Table Of ContentSolid Mechanics and Its Applications
M. Reza Eslami
Finite
Elements
Methods in
Mechanics
Solid Mechanics and Its Applications
Volume 216
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M. Reza Eslami
Finite Elements Methods
in Mechanics
123
M.Reza Eslami
Mechanical EngineeringDepartment
AmirkabirUniversity ofTechnology
Tehran
Iran
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ISSN 0925-0042 ISSN 2214-7764 (electronic)
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Preface
TheauthorispleasedtopresentFiniteElementMethodinMechanics.Thisbookwill
serve a wide range of readers, in particular, graduate students, Ph.D. candidates,
professors, scientists, researchers in various industrial and government institutes,
andengineers.Thus,thebookshouldbeconsiderednotonlyasagraduatetextbook,
butalsoasareferencebooktothoseworkingorinterestedinareasoffiniteelement
modellingofsolidmechanics,heatconduction,andfluidmechanics.
The book is self-contained, so that the reader should not need to consult other
sources while studying the topic. The necessary mathematical concepts and
numericalmethodsarepresentedinthebookandthereadermayeasilyfollowthe
subjectsbasedonthesebasictools.Itisexpected,however,thatthereadershould
have some basic knowledge in the classical mechanics, theory of elasticity, and
fluid mechanics.
The book contains 17 chapters, where the chapters cover the finite element
modeling of all major areas of mechanics.
Chapter1 presents the history of development offinite element method, where
the key references are given and the progress of this science is discussed.
Chapter 2 is devoted to the basic mathematical concepts of finite element
method. The method of calculus of variation is discussed and the distinction of
boundary value problems versus the variational formulation is presented and
several examples are given to make the reader familiar with the concepts of
functionandfunctional.Thematerialthenfollowsintothediscussionoftraditional
Ritz and Galerkin methods. Numerical examples show the powerful nature of
these numerical techniques.
IntroductiontothefiniteelementmethodisgiveninChap.3withthediscussion
of elastic membrane. The subject of elastic membrane is selected because the
height of membrane is approximated with finite element method. This gives a
physical feeling for the finite element approximation to the reader.
Sincealineartriangularelementisemployedtomodeltheelasticmembranein
Chap. 3, Chap. 4 discuss the one-, two-, and three-dimensional elements with
linearandhigherorderapproximation.Thediscussiongivesafeelingtothereader
that there is no limitation in the type of elements and the order of approximation,
geometrically and mathematically. The subparametric, isoparametric, and super-
parametric elements are discussed and the natural coordinates are presented.
vii
viii Preface
The finite element approximation of the field problems, harmonic and
biharmonic, are given in Chap. 5.
Chapter 6 deals with the finite element approximation of the heat conduction
equations. One-, two-, and three-dimensional conduction in solids are discussed
and the transient heat conduction problems are presented. Both variational and
Galerkin techniques are presented.
Up to this point, the reader learns how to obtain the element stiffness, capa-
citance, and force matrices for one element. He questions how a solution domain
with many number of elements should be modeled and solved to obtain the
requireddomainunknowns.AcomprehensivetreatmentisgiveninChap.7togive
a proper tool to the reader to write his own computer program. Many numerical
examples are solved to show the numerical scheme, and proper algorithms are
given. The assembly of global matrices, bandwidth calculation, the method to
apply the boundary conditions, and the Gauss elimination method are presented.
The method of solution of the transient and dynamic finite element equations are
thenpresented.Thecentraldifferencemethod,theHouboltMethod,theNewmark
Method, and the Wilson-h method are presented. At this stage, the reader learns
how to write his own computer problem. Now, he should learn how different
problemsofmechanicsareformulatedbythefiniteelementapproximation.These
techniques are discussed in the following chapters.
Chapter 8 deals with the finite element approximation of beams. Static beam
deflectionequation,basedontheEulerbeamtheory,ispresentedandtheGalerkin
and variational formulations are obtained. The axial, torsional, and lateral vibra-
tions of beams are modeled. Finally, the vibrations of Timoshenko beam are
presented.
Chapters9and10presentthefiniteelementformulationsofelasticityproblems
based on Galerkin and variational formulations.
