Schaum's Outline of Finite Element Analysis

By George R. Buchanan

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• Language: English
• Publisher: McGraw-Hill Education
• Release date: Nov 22, 1994
• ISBN: 9780071502887

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GEORGE R. BUCHANAN is Professor of Civil Engineering at the Tennessee Technological University, Cookeville. He received B.S.and M.S. degrees in Civil Engineering from the University of Kentucky and the Ph.D. from the Virginia Polytechnic Institute. He began his university career in the Department of Engineering Science and Mechanics at Tennessee Tech, where he served for 16 years including four years as chairperson. After a year as a visiting scientist at the Los Alamos National Laboratory he returned to Tennessee Tech with the Department of Civil Engineering. He is the author of a textbook, Mechanics of Materials (Saunders). He has served as a consultant with the U.S. Army Missile Command, Tennessee Valley Authority, and Los Alamos National Laboratory.

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To Chris,

my trusted friend and beloved wife,

who made me complete

Preface

During my academic career the topic of finite element analysis has literally grown from a mere concept into one of the most powerful methods of numerical analysis that exist today. My first significant work with finite element modeling and computer coding of those models was in the early 1970s, and I was not convinced then that the results that could be obtained were worth the effort that went into using the method. Isoparametric finite elements, Galerkin methods, and the use of the more powerful numerical integration techniques were new and emerging. At that time I was not yet aware of the impact these ideas were having on the use and development of the finite element method. As time passed, I became a serious student of the method and developed some capability for successful application of the method.

The Outline is based upon notes that I developed, over a period of several years, for a first course in finite element analysis. My own introduction to finite element methods was somewhat naive, and I think that is often the case for someone with prior knowledge of matrix analysis of structures. I must emphasize that the finite element analysis is not an extension of either the stiffness method or matrix theory of structures. Note that the method is not used to solve engineering problems but is used to solve differential equations, a subtle but significant difference.

The Outline is written with emphasis on applied techniques rather than theoretical justification of the techniques and methods. Each chapter of the Outline, especially Chapters 2 through 6, serves a specific purpose. Chapter 1 contains a brief review of specific mathematical topics. Chapter 2 begins with the Rayleigh-Ritz method and a variational statement of a standard second-order, one-dimensional differential equation that appears in numerous applications in applied physics. It is illustrated that the finite element method is an organized application of the Rayleigh-Ritz method of numerical analysis. Chapter 3 is an extension to two dimensions. The problems are formulated in the standard cartesian coordinate system with emphasis on the formulation of the element, area integration, and subsequent formulation of the global finite element model. Chapters 2 and 3 are academic, but very necessary since they serve as an introduction to the more powerful modern applications of the finite element method. Chapter 4 is intended to show the connection between finite element analysis and matrix analysis of structures and can be omitted by the reader who is not interested in beam and column structures. Chapter 5 is important for the reader who intends to use the finite element method for solving problems that involve coupled partial differential equations. There is an overview of the underlying mathematics that supports the use of the variational functions that were introduced in Chapters 2 and 3. The very powerful Galerkin method of numerical analysis is introduced in Chapter 5 and used to derive finite element models of partial differential equations that govern several different physical phenomena. Chapter 6 is devoted to isoparametric finite elements and the coordinate transformations and numerical integrations that pertain to that topic. Chapter 7 is a collection of several applied topics. A computer code is included as an Appendix for readers desiring some connection between theory and computer application. An index of solved problems is included that will assist the reader who is searching for a particular application.

Several people should be acknowledged for their assistance with the preparation of this book. Professor John Peddieson, Jr., Tennessee Technological University, for many discussions, over the years, concerning applied mathematics; Jeffery Abston for the analytical solution given in Problem 7.23; Mean-Fun Cheng for the numerical results given in Problem 7.11; and Satya P. Narimetla for the numerical solutions given in Problem 7.18. I wish to thank Ms. Yvette Clark for her very gracious assistance with computers, software, and printers. The draft copy was meticulously reviewed by Abraham J. Rokach, Hypermedia Systems Inc., Chicago. I also wish to thank the staff of editors of the Schaum’s Division of McGraw-Hill, John Aliano, David Beckwith, and Arthur Biderman for their patience and encouragement.

