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有限元经典Fundamental of finite element analysis
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有限元经典教程《Fundamental of finite element analysis 》
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Hutton: Fundamentals of
Finite Element Analysis
Front Matter Preface © The McGraw−Hill
Companies, 2004
F
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
xi
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
Front Matter Preface © The McGraw−Hill
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
Hutton: Fundamentals of
Finite Element Analysis
1. Basic Concepts of the
Finite Element Method
Text © The McGraw−Hill
Companies, 2004
1
Basic Concepts of the
Finite Element Method
1.1 INTRODUCTION
The finite element method (FEM), sometimes referred to as finite element
analysis (FEA), is a computational technique used to obtain approximate solu-
tions of boundary value problems in engineering. Simply stated, a boundary
value problem is a mathematical problem in which one or more dependent vari-
ables must satisfy a differential equation everywhere within a known domain of
independent variables and satisfy specific conditions on the boundary of the
domain. Boundary value problems are also sometimes called field problems. The
field is the domain of interest and most often represents a physical structure.
The field variables are the dependent variables of interest governed by the dif-
ferential equation. The boundary conditions are the specified values of the field
variables (or related variables such as derivatives) on the boundaries of the field.
Depending on the type of physical problem being analyzed, the field variables
may include physical displacement, temperature, heat flux, and fluid velocity to
name only a few.
1.2 HOW DOES THE FINITE ELEMENT
METHOD WORK?
The general techniques and terminology of finite element analysis will be intro-
duced with reference to Figure 1.1. The figure depicts a volume of some material
or materials having known physical properties. The volume represents the
domain of a boundary value problem to be solved. For simplicity, at this point,
we assume a two-dimensional case with a single field variable
(x, y) to be
determined at every point P(x, y) such that a known governing equation (or equa-
tions) is satisfied exactly at every such point. Note that this implies an exact
CHAPTER 1
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