Cantilever Beams
Exercises
References
CHAPTER 6: STRESS ANALYSIS OF A RECTANGULAR PLATE WITH A CIRCULAR HOLE
6.1 Introduction
6.2 A Rectangular Plate with a Circular Hole
Exercises
References
CHAPTER 7: THERMAL ANALYSIS
7.1 Introduction
7.2 Procedure of Finite Element Analysis (Related to Thermal Problems)
7.3 One-Dimensional Heat Conduction
7.4 Two-Dimensional Problem with Conduction and with Convection Boundary Conditions
Exercises
References
CHAPTER 8: FLUID FLOW ANALYSIS
8.1 Introduction
8.2 Procedure of Finite Element Analysis (Related to Fluid Flow Problems)
8.3 Potential Flow Over a Cylinder
8.4 Potential Flow Around an Airfoil
Exercises
References
CHAPTER 9: DYNAMIC ANALYSIS
9.1 Introduction
9.2 Procedure of Finite Element Analysis (Related to Dynamic Problems)
9.3 Fixed-Fixed Beam for Natural Frequency Determination
9.4 Transverse Vibrations of a Cantilever Beam
9.5 Fixed-Fixed Beam Subjected to Forcing Function
9.6 Axial Vibrations of a Bar
9.7 Bar Subjected to Forcing Function
Exercises
References
CHAPTER 10: ENGINEERING ELECTROMAGNETICS ANALYSIS
10.1 Introduction to Electromagnetics
10.2 Maxwell’s Equations and Continuity Equation
10.2.1 Maxwell’s Equations and Continuity Equation in Differential Form
10.2.2 Maxwell’s Equations and Continuity Equation in Integral Form
10.2.3 Divergence and Stokes Theorems
10.2.4 Maxwell’s Equations and Continuity Equation in Quasi-Statics Case
10.2.5 Maxwell’s Equations and Continuity Equation in Statics Case
10.2.6 Maxwell’s Equations and Continuity Equation in Source-Free Regions of Space Case
10.2.7 Maxwell’s Equations and Continuity Equation in Time-Harmonic Fields Case
10.3 Lorentz Force Law and Continuity Equation
10.4 Constitutive Relations
10.5 Potential Equations
10.6 Boundary Conditions
10.7 Laws for Static Fields in Unbounded Regions
10.7.1 Coulomb’s Law and Field Intensity
10.7.2 Bio-Savart’s Law and Field Intensity
10.8 Electromagnetic Energy and Power Flow
10.9 Loss in Medium
10.10 Skin Depth
10.11 Poisson’s and Laplace’s Equations
10.12 Wave Equations
10.13 Electromagnetic Analysis
10.13.1 One-Dimensional Elements
10.13.1.1 The Approach to FEM Standard Steps Procedure
10.13.1.2 Application to Poisson’s Equation in One-Dimension
10.13.1.3 Natural Coordinates in One Dimension
10.13.2 Two-Dimensional Elements
10.13.2.1 Applications of FEM to Electrostatic Problems
10.14 Automatic Mesh Generation
10.14.1 Rectangular Domains
10.14.2 Arbitrary Domains
10.15 Higher-Order Elements
10.15.1 Pascal Triangle
10.15.2 Local Coordinates
10.15.3 Shape Functions
10.15.4 Fundamental Matrices
10.16 Three-Dimensional Element
10.17 Finite Element Methods for External Problems
10.17.1 Infinite Element Method
10.17.2 Boundary Element Method
10.17.3 Absorbing Boundary Conditions
10.18 Modeling and Simulation of Shielded Microstrip Lines with COMSOL Multiphysics
10.18.1 Rectangular Cross-Section Transmission Line
10.18.2 Square Cross-Section Transmission Line
10.18.3 Rectangular Line with Diamondwise Structure
10.18.4 A Single-Strip Shielded Transmission Line
10.19 Multistrip Transmission Lines
10.19.1 Double-Strip Shielded Transmission Line
10.19.2 Three-Strip Line
10.19.3 Six-Strip Line
10.19.4 Eight-Strip Line
10.20 Solenoid Actuator Analysis with ANSYS
Exercises
References
APPENDIX A: ANSYS
APPENDIX B: MATLAB
APPENDIX C: COMSOL MULTIPHYSICS
APPENDIX D: 4-COLOR FIGURES FROM THE TEXT(On the companion files)
INDEX
PREFACE
Today, the finite element method (FEM) has become a common and a very powerful computational tool for solving engineering problems in industries for the obvious reasons of its versatility and affordability. To expose an undergraduate student in engineering to this powerful method, most universities have included this subject in the undergraduate curriculum. This book contains materials applied to mechanical engineering, civil engineering, electrical engineering, and physics. This book is written primarily to help the students and educators as a simple introduction to the practice of FEM analysis in engineering and physics. This book contains many 1D and 2D problems solved by the analytical method, by FEM using hand calculations, and by using ANSYS academic teaching software, COMSOL, and MATLAB. Results of all the methods have been compared. This book compromises 10 chapters and 3 appendices.