GEORGE R. BUCHANAN

Cookeville, Tennessee

Contents

Chapter 1 MATHEMATICAL BACKGROUND

1.1   Introduction

1.2   Vector Analysis

1.3   Matrix Theory

1.4   Differential Equations

1.5   Cartesian Tensors

Solved Problems

---

Chapter 2 ONE-DIMENSIONAL FINITE ELEMENTS

2.1   Introduction

2.2   Mathematical Equations of Engineering

2.3   Variational Functions

2.4   Interpolation Function

2.5   Shape Functions

2.6   Stiffness Matrix

2.7   Connectivity

2.8   Boundary Conditions

2.9   Problems in Cylindrical Coordinates

2.10 The Direct Method

Solved Problems

---

Chapter 3 TWO-DIMENSIONAL FINITE ELEMENTS

3.1   Introduction

3.2   Two-Dimensional Boundary-Value Problems

3.3   Connectivity and Nodal Coordinates

3.4   Theory of Elasticity

3.5   Variational Functions

3.6   Triangular Elements and Area Coordinates

3.7   Transformations

3.8   Cylindrical Coordinates

Solved Problems

---

Chapter 4 BEAM AND FRAME FINITE ELEMENTS

4.1   Introduction

4.2   Governing Differential Equation

4.3   The Displacement Method for Beam Analysis

4.4   Beam Finite Elements

4.5   Matrix Transformations

Solved Problems

---

Chapter 5 VARIATIONAL PRINCIPLES, GALERKIN APPROXIMATION, AND PARTIAL DIFFERENTIAL EQUATIONS

5.1   Introduction

5.2   Variational Principles

5.3   Galerkin Approximation

5.4   Coupled Partial Differential Equations

5.5   Initial-Value Problems

Solved Problems

---

Chapter 6 ISOPARAMETRIC FINITE ELEMENTS

6.1   Introduction

6.2   Numerical Integration

6.3   Interpolation Formulas and Shape Function Formulas

6.4   Generalized Coordinates

6.5   Isoparametric Elements

6.6   Axisymmetric Formulations

Solved Problems

---

Chapter 7 SELECTED TOPICS IN FINITE ELEMENT ANALYSIS

7.1   Introduction

7.2   Initial-Value Problems

7.3   Eigenvalue Problems

7.4   Three-Dimensional Finite Elements

7.5   Higher-Order Finite Elements

7.6   Element Continuity

7.7   Plate Finite Elements

Solved Problems

---

Appendix  COMPUTER CODE FOR COUPLED STEADY-STATE THERMOELASTICITY

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BIBLIOGRAPHY

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SOLVED-PROBLEM INDEX OF APPLICATIONS

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INDEX

Chapter 1

Mathematical Background

1.1. INTRODUCTION

The mathematics required for the study of finite element analysis can vary from elementary to sophisticated. Fortunately, most concepts can be mastered with a reasonable knowledge of vector analysis, matrix theory, and differential equations. Pertinent mathematical concepts will be reviewed in this chapter, and the reader who needs more information may consult the references listed in the bibliography. The review of vector analysis ranges from elementary definitions to more advanced integral theorems. The matrix theory that is covered consists of elementary definitions, matrix manipulations, and the solution of simultaneous equations. A brief treatment of differential equations is also included. Differential equations are required for solving boundary-value problems that can be used as a check on numerical solutions obtained using the finite element method. Finally, a discussion of tensor analysis is included but is limited to cartesian tensor notation. The formulation of problems using cartesian tensor (subscript) notation occurs in the literature of finite elements and usually streamlines the mathematical presentation.

