One-Dimensional Finite Elements

432 Pages·2018·12.012 MB·English

by  Andreas Öchsner, Markus Merkel

Andreas Öchsner · Markus Merkel
One-Dimensional
Finite Elements
An Introduction to the FE Method
Second Edition
One-Dimensional Finite Elements
Andreas Öchsner Markus Merkel
(cid:129)
One-Dimensional Finite
Elements
An Introduction to the FE Method
Second Edition
123
Andreas Öchsner Markus Merkel
Faculty of MechanicalEngineering Institute for Virtual Product Development
Esslingen University of Applied Sciences Aalen University of AppliedSciences
Esslingen Aalen
Germany Germany
ISBN978-3-319-75144-3 ISBN978-3-319-75145-0 (eBook)
https://doi.org/10.1007/978-3-319-75145-0
LibraryofCongressControlNumber:2018934874
1stedition:©Springer-VerlagBerlinHeidelberg2013
2ndedition:©SpringerInternationalPublishingAG,partofSpringerNature2018
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Preface
Thetitleofthisbook‘One-DimensionalFiniteElements:AnIntroductiontotheFE
Method’standsforcontentandfocus.Nowadays,muchliteratureregardingthetopic
of the finite element method exists. The different works reflect the multifaceted
perceptionsandapplicationpossibilities.Thebasicideaofthisintroductionintothe
finiteelementmethodreliesontheconceptofexplainingthecomplexmethodwith
thehelpofone-dimensionalelements.Itisthegoaltointroducethemanifoldaspects
of the finite element method and to enable the reader to get a methodical under-
standingofimportantsubjectareas.Thereaderlearnstounderstandtheassumptions
andderivationsatdifferentphysicalproblemsinstructuralmechanics.Furthermore,
he/shelearnstocritically evaluatepossibilitiesandlimitations ofthefiniteelement
method. Additional comprehensive mathematical descriptions, which solely result
from advanced illustrations for two- or three-dimensional problems, are omitted.
Hence,themathematicaldescriptionlargelyremainssimpleandclear.Thefocuson
one-dimensional elements, however, is not just a pure limitation on a more simple
and clearer formal illustration of the necessary equations. Within structural engi-
neering, there are various structures—for example bridges or high transmission
towers—whichareusuallymodeledviaone-dimensionalelements.Therefore,this
work also contains a‘set oftools’, which can also be applied in practice.
Theconcentrationonone-dimensionalelementsisnewforatextbookandallows
the treatment of various basic and demanding physical questions of structural
mechanics within one single textbook. This new concept therefore allows a
methodical understanding of important subject areas (e.g., plasticity or composite
materials),whichoccurtoaprospectiveengineerduringprofessionalwork,which,
however, are seldomly treated in this way at universities. Consequently, simple
access is possible, also in supplementary areas of application of the finite element
method.
Thisbookoriginatesfromacollectionoflecturenoteswhichweredevelopedas
written material for lectures and training documents for specialized courses on the
finite element method. Especially, the calculated examples and the supplementary
problems refer to typical questions which are raised by students and course
participants.
vii
viii Preface
A prerequisite for a good understanding is the basics in linear algebra, physics,
materials science and strength of materials, the way they are typically communi-
cated in the basic studies of a technical subject in the field of mechanical
engineering.
Within the first chapters, the one-dimensional elements will be introduced, due
to which the basic load cases of tension/compression, torsion, and bending can be
illustrated.Ineachcase,thedifferentialequationaswellasthebasicequationsfrom
the strength of materials (this is the kinematic relationship, the constitutive rela-
tionship, and the equilibrium equation) are being derived. Subsequently, the finite
elements with the usual definitions for force and displacement parameters are
introduced. With the help of examples, the general procedure is illustrated. Short
solutions for supplementary problems are attached in the appendix.
Chapter 6 deals with questions which are independent of the loading type and
the therewith connected element formulation. A general one-dimensional finite
element, which can be constructed from the combination of basic elements, the
transformation of elements in the general three-dimensional space and the numer-
ical integration as an important tool for the implementation of the finite element
method is dealt with.
The complete analysis of an entire structure is introduced in Chap. 7. The total
stiffnessrelationresultsfromthesinglestiffnessrelationofthebasicelementsunder
consideration of the relations to each other. A reduced system results due to the
boundaryconditions.Unknownparametersarederivedfromthereducedsystem.The
procedure will be introduced as examples on plane and general three-dimensional
structures.
Chapters 8–12 deal with topics which are usually not part of a basic textbook.
The beam element with shear consideration is introduced in Chap. 8. The