Chapter 1 contains mathematical preliminaries needed for understanding the chapters of the book. Chapter 2 provides a brief introduction to FEA, a theoretical background, and its applications. Chapter 3 contains the linear static analysis of bars of constant cross-section, tapered cross-section, and stepped bar. In each section, a different variety of exercise problems are given. Chapter 4 contains the linear static analysis of trusses. Trusses problems are also selected in such a way that each problem has different boundary conditions to apply. Chapter 5 provides the linear static analysis of simply supported and cantilever beams. In Chapters 3 to 5, all the problems are considered as one dimensional in nature. Indeed, stress analysis of a rectangular plate with a circular hole is covered in Chapter 6. In this chapter, emphasis is given on the concept of exploiting symmetric geometry and symmetric loading conditions. Also, stress and deformation plots are given. Chapter 7 introduces the thermal analysis of cylinders and plates. Here both one dimensional and two-dimensional problems are considered. Chapter 8 contains the problems of potential flow distribution over a cylinder and over an airfoil. Chapter 9 provides the dynamic analysis (modal and transient analysis) of bars and beams. Chapter 10 provides the engineering electromagnetics analysis. The chapter gives an overview of electromagnetics theory and provides the finite element method analysis toward electromagnetics; some models are demonstrated using COMSOL multiphysics and ANSYS.
Appendices (available on the companion files)
Appendix A contains the introduction to Classic ANSYS and the ANSYS Workbench. Appendix B contains an overview of a computation in MATLAB. Appendix C contains an overview of COMSOL Multiphysics. Appendix D contains the color figures from the book.
Acknowledgments
It is my pleasure to acknowledge the outstanding help and support of the team at Mercury Learning and Information in preparing this book, especially from David Pallai and Jennifer Blaney.
Sarhan M. MusaSeptember 2023
CHAPTER 1
MATHEMATICAL PRELIMINARIES
1.1 INTRODUCTION
This chapter introduces matrix and vector algebra which is essential in the formulation and solution of finite element problems. Finite element analysis procedures are most commonly described using matrix and vector notations. These procedures eventually lead to the solutions of a large set of simultaneous equations. This chapter will be a good help in understanding the remaining chapters of the book.
1.2 MATRIX DEFINITION
A matrix is an array of numbers or mathematical terms arranged in rows (horizontal lines) and columns (vertical lines). The numbers, or mathematical terms, in the matrix, are called the elements of the matrix. We denote the matrix through this book, by a boldface-letter, a letter in brackets [], or a letter in braces {}. We sometimes use {} for a column matrix. Otherwise, we define the symbols of the matrices.
EXAMPLE 1.1
The following are matrices.
,,,,
The size (dimension or order) of the matrices varies and is described by the number of rows (m) and the number of columns (n). Therefore, we write the size of a matrix as ( by ). The sizes of the matrices in Example 1.1 are , , , , and , respectively.
We use to denote the element that occurs in row and column of matrix A. In general, matrix A can be written
(1.1)
EXAMPLE 1.2
Location of an element in a matrix.
Let
Find (a) size of the matrix
(b) location of elements , , , and
Solution:
(a) Size of the matrix is
(b) is element a at row 1 and column 1
is element a at row 1 and column 2
is element a at row 3 and column 2
is element a at row 3 and column 3
Note that, two matrices are equal if they have the same size and their corresponding elements in the two matrices are equal. For example, let, , then since and are not the same size. Also, since the corresponding elements are not all equal.
1.3 TYPES OF MATRICES
The types of matrices are based on the number of rows (m) and the number of columns (n) in addition to the nature of elements and the way the elements are arranged in the matrix.
(a)Rectangular matrix is a matrix of different number of rows and columns, that is, . For example, the matrix
, is rectangular matrix.
(b)Square matrix is a matrix of equal number of rows and columns, that is, . For example, the matrix
, is square matrix.
(c)Row matrix is a matrix that has one row and has more than one column, that is, . For example, the matrix
, is row matrix.
(d)Column matrix is a matrix that has one column and has more than one row, that is, . For example, the matrix
, is column matrix.
(e)Scalar matrix is a matrix that has the number of columns and the number of rows equal to 1, that is, . For example, the matrix
, is a scalar matrix; we can write it as 7 without bracket.