Torsion of prismatic bars and rods are given in Chap. 11 and quasistatic
thermoelasticity theory is discussed in Chap. 12.
Chapter 13 is devoted to the finite element solution of viscous fluid mechanic
problems. Derivation of the Navier-Stokes equations is presented and the finite
element formulation of the two-dimensional fluid flow based on the velocity
components and pressure are derived. In the following section, the vorticity
transportmodeloftheNavier-Stokesequationsareobtainedandthefiniteelement
formulations are derived. The method of solution of the resulting nonlinear finite
element equation is presented.
Chapter 14 presents one-dimensional higher order elements. The local natural
coordinate for the quadratic and cubic elements are derived and the Jacobian
matrixisobtained.Todescribetheapplication,fieldproblemforone-dimensional
case is discussed and the element of the matrices are calculated. The chapter is
completed with a discussion of layerwise theory for composite beams, where
one-dimensional higher order element is used to discuss the problem.
The higher order element in two dimension in discussed in Chap. 15. The
triangular element with quadratic and cubic shape functions are given in terms of
the area element and the Jacobian matrix is calculated. The quadratic element is
Preface ix
employed to obtain the element matrices for a two-dimensional field problem.
In the following, the quadrilateral element is discussed and its application to the
field problem is presented.
Chapter 16 presents the linear coupled thermoelasticity problems, and their
method of solution by finite element method. This chapter is unique in the
literature offinite element analysis of solid elastic continuum. The most general
form of the three-dimensional classical coupled thermoelasticity equations are
considered and the finite element formulations are presented.
ComputerprogramsforthreedifferenttypesofproblemsaregiveninChap.17.
The first program is related to the elastic membrane problem, where Poisson’s
equation is solved. This program may be used for any other application of
Poisson’s equation, such as the steady-state heat conduction, torsion of prismatic
bars, inviscid incompressible fluid flow problems, and the pressure in porous
media. The second computer program handles two-dimensional elasticity
problems, and the third computer program presents three-dimensional transient
heat conduction problems. The programs are written in C++ environment.
At the end of all the chapters, except Chap. 1, there are a number of problems
for students to solve. Also, at the end of each chapter, there is a list of relevant
references.
The book was prepared over some 40 years of teaching the graduate finite
element course. During this long period of time, the results of classwork
assignmentsandstudentresearcharecarefullygatheredandputintothisvolumeof
work. The author takes this opportunity to thank all his students who made
possible to provide this piece of work.
October 2013 M. Reza Eslami
Contents
1 Introduction and History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Statement of Extremum Principle. . . . . . . . . . . . . . . . . . . . 8
2.3 Method of Calculus of Variation . . . . . . . . . . . . . . . . . . . . 9
2.4 Function of One Variable, Euler Equation. . . . . . . . . . . . . . 10
2.5 Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Minimization of Functions of Several Variables. . . . . . . . . . 14
2.7 Cantilever Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Approximate Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.1 A: Weighted Residual Methods . . . . . . . . . . . . . . . 21
2.8.2 B: Stationary Functional Method . . . . . . . . . . . . . . 23
2.9 Further Notes on the Ritz and Galerkin Methods . . . . . . . . . 23
2.10 Application of the Ritz Method . . . . . . . . . . . . . . . . . . . . . 26
2.10.1 Non-homogeneous Boundary Conditions. . . . . . . . . 28
2.11 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Finite Element of Elastic Membrane. . . . . . . . . . . . . . . . . . . . . . 35
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Weightless Elastic Membrane (Method I) . . . . . . . . . . . . . . 38
3.4 Membrane Analysis (Method II). . . . . . . . . . . . . . . . . . . . . 39
3.5 Strain Energy of Elastic Membrane. . . . . . . . . . . . . . . . . . . 41
3.6 Application of Calculus of Variation. . . . . . . . . . . . . . . . . . 43
3.7 Introduction to the Finite Element Method. . . . . . . . . . . . . . 44
3.7.1 The Elastic Membrane. . . . . . . . . . . . . . . . . . . . . . 45
3.7.2 Boundary Value Problem. . . . . . . . . . . . . . . . . . . . 45
xi
