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Hutton: Fundamentals of

Finite Element Analysis

Front Matter Preface © The McGraw−Hill

Companies, 2004

undamentals of Finite Element Analysis is intended to be the text for a

senior-level ﬁnite element course in engineering programs. The most

appropriate major programs are civil engineering, engineering mechan-

ics, and mechanical engineering. The ﬁnite element method is such a widely used

analysis-and-design technique that it is essential that undergraduate engineering

students have a basic knowledge of the theory and applications of the technique.

Toward that objective, I developed and taught an undergraduate “special topics”

course on the ﬁnite element method at Washington State University in the sum-

mer of 1992. The course was composed of approximately two-thirds theory and

one-third use of commercial software in solving ﬁnite element problems. Since

that time, the course has become a regularly offered technical elective in the

mechanical engineering program and is generally in high demand. During

the developmental process for the course, I was never satisﬁed with any text that

was used, and we tried many. I found the available texts to be at one extreme or

the other; namely, essentially no theory and all software application, or all theory

and no software application. The former approach, in my opinion, represents

training in using computer programs, while the latter represents graduate-level

study. I have written this text to seek a middle ground.

Pedagogically, I believe that training undergraduate engineering students to

use a particular software package without providing knowledge of the underlying

theory is a disservice to the student and can be dangerous for their future employ-

ers. While I am acutely aware that most engineering programs have a speciﬁc

ﬁnite element software package available for student use, I do not believe that the

text the students use should be tied only to that software. Therefore, I have writ-

ten this text to be software-independent. I emphasize the basic theory of the ﬁnite

element method, in a context that can be understood by undergraduate engineer-

ing students, and leave the software-speciﬁc portions to the instructor.

As the text is intended for an undergraduate course, the prerequisites required

are statics, dynamics, mechanics of materials, and calculus through ordinary dif-

ferential equations. Of necessity, partial differential equations are introduced

but in a manner that should be understood based on the stated prerequisites.

Applications of the ﬁnite element method to heat transfer and ﬂuid mechanics are

included, but the necessary derivations are such that previous coursework in

those topics is not required. Many students will have taken heat transfer and ﬂuid

mechanics courses, and the instructor can expand the topics based on the stu-

dents’ background.

Chapter 1 is a general introduction to the ﬁnite element method and in-

cludes a description of the basic concept of dividing a domain into ﬁnite-size

subdomains. The ﬁnite difference method is introduced for comparison to the

PREFACE

Hutton: Fundamentals of

Finite Element Analysis

Front Matter Preface © The McGraw−Hill

Companies, 2004

xii Preface

ﬁnite element method. A general procedure in the sequence of model deﬁnition,

solution, and interpretation of results is discussed and related to the generally

accepted terms of preprocessing, solution, and postprocessing. A brief history of

the ﬁnite element method is included, as are a few examples illustrating applica-

tion of the method.

Chapter 2 introduces the concept of a ﬁnite element stiffness matrix and

associated displacement equation, in terms of interpolation functions, using the

linear spring as a ﬁnite element. The linear spring is known to most undergradu-

ate students so the mechanics should not be new. However, representation of

the spring as a ﬁnite element is new but provides a simple, concise example of

the ﬁnite element method. The premise of spring element formulation is ex-

tended to the bar element, and energy methods are introduced. The ﬁrst theorem

of Castigliano is applied, as is the principle of minimum potential energy.

Castigliano’s theorem is a simple method to introduce the undergraduate student

to minimum principles without use of variational calculus.

Chapter 3 uses the bar element of Chapter 2 to illustrate assembly of global

equilibrium equations for a structure composed of many ﬁnite elements. Trans-

formation from element coordinates to global coordinates is developed and

illustrated with both two- and three-dimensional examples. The direct stiffness

method is utilized and two methods for global matrix assembly are presented.

Application of boundary conditions and solution of the resultant constraint equa-

tions is discussed. Use of the basic displacement solution to obtain element strain

and stress is shown as a postprocessing operation.

Chapter 4 introduces the beam/ﬂexure element as a bridge to continuity

requirements for higher-order elements. Slope continuity is introduced and this

requires an adjustment to the assumed interpolation functions to insure continuity.

Nodal load vectors are discussed in the context of discrete and distributed loads,

using the method of work equivalence.

Chapters 2, 3, and 4 introduce the basic procedures of ﬁnite-element model-

ing in the context of simple structural elements that should be well-known to the

student from the prerequisite mechanics of materials course. Thus the emphasis

in the early part of the course in which the text is used can be on the ﬁnite ele-

ment method without introduction of new physical concepts. The bar and beam

elements can be used to give the student practical truss and frame problems for

solution using available ﬁnite element software. If the instructor is so inclined,

the bar and beam elements (in the two-dimensional context) also provide a rela-

tively simple framework for student development of ﬁnite element software

using basic programming languages.

Chapter 5 is the springboard to more advanced concepts of ﬁnite element

analysis. The method of weighted residuals is introduced as the fundamental

technique used in the remainder of the text. The Galerkin method is utilized

exclusively since I have found this method is both understandable for under-

graduate students and is amenable to a wide range of engineering problems. The

material in this chapter repeats the bar and beam developments and extends the

ﬁnite element concept to one-dimensional heat transfer. Application to the bar

Hutton: Fundamentals of

Finite Element Analysis

Front Matter Preface © The McGraw−Hill

Companies, 2004

Preface xiii

and beam elements illustrates that the method is in agreement with the basic me-

chanics approach of Chapters 2–4. Introduction of heat transfer exposes the stu-

dent to additional applications of the ﬁnite element method that are, most likely,

new to the student.