(f)Null matrix is a matrix whose elements are all zero. For example, the matrix
, is a null matrix.
(g)Diagonal matrix is a square matrix that has zero elements everywhere except on its main diagonal. That is, for diagonal matrix , when and not all are zero for when . For example, the matrix
Main diagonal elements have equal row and column subscripts—the main diagonal runs from the upper-left corner to the lower-right corner. The main diagonal of the matrix here is and .
(h)Identity(unit)matrix or , is a diagonal matrix whose main diagonal elements are equal to unity (1’s) for any square matrix. That is, if the elements of an identity matrix are denoted as , then
. (1.2)
For example, the matrix
, is an identity matrix.
(i)Banded matrix is a square matrix that has a band of nonzero elements parallel to its main diagonal. For example, the matrix
, is a banded matrix.
(j)Symmetric matrix is a square matrix whose elements satisfy the condition for . For example, the matrix
, is a symmetric matrix.
(k)Anti-symmetric (Skew-symmetric) matrix is a square matrix whose elements for , and . For example, the matrix
, is an anti-symmetric matrix.
(l)Triangular matrix is a square matrix whose elements on one side of the main diagonal are all zero. There are two types of triangular matrices; first, an upper triangular matrix whose elements below the main diagonal are zero, that is, ; second, a lower triangular matrix whose elements above the main diagonal are all zero, that is . For example, the matrix
, is an upper triangular matrix.
While the matrix
, is a lower triangular matrix.
(m)Partitioned matrix (Super-matrix) is a matrix that can be divided into smaller arrays (submatrices) by horizontal and vertical lines; that is, the elements of the partitioned matrix are matrices. For example, the matrix is partitioned matrix with four smaller matrices, where
, , , and . For example, the matrix
, is a partitioned matrix, where , , , and .
1.4 ADDITION OR SUBTRACTION OF MATRICES
1.5 MULTIPLICATION OF A MATRIX BY SCALAR
A matrix is multiplied by a scalar, , by multiplying each element of the matrix by this scalar. That is, the multiplication of a matrix by a scalar is defined as
. (1.5)
The scalar multiplication is commutative.
For example,
Let , then .
1.6 MULTIPLICATION OF A MATRIX BY ANOTHER MATRIX
The product of two matrices is , if and only if, the number of columns in is equal to the number of rows in B. The product of matrix of size and matrix of size , the result in matrix has size .
(1.6)
and , (1.7)
where the ()th component of matrix is obtained by taking the dot product
.
That is, to find the element in row and column of , you need to single out row from and column from , then multiply the corresponding elements from the row and column together and add up the resulting products.
For example,
let and , then
Size of .
1.7 RULES OF MATRIX MULTIPLICATIONS
1.8 TRANSPOSE OF A MATRIX MULTIPLICATION
The transpose of a matrix is denoted as . It is obtained by interchanging the rows and columns in matrix A. Thus, if a matrix is of order , then will be of order .
For example,
let , then .
Note that it is valid that,, , , and . Also note, if , then is a symmetric matrix.
EXAMPLE 1.4
Consider that matrix and .
Show that .
Solution:
,
Therefore, .
1.9 TRACE OF A MATRIX
1.10 DIFFERENTIATION OF A MATRIX
Differentiation of a matrix is the differentiation of every element of the matrix separately. For example, if the elements of the matrix A are a function of , then
. (1.16)
EXAMPLE 1.6
Consider the matrix , find the derivative .
Solution:
1.11 INTEGRATION OF A MATRIX
Integration of a matrix is the integration of every element of the matrix separately. For example, if the elements of the matrix A are a function of , then
. (1.17)
EXAMPLE 1.7
Consider the matrix , find the derivative .
Solution:
1.12 EQUALITY OF MATRICES
1.13 DETERMINANT OF A MATRIX
The determinant of a square matrix A is a scalar number denoted by or det . It is the sum of the products , where are the elements along any one row or column and are the deleted elements of ith row and jth column from the matrix .
For example, the value of the determinant of matrix is and can be obtained by expanding along the first row as:
(1.18)
where the minor is a determinant of the matrix formed by removing the th row and th column.
Also, the value can be obtained by expanding along the first column as:
. (1.19)
Now, the value of a second-order determinant of () matrix is calculated by
. (1.20)
The value of a third-order determinate of () matrix is calculated by
=
=
= .(1.21)
EXAMPLE 1.9
Evaluate the following determinants:
a.
b.
Solution:
a.
b.
An alternative method of obtaining the determinant of a () matrix is by using the sign rule of each term that is determined by the first row in the diagram as follows:
, or by repeating the first two rows and multiplying the terms diagonally as follows:
.