1.2. VECTOR ANALYSIS

A vector is defined as a physical quantity that can be described by a single magnitude and a direction that is related to a coordinate reference frame. A fundamental concept, which justifies the use of vector analysis, is that physical quantities that are arbitrarily directed in space can be resolved into orthogonal components corresponding to the reference frame. Once the components are found, they can be manipulated using standard algebraic operations. Several specialized vector operations required throughout this text will be reviewed in this chapter. Vectors, in this chapter, will be written using boldface lowercase letters. The vector of Fig. 1-1 is

Fig. 1-1

where i, j, and k are unit vectors directed along the x, y, and z axes, respectively.

The vector differential operator del ∇ is defined as

This operator, by definition, has vector properties and is used to define three fundamental vector operations, the gradient, the divergence, and the curl. These vector operations are useful when defining integral vector theorems such as the divergence theorem and Green’s theorem, which is sometimes called the Green-Gauss theorem.

1.3. MATRIX THEORY

Matrices

A rectangular array of numbers with a definite number of rows and columns is a matrix. Once an array has been defined as a matrix, it has certain mathematical properties that can be classified within the context of matrix theory. A comprehensive knowledge of matrix theory is not required for finite element analysis; however, certain fundamental concepts are necessary for the study of finite element theory and for its subsequent application.

The array of numbers can be written in the abstract as

The notation [A] will be used in this text to indicate a matrix. The terms within the matrix are called elements, and when an element or a group of elements is referred to, subscript notation will be used, such as aij, where i indicates a row number and j indicates a column number. The matrix of Eq. (1.3) is called an m by n matrix or simply an matrix, and is referred to as the order of the matrix. A row matrix is defined as a 1 × m matrix and, similarly, a column matrix is defined as an matrix. The column matrix is often written {A}.

A matrix [A] and a matrix [B] can be added or subtracted, element by element, as long as both are matrices. Proper addition and subtraction are not defined for matrices of unequal order. Matrix multiplication is the process of multiplying one matrix by a second matrix and is written [A][B]; in general, . In matrix multiplication [A][B], [B] is said to be premultiplied by [A] or [A] is said to be postmultiplied by [B].

The division, element by element, of one matrix by a second matrix is not defined. However, the inverse of a matrix, written as [A]–1, serves a similar purpose and will be discussed later.

The transpose of a matrix is obtained by interchanging its rows and columns. The transpose of [A] is written [A]r, and in subscript notation the element interchange is

It follows that

A symmetric matrix is defined as a square matrix with property for . A diagonal matrix has all elements of a square matrix equal to zero except those on the principal diagonal, which is the diagonal from upper left to lower right. A unit matrix is a special case of a diagonal matrix; all diagonal elements are equal to 1, and all off-diagonal elements are equal to 0.

Determinants

An understanding of selected topics from the theory of determinants is necessary for successful solution of the simultaneous equations that result in finite element analysis. The determinant used throughout this text is of course a square matrix and as a square matrix has certain mathematical properties an ordinary matrix does not have. The determinant is used in Prob. 1.4 to define the vector product. Determinants of order 2 or 3 can generally be used to illustrate all the concepts required for understanding the manipulation of determinants. A determinant is usually symbolized by enclosing the array of numbers within vertical lines rather than brackets. For a matrix denoted [A], the notation for the determinant might be |A|, det [A], or |det A| and indicates the determinant of the matrix [A].

Every determinant has a determinantal equation. For higher-order determinants it can be quite formidable from a computational standpoint to obtain that equation. The determinantal equation of a determinant of order 3 is obtained as follows:

and can be described as the product of the principal diagonal terms minus the product of the secondary diagonal terms. This elementary concept can be applied to determinants of order 2 or 3 but fails for higher-order determinants.