Timoshenko beam theory serves as a basis for this.
WithinChap.9,aspecialclassofmaterial—compositematerials—isintroduced
into a finite element formulation. First, various ways of description for direction-
dependentmaterialbehaviorareintroduced.Fibercompositesareaddressedbriefly.
A composite element is demonstrated by examples of a composite bar and a
composite beam.
Chapters10,11,and12dealwithnonlinearities.InChap.10,thedifferenttypes
of nonlinearities are introduced. The case of the nonlinear elasticities is dealt with
more closely. The problem is illustrated for bar elements. First, the principal finite
element equation is derived under consideration of the strain dependency. The
direct iteration as well as the complete and modified Newton–Raphson iterations
are derived for the solution of a nonlinear system of equations. In addition, many
examples serve as a demonstration of this issue.
Chapter 11considers elasto-plasticbehavior,oneofthemostcommon forms of
material nonlinearities. First, the continuum-mechanical basics for plasticity in the
caseoftheone-dimensional continuumbararecomposed.Theyieldcondition,the
flow rule, the hardening law, and the elasto-plastic modulus are introduced for
uniaxial, monotonic load cases. Within the hardening behavior, the description is
limited to isotropic hardening. For the integration of the elasto-plastic constitutive
Preface ix
equation, the incremental predictor–corrector method is generally introduced and
derived for the case of the fully implicit and the semi-implicit backward-Euler
algorithm. On crucial points, the difference between one- and three-dimensional
descriptions will be pointed out, to guarantee a simple transfer of the derived
methods to general problems.
Chapter 12 deals with stability, which is an issue that is especially relevant for
thedesignanddimensioningoflightweightcomponents.Finiteelementsdeveloped
forthistypeofnonlinearitiesareusedforthesolvingoftheEuler’sbucklingloads.
Chapter 13 serves to introduce a finite element formulation for dynamic prob-
lems. Stiffness matrices as well as mass matrices will be established. Different
assumptions for the distribution of the masses, whether continuously or concen-
trated, lead to different formulations. The issue is discussed by example of axial
vibrations of the bar.
As an illustration, each chapter is recessed both with precisely calculated and
commented examples as well as with supplementary problems—including short
solutions. Each chapter concludes with an extensive bibliography.
Preface to the 2nd Edition
The basic concept for the treatment of the finite element method with one-
dimensionalquestionshasbeenpreservedinthe2ndedition.Additionallyincluded
isthestationaryheat conduction, andtheprinciple ofvirtual workbecameanother
method for the derivation offinite element formulations. Added is Chap. 14 with
special elements for the modeling of elastic foundations, singularities, and infinite
extension of elements.
Bexbach, Hüttlingen Andreas Öchsner
December 2017 Markus Merkel
Acknowledgements
Critical questions and comments of students and course participants are gratefully
acknowledged. Their input contributed to the actual form of this book.
Finally,we liketothank ourfamiliesfor theunderstandingandpatienceduring
the preparation of this book.
xi
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Finite Element Method at a Glance. . . . . . . . . . . . . . . . . . 1
1.2 Foundations of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Motivation for the Finite Element Method . . . . . . . . . . . . . . . . . . . 5
2.1 From the Engineering Perspective Derived Methods . . . . . . . . . 5
2.1.1 The Matrix Stiffness Method. . . . . . . . . . . . . . . . . . . . 6
2.1.2 Transition to the Continuum . . . . . . . . . . . . . . . . . . . . 10
2.2 Integral Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Weighted Residual Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Procedure on Basis of the Inner Product . . . . . . . . . . . 18
2.3.2 Procedure on Basis of the Weak Formulation. . . . . . . . 22
2.3.3 Procedure on Basis of the Inverse Formulation . . . . . . 23
2.4 Sample Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Bar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Basic Description of the Bar Element. . . . . . . . . . . . . . . . . . . . 31
3.2 The Finite Element Tension Bar . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Derivation Through Potential . . . . . . . . . . . . . . . . . . . 37
3.2.2 Derivation Through Castigliano’s Theorem . . . . . . . . . 38
3.2.3 Derivation Through the Weighted Residual
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.4 Derivation Through Virtual Work . . . . . . . . . . . . . . . . 43
3.3 Sample Problems and Supplementary Problems . . . . . . . . . . . . 45
3.3.1 Sample Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Supplementary Problems. . . . . . . . . . . . . . . . . . . . . . . 49
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
xiii