Chapter 6 is a stand-alone description of the requirements of interpolation

functions used in developing ﬁnite element models for any physical problem.

Continuity and completeness requirements are delineated. Natural (serendipity)

coordinates, triangular coordinates, and volume coordinates are deﬁned and used

to develop interpolation functions for several element types in two- and three-

dimensions. The concept of isoparametric mapping is introduced in the context of

the plane quadrilateral element. As a precursor to following chapters, numerical

integration using Gaussian quadrature is covered and several examples included.

The use of two-dimensional elements to model three-dimensional axisymmetric

problems is included.

Chapter 7 uses Galerkin’s ﬁnite element method to develop the ﬁnite ele-

ment equations for several commonly encountered situations in heat transfer.

One-, two- and three-dimensional formulations are discussed for conduction and

convection. Radiation is not included, as that phenomenon introduces a nonlin-

earity that undergraduate students are not prepared to deal with at the intended

level of the text. Heat transfer with mass transport is included. The ﬁnite differ-

ence method in conjunction with the ﬁnite element method is utilized to present

methods of solving time-dependent heat transfer problems.

Chapter 8 introduces ﬁnite element applications to ﬂuid mechanics. The

general equations governing ﬂuid ﬂow are so complex and nonlinear that the

topic is introduced via ideal ﬂow. The stream function and velocity potential

function are illustrated and the applicable restrictions noted. Example problems

are included that note the analogy with heat transfer and use heat transfer ﬁnite

element solutions to solve ideal ﬂow problems. A brief discussion of viscous

ﬂow shows the nonlinearities that arise when nonideal ﬂows are considered.

Chapter 9 applies the ﬁnite element method to problems in solid mechanics

with the proviso that the material response is linearly elastic and small deﬂection.

Both plane stress and plane strain are deﬁned and the ﬁnite element formulations

developed for each case. General three-dimensional states of stress and axisym-

metric stress are included. A model for torsion of noncircular sections is devel-

oped using the Prandtl stress function. The purpose of the torsion section is to

make the student aware that all torsionally loaded objects are not circular and the

analysis methods must be adjusted to suit geometry.

Chapter 10 introduces the concept of dynamic motion of structures. It is not

presumed that the student has taken a course in mechanical vibrations; as a re-

sult, this chapter includes a primer on basic vibration theory. Most of this mater-

ial is drawn from my previously published text Applied Mechanical Vibrations.

The concept of the mass or inertia matrix is developed by examples of simple

spring-mass systems and then extended to continuous bodies. Both lumped and

consistent mass matrices are deﬁned and used in examples. Modal analysis is the

basic method espoused for dynamic response; hence, a considerable amount of

Hutton: Fundamentals of

Finite Element Analysis

Companies, 2004

xiv Preface

text material is devoted to determination of natural modes, orthogonality, and

modal superposition. Combination of ﬁnite difference and ﬁnite element meth-

ods for solving transient dynamic structural problems is included.

The appendices are included in order to provide the student with material

that might be new or may be “rusty” in the student’s mind.

Appendix A is a review of matrix algebra and should be known to the stu-

dent from a course in linear algebra.

Appendix B states the general three-dimensional constitutive relations for

a homogeneous, isotropic, elastic material. I have found over the years that un-

dergraduate engineering students do not have a ﬁrm grasp of these relations. In

general, the student has been exposed to so many special cases that the three-

dimensional equations are not truly understood.

Appendix C covers three methods for solving linear algebraic equations.

Some students may use this material as an outline for programming solution

methods. I include the appendix only so the reader is aware of the algorithms un-

derlying the software he/she will use in solving ﬁnite element problems.

Appendix D describes the basic computational capabilities of the FEPC

software. The FEPC (FEPﬁnite element program for the PCpersonal computer)

was developed by the late Dr. Charles Knight of Virginia Polytechnic Institute

and State University and is used in conjunction with this text with permission of

his estate. Dr. Knight’s programs allow analysis of two-dimensional programs

using bar, beam, and plane stress elements. The appendix describes in general

terms the capabilities and limitations of the software. The FEPC program is

available to the student at www.mhhe.com/hutton.

Appendix E includes problems for several chapters of the text that should be

solved via commercial ﬁnite element software. Whether the instructor has avail-

able ANSYS, ALGOR, COSMOS, etc., these problems are oriented to systems

having many degrees of freedom and not amenable to hand calculation. Addi-

tional problems of this sort will be added to the website on a continuing basis.

The textbook features a Web site (www

.mhhe.com/hutton) with ﬁnite ele-

ment analysis links and the FEPC program. At this site, instructors will have

access to PowerPoint images and an Instructors’ Solutions Manual. Instructors

can access these tools by contacting their local McGraw-Hill sales representative

for password information.

I thank Raghu Agarwal, Rong Y. Chen, Nels Madsen, Robert L. Rankin,

Joseph J. Rencis, Stephen R. Swanson, and Lonny L. Thompson, who reviewed

some or all of the manuscript and provided constructive suggestions and criti-

cisms that have helped improve the book.

I am grateful to all the staff at McGraw-Hill who have labored to make this

project a reality. I especially acknowledge the patient encouragement and pro-

fessionalism of Jonathan Plant, Senior Editor, Lisa Kalner Williams, Develop-

mental Editor, and Kay Brimeyer, Senior Project Manager.

David V. Hutton

Pullman, WA

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