The minor of a determinant is the determinant that remains after a row and a column are removed from the original determinant. The minor can be referenced to a particular element of the determinant using the notation aij. The minor of |a22| of Eq. (1.6) is

a determinant of order 2. The cofactor of an element of a determinant is defined as , where |Mij| is the minor of the element aij. It is now possible to define a cofactor matrix as the square matrix constructed by replacing each element of a square matrix by the cofactor of the determinant corresponding to the original square matrix. The adjoint matrix is defined as the transpose of the cofactor matrix. The adjoint matrix is used to compute the inverse of a matrix in Prob. 1.14.

Simultaneous Equations

Numerous methods have been proposed for the solution of a set of simultaneous equations, and two of these procedures will be emphasized in this text. A set of simultaneous equations can be written in matrix form as

The matrix [A] represents the matrix of coefficients that are multiplied by the unknown quantities {x}. The column matrix on the right-hand side contains the known quantities f. Multiplying by the inverse of [A] gives

The use of the inverse for solving a set of simultaneous equations is inefficient for large sets of equations.

A method that is sometimes called gaussian elimination is faster and hence more efficient than the inverse method. Gaussian elimination is an organized method of substituting each equation into the previous equation until the last equation contains only one unknown. The unknowns are determined sequentially, starting with the last equation and proceeding upward. The method is sometimes referred to as upper trianglization and is best illustrated by example, as in Prob. 1.16.

1.4. DIFFERENTIAL EQUATIONS

Finite element analysis is a method for the numerical solution of a differential equation. It follows that without differential equations there would not be a finite element method. Many practicing engineers and scientists learned the finite element method as an application of structural analysis for civil engineering or aircraft structures. The state of the art of finite element analysis several decades ago was responsible for that situation. The classical stiffness method of structural analysis, as discussed in Chap. 4, can be derived without mention of the governing differential equations. That is, the fundamental relationships for deriving the stiffness method are based upon solutions of differential equations, but the user can easily lose sight of the origin of the analysis. Finite element analysis for beam and frame structures can be based upon energy theorems without considering differential equations. Again, the fault is not with the engineer or scientist, but historically the connection between energy methods in structural analysis and the governing differential equation has not been emphasized.

Differential equations are emphasized in this text. Beginning in Chap. 2 the differential equation is associated with the corresponding variational function (energy theorem). The finite element method can be derived in a variety of ways, but regardless of the derivation, the method is the numerical solution of a differential equation. The differential equations in this text are for the most part elementary. A very basic differential equation is considered in Chap. 2, where it is shown that the same equation governs numerous physical theories. The most elementary equation will probably allow the reader to become acquainted with the connection between finite element theory and differential equations, and for this reason Chap. 2 is a very important chapter. From a practical viewpoint finite element analysis would not be used to solve a one-dimensional second-order differential equation, however, Chap. 2 is an absolute necessity for understanding the more complicated analysis problems.

The analytical solution of the differential equation is important as a check on the numerical solution obtained using the finite element method. How else will the user know if the computer code that generates a numerical solution is correct? The fundamental differential equations of Chap. 2 are of the form

where a(x) is a material parameter that can be a function of x, C(x) is an external source, and A(x) is the cross-sectional area. If material parameters, external source terms, and area are functions of x, they are allowed to change from element to element. In other words, the finite element is not expected to model a variable area or material parameter within the element since they can be modeled from element to element. The functional form is usually disregarded, and Eq. (1.9) is written in a more elementary form as

The analytical solution is elementary. Nevertheless, such elementary solutions are invaluable for successful application of the finite element method. The most general form of Eq. (1.10) is given as Eq. (2.19) and appears as

Solutions for one-dimensional differential equations are given in Probs. 2.1, 2.3, 2.17, 2.18, and 2.28. Problem 2.18 provides a solution for an equation of the general form of Eq. (1.11), and Prob. 2.28 represents a solution for a differential equation in one-dimensional cylindrical coordinates with a change in material properties. The one-dimensional counterpart of Eq. (1.11) in cylindrical coordinates is

The two-dimensional counterpart of Eq. (1.11) is a partial differential equation and is discussed