1.14 DIRECT METHODS FOR LINEAR SYSTEMS
Many engineering problems in finite element analysis will result in a set of simultaneous equations represented by .
For a set of simultaneous equations having the form
(1.22)
where there are unknown to be determined. These equations can be written in matrix form as
.
This matrix equation can be written in a compact form as
, (1.23)
where is a square matrix with order , while are column matrices defined as
There are several methods for solving a set of simultaneous equations such as by substitution, Gaussian elimination, Cramer’s rule, matrix inversion, and numerical analysis.
1.15 GAUSSIAN ELIMINATION METHOD
In the argument matrix of a system, the variables of each equation must be on the left side of the equal sign (vertical line) and the constants on the right side. For example, the argument matrix of the system
is .
The argument matrix is used in the Gaussian elimination method. The Gaussian elimination method is summarized by the following steps:
1.Write the system of equations in the argument matrix form.
2.Perform elementary row operations to get zeros below the main diagonal.
a. interchange any two rows
b. replace a row by a nonzero multiply of that row
c. replace a row by the sum of that row and a constant nonzero multiple of some other row
3.Use back substitution to find the solution of the system.
We demonstrate the Gaussian elimination method in Example 1.10.
EXAMPLE 1.10
Solve the linear system using the Gaussian elimination method.
Solution:
We use to represent the th row. Write the argument matrix of the system as:
.
Interchange and , this gives: .
, this gives: .
, this gives: .
, this gives: .
, this gives: .
gives , substitute the value of in and , this gives , and , respectively.
1.16 CRAMER’S RULE
Cramer’s rule can be used to solve the simultaneous equations for as
(1.24)
where are the determinations expressed as
(1.25)
.
It is worth noting that is the determinant of matrix and is the determinant of the matrix formed by replacing the nth column of by . Also, Cramer’s rule applies only when , but when , the set of questions has no unique solution because the equations are linearly dependent.
Summary of Cramer’s Rule
1.17 INVERSE OF A MATRIX
1.18 VECTOR ANALYSIS
1.19 EIGENVALUES AND EIGENVECTORS
Eigenvalues problems arise from many branches of engineering, especially in the analysis of the vibration of elastic structures and electrical systems.
The eigenvalue problem is presented in linear equations in the form
. (1.85)
where is a square matrix; is a scalar and called eigenvalue of matrix ; is eigenvector of matrix corresponding to .
To find the eigenvalues of a square matrix [A], we rewrite the equation (1.55) as
(1.86)
or
. (1.87)
There must be a nonzero solution of equation (1.87) in order for to be an eigenvalue. However, equation (1.87) can have a nonzero solution if and only if
. (1.88)
Equation (1.88) is called the characteristic equation of matrix , and the scalars satisfy the equation (1.88) are the eigenvalues of matrix . If matrix has the form
, then equation (1.88) can be written as
. (1.89)
The equation (1.89) can be expanded to a polynomial equation in as
. (1.90)
Thus, the nth-degree polynomial is
. (1.91)
Equation (1.91) is called a characteristic polynomial of matrix . Indeed, the nth roots of the polynomial equation are the nth eigenvalues of matrix . The solutions of equation (1.87) with the eigenvalues substituted on the equation are called eigenvectors.
EXAMPLE 1.19
Find the eigenvalues and eigenvectors of the matrix
Solution:
Since
,
the characteristic polynomial of matrix is
.
And the characteristic equation of matrix is
.
The solutions of this equation are and ; these values are the eigenvalues of matrix .
The eigenvectors for each of the above eigenvalues are calculated using equation (1.87).
For , we obtain
.
The above equation yields two simultaneous equations for and , as follows:
gives
gives .
Thus, choosing , we obtain the eigenvector , where is an arbitrary constant.
For , we obtain
.
The above equation yields two simultaneous equations for and , as follows:
gives
gives .
Thus, choosing , we obtain the eigenvector , where is an arbitrary constant.
1.20 USING MATLAB
EXERCISES
REFERENCES
1.R. Butt, “Applied Linear Algebra and Optimization using MATLAB,” Mercury Learning and Information, 2011.
2.H. Anton, “Elementary Linear Algebra,” 6th Edition, John Wiley and Sons, INC., 1991.
3.S. Nakamura, “Applied Numerical Methods with Software,” Prentice-Hall, 1991.
4.B. Kolman, “Introductory Linear Algebra with Applications,” 6th Edition, John, Prentice Hall, 1997.
5.H. Schneider and G. P. Barker, “Matrix and Linear Algebra with Applications,” 2nd Edition, Holt, Rinehart and Winston, Inc., 1973.